Keywords: group quantique déroulé, algèbre topologique localement convexe, TQFT, super-symétries, invariant de 3-variétés, trace modifiée. v Topological quantum field theory for Lie superalgebra sl unrolled quantum group, locally convex topological algebra, TQFT, super-symmetries, invariant of 3-manifolds, modified trace. vi MSC: 57M27, 17B37 Lie superalgebra, quantum group, link invariant, 3-manifold. 7 57M27, 17B37 Lie superalgebra, unrolled quantum group, G-integral, invariant of 3-manifolds, Hennings invariant, topological Hopf superalgebra

Théorie topologique des champs quantiques pour la superalgèbre de Lie sl(2|1)

Résumé

Ce texte étudie le groupe quantique U H ξ sl(2|1) associé à la superalgèbre de Lie sl(2|1) et une catégorie de ses représentations de dimension finie. L'objectif est de construire des invariants topologiques de 3-variétés en utilisant la notion de trace modifiée. D'abord nous prouvons que la catégorie C H des modules de poids nilpotents sur U H ξ sl(2|1) est enrubannée et qu'il existe une trace modifiée sur son idéal des modules projectifs. De plus C H possède une structure relativement G-prémodulaire ce qui est une condition suffisante pour construire un invariant de 3-variétés à la Costantino-Geer-Patureau. Cet invariant est le coeur d'une 1 + 1 + 1-TQFT (Topological Quantum Field Theory). D'autre part Hennings a proposé à partir d'une algèbre de Hopf de dimension finie une construction d'invariants qui dispense de considérer la catégorie de ses représentations. Nous montrons que le groupe quantique déroulé U H ξ sl(2|1)/(e 1 , f 1 ) possède une complétion qui est une algèbre de Hopf enrubannée topologique. Nous construisons un invariant de 3-variétés à la Hennings en utilisant cette structure algébrique, une transformation de Fourier discrète et la notion de G-intégrales. L'intégrale dans une algèbre de Hopf est centrale dans la construction de Hennings. La notion de trace modifiée dans une catégorie s'est récemment révélée être une généralisation des intégrales dans les algèbres de Hopf de dimension finie. Dans un contexte plus général d'algèbre de Hopf de dimension infinie nous prouvons la relation formulée entre la trace modifiée et la G-intégrale.

Pendant plus de trois ans de thèse, j'ai eu la chance de participer à des activités de recherches. Je remercie Gaël, Guillaume, Michèle, Sylvain, . . . et les membres de l'équipe de recherche Géométrie et Topologie du LMBA pour le Séminaire Quimpériodique. Je me souviens des Journées TQFT, groupes quantiques et invariants non commutatifs , je remercie Christian et Nathan pour ces journées à Paris. Je n'oublie pas les jours où j'étais au Meeting on non-semisimple TFT and logarithmic CFT à l'Universität Hamburg, merci à Ingo Runkel pour ces jours-là. Je remercie encore Christian, Hoel et la Fondation Louis D. pour TQFT and categorification à Cargèse. Merci aux membres du LMBA pour un bon environnement mathématique et pour la gentillesse des gens ici.

Je me souviens de bon moments où j'en ai bien profité, c'est le foot le vendredi avec Sylvain, Ronan et Du, l'algèbre de Hopf avec Salim, du café, des gâteaux avec les autres . . . Je vous remercie pour ces bons moments. Je tiens à remercier mes amis et les doctorants dans le labo pour leur aide dans la vie : Merci à Du, à Kévin (pour sa présentation de la culturel française : du vin, du café, . . . ), à Jamila, à Thuy, à Erwan, à Phuc, à la famille Tan iii Tra, à la famille Le Tho -Jean, à la famille bac Hung et ba noi, à Ronan, à Éric, à Jonathan, à Tarik, à Hui, à Cuong, à Thong et Vy, à Jean François, à Lucien, ... Ils m'ont aidé à garder l'équilibre entre la vie quotidienne et le travail de recherche.

Merci à mes collègues à l'Université de Hung Vuong, Phu Tho, Vietnam m'avoir remplacé dans l'enseignement pendant le temps où je faisais ma thèse en France.

Enfin, je me souviens toujours de ma fille Phuong Quynh, ma femme Huong, de mes parents, de ma soeur et mon frère au Vietnam. Ils ont été à mes côtés tout le temps de ma thèse. Je vous remercie beaucoup.

Chapitre 1 Introduction 1.1 Contexte

Depuis les années 80, la naissance du polynôme de Jones (voir [START_REF] Jones | A polynomial invariant for knots via von Neumann algebras[END_REF]) a ouvert une nouvelle direction de recherche pour les invariants topologiques d'entrelacs et de 3-variétés. Quelques années après, beaucoup d'invariants d'entrelacs ont été découverts qui sont des généralisations du polynôme de Jones. D'abord le polynôme à deux variables appelé "HOMFLY" qui est une généralisation du polynôme de Jones (le nom HOMFLY provient des noms de six mathématiciens Hoste, Ocneanu, Millett, Freyd, Lickorish, et Yetter qui découvrent simultanément ce polynôme) (voir [START_REF] Freyd | A new polynomial invariant of knots and links[END_REF]). Puis Kauffman a défini une autre généralisation et a construit un invariant d'entrelacs en bande indépendant de l'orientation, ... ( [START_REF] Ohtsuki | Colored ribbon Hopf algebras and universal invariants of framed links[END_REF]). Ensuite, dans les deux articles [START_REF] Reshetikhin | Ribbon graphs and their invariants derived from quantum groups[END_REF] en 1990, et [START_REF] Reshetikhin | Invariants of 3-manifolds via link polynomials and quantum groups[END_REF] en 1991 N. Reshetikhin, V.G. Turaev et E. Witten ont introduit une méthode de construction d'invariant d'entrelacs (nommé invariant RT) et de 3-variétés (nommé invariant WRT). Le premier article a présenté la construction d'un foncteur F d'une catégorie des graphes en rubans vers une catégorie enrubannée C . Ces graphes en rubans sont composés par les parties élémentaires comme des bandes, des coupons, des anneaux, ... Ils sont coloriés par des objets et des morphismes de la catégorie C . Le foncteur F ne dépend que la classe d'isotopie des graphes plongés et il détermine un invariant des entrelacs. En utilisant des représentations du groupe quantique U q sl(2) on retrouve le polynôme de Jones. Dans leur deuxième article, inspiré par les idées de E. Witten (voir [START_REF] Witten | Quantum field theory and the Jones polynomial[END_REF]) ils ont utilisé une catégorie modulaire enrubannée C pour construire un invariant de 3-variétés.

Dans certains contextes, l'invariant RT se révèle être trivial, par exemple pour les représentations projectives du groupe quantique U q sl(2) où q est une racine de l'unité. La raison qui cause ce phénomène est la nullité de dimen-1 sion quantique de la représentation (e. g. [START_REF] Geer | An invariant supertrace for the category of representations of Lie superalgebras[END_REF]). Pour trouver des informations cachées dans cette situation N. Geer, B. Patureau-Mirand et V. Turaev ont proposé une méthode dont l'idée principale est le remplacement de la dimension quantique par la dimension modifiée dans la construction de l'invariant RT ( [START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF]). La dimension modifiée est déterminée par une famille des formes klinéaires nommée une trace modifiée. Ces notions leur permettent de trouver un invariant F non trivial même lorsque l'invariant RT est trivial. La trace modifiée et ses techniques fournissent un autre point de vue sur la construction des invariants topologiques. Avec F. Costantino ([8]) et F. Costantino et C. Blanchet ([4]) ils généralisent avec ces nouveaux invariants la construction WRT pour produire des invariants de 3-variétés et des TQFTs (Topological Quantum Field Theories).

Les superalgèbres de Lie ( [START_REF] Kac | Lie superalgebra[END_REF]) sont des généralisations des algèbres de Lie utilisées par les physiciens pour décrire les super symétries. Elles admettent, comme les algèbres de Lie une déformation et leurs représentations sont en partie connues. Par exemple, les représentations irréductibles de U q sl(2|1) aux racines de l'unité sont décrites dans [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]. La construction de Reshetikhin et Turaev repose sur l'existence d'une catégorie de représentations semi-simples des groupes quantiques. Cette propriété fait défaut dans le cas des groupes quantiques associés aux superalgèbres de Lie. Ceci suggère d'essayer d'utiliser des traces modifiées pour contourner cette difficulté et de tenter de développer une construction similaire à celle de [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF].

Dans une autre direction, M. Hennings a présenté une méthode de construction d'invariants de 3-variétés en utilisant une intégrale sur une algèbre de Hopf enrubannée de dimension finie ( [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF]). De plus, dans [START_REF] Turaev | Homotopy Quantum Field Theory[END_REF] V. G. Turaev a présenté une structure de π-cogèbre de Hopf, i.e. un ensemble d'algèbres indexées par les éléments d'un groupe π avec des applications nommées le produit, le coproduit, l'unité, la counité et l'antipode qui satisfont des axiomes de compatiblité. Puis A. Virelizier a démontré l'existence d'une intégrale et d'une trace sur π-structure dans [START_REF] Virelizier | Hopf group-coalgebra[END_REF]. L'intégrale sur une π-cogèbre de Hopf nommée π-intégrale est une généralisation de la notion de l'intégrale sur une algèbre de Hopf utilisée dans la construction de Hennings. En utilisant une π-cogèbre de Hopf unimodulaire enrubannée de type finie et une π-intégrale, ils ont construit un invariant de 3-variétés dans [START_REF] Virelizier | Algèbres de Hopf graduées et fibrés plats sur les 3-variétés[END_REF]. Récemment dans [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] une relation a été trouvé entre l'intégrale sur l'algèbre de Hopf H et la trace modifiée dans la catégorie correspondante H-mod : A. Beliakova, C. Blanchet et A. M. Gainutdinov ont notamment établi une formule reliant la trace modifiée et l'intégrale.

Présentation des objectifs

Motivé par la notion de la trace modifiée nous voulons développer ses techniques dans le contexte des représentations du groupe quantique U ξ sl(2|1) décrites dans [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]. On suppose pouvoir construire un invariant de l'entrelacs coloriés par ses représentations. Cela nous fournit le premier objectif : c'est la construction des invariants quantiques associés à la super algèbre de Lie sl(2|1). Pour faire cela : d'abord on démontre qu'il existe une structure enrubannée dans la catégorie C H des représentations nilpotentes des modules de poids sur U H ξ sl(2|1), ensuite on indique l'existence de la trace modifiée sur l'idéal Proj(C H ) des modules projectifs dans C H . Cette trace modifiée nous donne un invariant des graphes enrubannés. De plus la catégorie enrubannée C H a aussi une structure relativement G-prémodulaire, ce qui permet de construire un invariant de 3-variétés à la Witten-Reshetikhin-Turaev.

À partir d'un invariant de 3-variétés on sait avoir une chance de construire une famille des TQFTs. Par exemple, en utilisant la construction universelle présentée par C. Blanchet, N. Habegger, G. Masbaum and P. Vogel dans [START_REF] Blanchet | Topological quatum field theories derived from the Kauffman bracket[END_REF], une famille de TQFTs est construite dans [START_REF] Blanchet | Nonsemi-simple TQFTs, Reidemeister torsion and Kashaev's invariants[END_REF] à partir de l'invariant quantique trouvé par F. Costantino, N. Geer and B. Patureau-Mirand [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF]. Les TQFTs dans [START_REF] Blanchet | Nonsemi-simple TQFTs, Reidemeister torsion and Kashaev's invariants[END_REF] sont construites à partir de l'invariant CGP associé à sl(2) ( [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF]) qui est similaire à celui que l'on définit ici avec U H ξ sl(2|1). C'est la raison qui a motivé le deuxième objectif : la construction de 1 + 1 + 1-TQFTs à partir ces invariants de 3-variétés. Pour appliquer la construction de De Renzi ( [START_REF] Renzi | Non-semisimple extended topological quantum field theories[END_REF]) on montre que la catégorie C H est une catégorie relativement G-modulaire.

M. Hennings dans [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF] a proposé une manière de construire un invariant de 3-variétés à partir d'une algèbre de Hopf enrubannée de dimension finie à l'aide de l'intégrale. Inspiré par sa méthode, nous désirions construire un invariant de 3-variétés pour le groupe quantique U H ξ sl(2|1). Néanmoins, la dimension de la superalgèbre de Hopf U H ξ sl(2|1) n'est pas finie, cela cause des difficultés. Les travaux ont été motivés par les réflexions suivantes : Puisqu'il existe une trace modifiée sur idéal des modules projectifs dans C H produisant un invariant de 3-variétés, nous conjecturons qu'il existe quand même une chose analogue pour la superalgèbre U H = U H ξ sl(2|1)/(e 1 , f 1 ). En d'autres termes, nous pourrions construire un invariant de 3-variétés à la Hennings avec la superalgèbre U H .

Ceci est effectivement réalisé en remplaçant l'intégrale par une intégrale graduée. Donc, à partir de la superalgèbre U H ξ sl(2|1), on a deux approches pour construire cet invariant. La première manière utilise la structure de la catégorie enrubannée C H et la trace modifiée en dedans. L'autre manière utilise une structure d'algèbre topologique U H et l'intégrale graduée. Ceci suggère une relation entre les deux objets : la trace modifiée dans une catégorie et l'intégrale graduée d'une algèbre de Hopf. Dans un article récent [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] les auteurs ont montré que la trace modifiée dans la catégorie H-mod est l'intégrale symétrisée de l'algèbre de Hopf de dimension finie H. Inspiré par la suggestion ci-dessus nous nous fixons deux objectifs supplémentaires. Le troisième objectif est de trouver la relation entre la trace modifiée des catégories de représentations des groupes quantiques et les intégrales des G-cogèbres de Hopf pivotales correspondantes.

Finalement, le dernier objectif est la construction d'un invariant de 3variétés de type Hennings associé au groupe quantique déroulé U H ξ sl(2|1) malgré que ce soit une algèbre de Hopf de dimension infinie. Cet invariant est construit en trois étapes : 1) l'introduction d'une topologie sur l'algèbre déroulée, 2) une transformation de Fourier discrète et 3) la version G-graduée de l'invariant de Hennings dû à A. Virelizier ([48]).

Résultats principaux

Le texte est composé de quatre chapitres. Ses résultats principaux sont présentés dans les trois derniers chapitres. En particulier ils sont la reproduction des articles [START_REF] Ha | Topological invariants from quantum group U ξ sl(2|1) at roots of unity[END_REF], [START_REF] Ha | Modified trace from pivotal Hopf G-coalgebra[END_REF] et [START_REF] Ha | A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra[END_REF]. Dans le deuxième chapitre, nous démontrons que la catégorie paire C H des modules de poids nilpotents du groupe quantique U H ξ sl(2|1) est enrubannée par le théorème 2.4.4, la proposition 2.4.5 et qu'il existe une trace modifiée sur idéal des modules projectifs de C H par le théorème 2.5.4. On construit un invariant de graphes enrubannées dans S 3 par cette trace avec le théorème 2.5.5. De plus, cette catégorie possède une structure relativement G-prémodulaire avec G = (C/Z × C/Z, +), cela implique une construction d'invariants de 3-variétés similaire à celle dans [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF] par le théorème 2.6.4. Ses résultats sont présentés dans l'article Topological invariants from quantum groups U ξ sl(2|1) at roots of unity ([22]). Dans ce chapitre nous rajoutons aussi une partie complémentaire où on montre que la catégorie paire C H des modules de poids nilpotents du groupe quantique U H ξ sl(2|1) est relativement G-modulaire d'après le sens de De Renzi ( [START_REF] Renzi | Non-semisimple extended topological quantum field theories[END_REF]) par la proposition 2.7.2. Cela nous permet de construire une famille de 1 + 1 + 1-TQFTs étendues et graduées par Z × Z × Z/2Z.

Le troisième chapitre parle de la relation proche entre trace modifiée et intégrale. Soit H une algèbre de Hopf de dimension finie, il existe un élément de H * appelé intégrale sur H qui est utilisé pour construire l'invariant de 3-variétés de Hennings (voir [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF]). À partir de cette notion on peut définir la notion d'intégrale symétrisée et prouver une formule où chaque intégrale symétrisée sur H correspond à une trace modifiée sur l'idéal H-pmod des H-modules projectifs dans la catégorie H-mod (voir [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]). Pour généraliser ce résultat dans le contexte où la dimension de l'algèbre de Hopf H peut être infinie, on a défini la notion d'une G-cogèbre de Hopf pivotale. Une G-intégrale sur une G-cogèbre de Hopf pivotale nous permet de définir une G-intégrale symétrisée. Elle coïncide avec une trace modifiée sur l'idéal des Hmodules projectifs. Autrement dit, si H est une algèbre de Hopf (sa dimension peut être infinie) et G est un groupe, on peut parfois former une G-cogèbre de Hopf pivotale (H g ) g∈G à partir de quotients de H. Les relations entre les G-intégrales sur (H g ) g∈G et les traces modifiées sur l'idéal des H-modules projectifs de dimension finie sont établies par le théorème 3.1.1. Ceci nous a permit une autre approche de construction de l'invariant de 3-variétés [START_REF] Ha | Topological invariants from quantum group U ξ sl(2|1) at roots of unity[END_REF] à partir des intégrales symétrisées. Dans cette partie nous donnons aussi une application (voir Section 3.5) de la relation entre G-intégrale et trace modifiée par des calculs pour le groupe quantique associé à l'algèbre de Lie sl(2) et la catégorie correspondante. Ces résultats sont prépubliés sur arXiv ( [START_REF] Ha | Modified trace from pivotal Hopf G-coalgebra[END_REF]).

Le quatrième chapitre revient au groupe quantique déroulé U H générant la catégorie C H . Soit W l'espace vectoriel de dimension finie sur C avec une base {e p 1 e ρ 3 e σ 2 f p

1 f ρ 3 f σ 2 0 ≤ ρ, σ, ρ , σ ≤ 1, 0 ≤ p, p ≤ -1}. Le groupe quan- tique U H est isomorphe à W ⊗C[k ±1 1 , k ±1 2 , h 1 , h 2 ]
. Nous considérons l'injection de U H dans W ⊗H(h 1 , h 2 ) où H(h 1 , h 2 ) est l'espace vectoriel des fonctions holomorphes sur C 2 . On peut voir chaque élément de W ⊗H(h 1 , h 2 ) comme une fonction holomorphe à valeurs dans W . Puis on peut déterminer une topologie sur cet espace : c'est la topologie de la convergence uniforme sur les ensembles compacts. Nous démontrons que cette superalgèbre de Hopf possède une structure de superalgèbre de Hopf enrubannée au sens topologique. C'est à dire que cette topologie est compatible avec la structure d'algèbre de Hopf (cf. théorème 4.2.17). Sa bosonization est une algèbre de Hopf topologique enrubannée. Cette algèbre nous donne d'abord une construction d'invariant universel de l'entrelacs par le théorème 4.3.2 et puis une G-cogèbre de Hopf pivotale de type finie U σ par la proposition 4.4.2 où chaque composante de U σ est le quotient de l'algèbre par l'idéal engendré par k i -ξ α i pour i = 1, 2. Les G-intégrales sur U σ , l'invariant universel et une transformation de Fourier discrète nous permettent de construire un invariant de 3-variétés de type Hennings par le théorème 4.4.15. La méthode présentée dans ce chapitre pourrait se généraliser dans le contexte des groupes quantiques déroulés. Ces résultats sont prépubliés sur arXiv ( [START_REF] Ha | A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra[END_REF]).

Au début de chaque chapitre, nous redéfinissons les notions nécessaires et rappelons les résultats préliminaires. En conséquence chaque chapitre pourrait être lu indépendamment des autres.

Chapter 2

Topological invariants from quantum group U ξ sl(2|1) at roots of unity

This chapter contains two parts, the first one with six sections is the content of the paper [START_REF] Ha | Topological invariants from quantum group U ξ sl(2|1) at roots of unity[END_REF] in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, the second one is to prove the category C H is relative G-modular.

Résumé. Dans ce chapitre, nous construisons des invariants d'entrelacs et des invariants de 3-variétés à partir du groupe quantique associé à la superalgèbre de Lie sl(2|1). La construction est basée sur des représentations nilpotentes irréductibles finies du groupe quantique U ξ sl(2|1) où ξ est une racine de l'unité d'ordre impair. Ces constructions utilisent la notion de trace modifiée présentée par Geer, Kujawa et Patureau-Mirand [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF] et la catégorie relativement G-modulaire présentée par Costantino, Geer et Patureau-Mirand [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF].

Abstract. In this chapter we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra sl(2|1). The construction is based on nilpotent irreducible finite dimensional representations of quantum group U ξ sl(2|1) where ξ is a root of unity of odd order. These constructions use the notion of modified trace presented by Geer, Kujawa and Patureau-Mirand [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF] and relative G-modular category presented by Costantino, Geer and Patureau-Mirand [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF].

Introduction

The vanishing of the dimension of an object V in a ribbon category C is an obstruction when one studies the Reshetikhin-Turaev link invariant. If the dimension of a simple object V of C is zero, then the quantum invariants of all (framed oriented) links with components labelled by V are equal to zero, i.e. they are trivial. To overcome this difficulty, the authors N. Geer, B. Patureau-Mirand and V. Turaev introduced the notion of a modified dimension (see [START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF]). The modified dimension may be non-zero when dim C (V ) = 0. Using the modified dimension, for example on the class of projective simple objects, they defined an isotopy invariant F (L) (the renormalized Reshetikhin-Turaev link invariant) for any link L whose components are labelled with objects of C under the only assumption that at least one of the labels belongs to the set of projective ambidextrous objects. Here F (L) is a nontrivial link invariant (see [START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF]). This modified dimension is used to construct the quantum invariants in [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF], [START_REF] Geer | Multivariable link invariants arising from sl(2|1) and the Alexander polynomial[END_REF].

The existence of the modified dimension generalizes the definition of modified traces (see [START_REF] Geer | Generalized trace and modified dimension functions on ribbon categories[END_REF]). In the article [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF], the authors showed that a necessary and sufficient condition for the existence of a modified trace on an ideal generated by a simple object J is that J is an ambidextrous object. Recently the existence of an ambidextrous object has been shown in the context of factorizable finite tensor categories [START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF].

The Lie superalgebras (see [START_REF] Kac | Lie superalgebra[END_REF]) are the generalizations of Lie algebras in the category of super vector spaces. They are used among others by physicists to describe supersymmetry. Deformations of these superalgebras and their representations are partially known. For the Lie superalgebra sl(2|1) one can define a Hopf superalgebra U ξ sl(2|1) which is a deformation of the universal enveloping algebra. Its irreducible representations at roots of unity are described in [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]. Using these representations and developing the idea of modified traces open up the method for constructing a quantum invariant of framed links with components labelled by irreducible representations.

The aim of this chapter is to construct a link invariant and a 3-manifold invariant from quantum group U ξ sl(2|1) at a root of unity of odd order. Note that the Lie superalgebra sl(2|1) having superdimension zero, sl(2|1)-weight functions are trivial. Hence combining them with the Kontsevich integral or the LMO invariant also give trivial link and 3-manifold invariants. The chapter contains six sections. In Section 2.2, we recall the monoidal category, pivotal category, braided category, ribbon category and, Hopf superalgebra definitions. In Section 2.3 and 2.4, we describe the quantum superalgebra U ξ sl(2|1) where ξ is a root of unity of odd order and by adding two elements h 1 , h 2 to U ξ sl(2|1), we have the Hopf superalgebra U H ξ sl(2|1). Using this extension we can construct a non semi-simple ribbon category C H of the nilpotent simple finite dimensional representations of U H ξ sl(2|1). In Section 2.5 we prove that a typical module over U H ξ sl(2|1) is an ambidextrous module and that a modified trace exists on the ideal of projective modules Proj. This modified trace will be used to construct a link invariant. In Section 2.6, we prove that the category C H is relative G-premodular ( [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF]) and we construct a 3-manifold invariant using this property.

Preliminaries

Monoidal category

Definition 2.2.1 ([33, 29]). A monoidal category C is a category enhanced with a bifunctor called tensor product • ⊗ • : C × C → C and a unit object I such that there are natural isomorphisms

I ⊗ • ∼ = • ⊗ I ∼ = Id C and (• ⊗ •) ⊗ • ∼ = • ⊗ (• ⊗ •), (2.2.1)
fulfilling the Pentagon Axiom and the Triangle Axiom.

We call strict monoidal category a monoidal category C whose the isomorphisms (2.2.1) are identities. In our examples the morphisms in (2.2.1) are simply the morphisms of the underlying vector spaces and are in the following regarded as equality. We write V ∈ C to denote an object V in the category C and call Hom C (V, W ) the morphisms in

C from V ∈ C to W ∈ C and End C (V ) = Hom C (V, V ).
We say that C is a monoidal C-linear category if for all V, W ∈ C , the morphisms Hom C (V, W ) form a C-vector space and the composition and the tensor product are bilinear and End C (I)

∼ = C. An object V ∈ C is simple if and only if End C (V ) ∼ = C as a unitary C-algebra. An object W ∈ C is a direct sum of V 1 , ..., V n ∈ C if there is for i = 1, ..., n, f i ∈ Hom C (V i , W ), g i ∈ Hom C (W, V i ) such that g i •f i = Id V i , g i •f j = 0 for i = j and n i=1 f i •g i = Id W . An object W ∈ C is semi-simple if it
is a direct sum of simple objects. The category C is semi-simple if all objects are semi-simple and Hom C (V, W ) = {0} for any pair of non-isomorphic simple objects in C .

Pivotal category Definition 2.2.2. Let C be a monoidal category and A, B ∈ C . A duality between A and B is given by a pair of morphisms

(α ∈ Hom C (I, B ⊗ A), β ∈ Hom C (A ⊗ B, I)) such that (β ⊗ Id A ) • (Id A ⊗α) = Id A and (Id B ⊗β) • (α ⊗ Id B ) = Id B . (2.2.2)
A pivotal category (or sovereign) is a strict monoidal category C , with a unit object I, equipped with the data for each object V ∈ C of its dual object V * ∈ C and of four morphisms

-→ ev V : V * ⊗ V → I, -→ coev V : I → V ⊗ V * , ←- ev V : V ⊗ V * → I, ←- coev V : I → V * ⊗ V such that ( -→ ev V , -→ coev V ) and ( ←- ev V , ←- coev V )
are dualities which induce the same functor duality and the same natural isomorphism (V ⊗ W ) * ∼ = W * ⊗ V * . Thus, the right and left dual coincide in C : for every morphism h : V → W , we have

h * = ( -→ ev W ⊗ Id V * ) • (Id W * ⊗h ⊗ Id V * ) • (Id W * ⊗ -→ coev V ) = (Id V * ⊗ ←- ev W ) • (Id V * ⊗h ⊗ Id W * ) • ( ←- coev V ⊗ Id W * ) : W * → V * and for V, W ∈ C , the isomorphisms γ V,W : W * ⊗ V * → (V ⊗ W ) * are given by γ V,W = ( -→ ev W ⊗ Id (V ⊗W ) * ) • (Id W * ⊗ -→ ev V ⊗ Id W ⊗(V ⊗W ) * ) • (Id W * ⊗V * ⊗ -→ coev V ⊗W ) = (Id (V ⊗W ) * ⊗ ←- ev V ) • (Id (V ⊗W ) * ⊗V ⊗ ←- ev W ⊗ Id V * ) • ( ←- coev V ⊗W ⊗ Id W * ⊗V * ).
The family of isomorphisms

Φ = {Φ V = ( ←- ev V ⊗ Id V * * ) • (Id V ⊗ -→ coev V * ) : V → V * * } V ∈C
is a monoidal natural isomorphism called the pivotal structure.

Definition 2.2.3. Given a multiplicative group G, we call the category C pivotal G-graded k-linear if there exists a family of full subcategories

(C α ) α∈G of C such that 1. I ∈ C 1 . 2. ∀(α, β) ∈ G 2 , ∀(V, W ) ∈ C α × C β , Hom C (V, W ) = {0} ⇒ α = β. 3. ∀V ∈ C , ∃n ∈ N, ∃(α 1 , ..., α n ) ∈ G n , ∃V i ∈ C α i for i = 1, ..., n such that V V 1 ⊕ ... ⊕ V n . 4. ∀(V, W ) ∈ C α × C β , V ⊗ W ∈ C αβ .
5. ∀α ∈ G, C α does not reduce to null object.

Ribbon category

A braided category is a tensor category C provided with a braiding c : for all objects V and W of C , we have an isomorphism

c V,W : V ⊗ W → W ⊗ V.
These isomorphisms are natural and for all objects U, V and W of C , we have

c U,V ⊗W = (Id V ⊗c U,W )•(c U,V ⊗Id W ) and c U ⊗V,W = (c U,W ⊗Id V )•(Id U ⊗c V,W ).
If the category C is pivotal and braided, we can define a family of natural isomorphisms

θ V = ptr R (c V,V ) = (Id V ⊗ ←- ev V ) • (c V,V ⊗ Id V * ) • (Id V ⊗ -→ coev V ) : V → V.
We say that θ is a twist if it is compatible with the dual in the following sense

∀V ∈ C , θ V * = (θ V ) * which is equivalent to θ V = ptr L (c V,V ) = ( -→ ev V ⊗ Id V ) • (Id V * ⊗c V,V ) • ( ←- coev V ⊗ Id V ) : V → V.
A ribbon category is a braided pivotal category in which the family of isomorphisms θ is a twist.

Hopf superalgebras

We recall some notions (see also [START_REF] Geer | An invariant supertrace for the category of representations of Lie superalgebras[END_REF], [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]). A super space is a Z/2Zgraded vector space

V = V 0 ⊕ V 1 over C. An element x ∈ V is called even (resp. odd) if x ∈ V 0 (resp. x ∈ V 1 )
. For the super spaces U, V the set of the morphisms between them denoted by Hom C (U, V ) is the super space of linear maps given by

Hom C (U, V ) 0 = Hom C (U 0 , V 0 ) ⊕ Hom C (U 1 , V 1 ) and Hom C (U, V ) 1 = Hom C (U 0 , V 1 ) ⊕ Hom C (U 1 , V 0 ).
Denote ⊗ the usual tensor product in the category Vect C . We call even category SVect 0 the category whose the objects are the super spaces and the morphisms are the even morphisms. Category SVect 0 is monoidal with the operator ⊗: For U, V ∈ SVect 0 their tensor product is the vector space U ⊗V with the parity given by

(U ⊗ V ) 0 = U 0 ⊗V 0 ⊕ U 1 ⊗V 1 and (U ⊗ V ) 1 = U 0 ⊗V 1 ⊕ U 1 ⊗V 0 , for f ∈ Hom C (U, U ), g ∈ Hom C (V, V ) the tensor product f ⊗ g is given by f ⊗ g =    f ⊗g on U 0 ⊗V (-1) g f ⊗g on U 1 ⊗V . This means that f ⊗ g(x ⊗ y) = (-1) g.x f (x) ⊗ g(y).
Further, SVect 0 is also a symmetric monoidal category with symmetry isomorphisms τ U,V :

U ⊗ V V ⊗ U given by τ U,V (u ⊗ v) = (-1) u.v v ⊗ u.
Note that the category SVect of the super spaces with all morphisms is not a symmetric monoidal category because in general (Id ⊗g)

• (f ⊗ Id) = (f ⊗ Id) • (Id ⊗g).
We call Hopf superalgebra a Hopf algebra object in SVect 0 . That is a super C-vector space H endowed with five even C-linear maps called product, unit, coproduct, counit and antipode

m : H ⊗ H → H, η : C → H, ∆ : H → H ⊗ H, ε : H → C and S : H → H
satisfying the axioms:

1. the product m is associative on H admitting 1 H = η(1) as unity.

the coproduct

∆ is coassociative, i.e. (∆ ⊗ Id H ) • ∆ = (Id H ⊗∆) • ∆ and (ε ⊗ Id H ) • ∆ = (Id H ⊗ε) • ∆ = Id H .
3. ∆ and ε are algebra morphisms where the associative product in

H ⊗H is determined by (m ⊗ m) • (Id H ⊗τ H,H ⊗ Id H ). 4. m • (S ⊗ Id H ) • ∆ = m • (Id H ⊗S) • ∆ = η • ε.
Let H be a Hopf superalgebra. An even grouplike element

φ ∈ H is said a pivotal element if ∆(φ) = φ ⊗ φ, ε(φ) = 1 and for all h ∈ H, S 2 (h) = φhφ -1 .
The pair (H, φ) of a Hopf superalgebra and a pivot φ is called a pivotal Hopf superalgebra (see [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]). Let (H, φ) be a Hopf superalgebra, let H-mod 0 be the even category of finite dimensional modules over H. If V is an object of H-mod 0 we denote by ρ V : H → End C (V ) the representation of H in the module V .

Proposition 2.2.4 ([39]

). The category H-mod 0 has the structure of a pivotal category with dual morphisms given by

-→ ev V : e * i ⊗ e j → e * i (e j ) = δ j i , -→ coev V : 1 → i e i ⊗ e * i , ←- ev V : e j ⊗ e * i → (-1) deg e j e * i (φ 0 .e j ), ←- coev V : 1 → i (-1) deg e i e * i ⊗ (φ -1 0 .e i )
where (e i ) i is a basis of V and (e * i ) i is its basis dual. Proof. Let V be an object of H-mod 0 . Its dual is a C-vector space V * = Hom C (V, C) provided with the action of h ∈ H given by

(h, ϕ) → (-1) deg h deg ϕ ϕ • ρ V (S(h)).
First we show that four morphisms

-→ ev V , -→ coev V , ←- ev V , ←- coev V are invariant morphisms of H-mod 0 . It is clear for -→ ev V , -→ coev V , we prove ←- ev V is invariant morphism. The invariant of the morphism ←- coev V is proved similarly. For h ∈ H, using the Sweedler notation ∆(h) = h (1) ⊗ h (2) and denote deg x = |x|, one computes ←- ev V (h.(e j ⊗ e * i )) = (-1) |h (2) ||e j | ←- ev V (h (1) e j ⊗ h (2) e * i ) = (-1) |h (2) ||e j |+|e * i ||h (2) | ←- ev V (h (1) e j ⊗ e * i • S(h (2) )) = (-1) |h (2) ||e j |+|e * i ||h (2) |+(|e j |+|h (1) |)(|h (2) |+|e * i |) e * i S(h (2) )φh (1) e j = (-1) |h (2) ||e * i |+|e j ||e * i |+|h (1) ||h (2) |+|h (1) ||e * i | e * i φS -1 (h (2) )h (1) e j = (-1) |h (2) ||e * i |+|e j ||e * i |+|h (1) ||e * i | e * i φS -1 (S(h (1) )h (2) )e j = (-1) |h (2) ||e * i |+|e j ||e * i |+|h (1) ||e * i | ε(h)e * i (φe j ) = (-1) |h||e * i |+|e j ||e * i | ε(h)e * i (φe j ) = (-1) |h||e * i | ε(h) ←- ev V (e j ⊗ e * i ). If |h| = 1 then ε(h) = 0. This implies that ←- ev V (h.(e j ⊗ e * i )) = ε(h) ←- ev V (e j ⊗ e * i )
. The duality of the pair (

-→ ev V , -→ coev V ) is clear by definition. For ( ←- ev V , ←- coev V ), one checks ( ←- ev V ⊗ Id V ) • (Id V ⊗ ←- coev V ) = Id V .
For each e j we have

( ←- ev V ⊗ Id V ) • (Id V ⊗ ←- coev V )(e j ⊗ 1) = i (-1) 2|e i | e * i (ρ V (φ)e j ) ⊗ ρ V (φ -1 )e i = i e * i (ρ V (φ)e j ) ⊗ ρ V (φ -1 )e i .
Suppose A = (a st ) s,t is the matrix of ρ V (φ) then the matrix of ρ V (φ -1 ) is

A -1 = (b st ) s,t in the basis (e i ) i , one gets i e * i (ρ V (φ)e j ) ⊗ ρ V (φ -1 )e i = i e * i s a sj e s ⊗ t b ti e t = i s a sj e * i (e s ) ⊗ t b ti e t = i a ij ⊗ t b ti e t = t i b ti a ij e t = t δ t j e t = e j .
By similar calculations one gets the equality

(Id V * ⊗ ←- ev V ) • ( ←- coev V ⊗ Id V * ) = Id V * .
Thus the pair of morphisms (

←- ev V , ←- coev V ) are dualities.

Quantum superalgebra U ξ sl(2|1)

In this section we define the superalgebra U ξ sl(2|1) and we prove that it is a pivotal Hopf superalgebra. We also show that the Borel part of U ξ sl(2|1) is a Nichols algebra.

Hopf superalgebra U ξ sl(2|1)

Definition 2.3.1. Let ≥ 3 be an odd integer and ξ = exp( 2πi ). The superalgebra U ξ sl(2|1) is an associative superalgebra on C generated by the elements

k 1 , k 2 , k -1 1 , k -1
2 , e 1 , e 2 , f 1 , f 2 and the relations

k 1 k 2 = k 2 k 1 , (2.3.1) k i k -1 i = 1, i = 1, 2, (2.3.2) k i e j k -1 i = ξ a ij e j , k i f j k -1 i = ξ -a ij f j i, j = 1, 2, (2.3.3) e 1 f 1 -f 1 e 1 = k 1 -k -1 1 ξ -ξ -1 , e 2 f 2 + f 2 e 2 = k 2 -k -1 2 ξ -ξ -1 , (2.3.4) [e 1 , f 2 ] = 0, [e 2 , f 1 ] = 0, (2.3.5) e 2 2 = f 2 2 = 0, (2.3.6) e 2 1 e 2 -(ξ + ξ -1 )e 1 e 2 e 1 + e 2 e 2 1 = 0, (2.3.7) f 2 1 f 2 -(ξ + ξ -1 )f 1 f 2 f 1 + f 2 f 2 1 = 0. (2.3.8)
The last two relations are called the Serre relations. The matrix (a ij ) is given by a 11 = 2, a 12 = a 21 = -1, a 22 = 0. The odd generators are e 2 , f 2 .

We define ξ x := exp( 2πix ), afterwards we will use the concepts

{x} = ξ x -ξ -x , [x] = ξ x -ξ -x ξ -ξ -1 .
Let define the odd elements

e 3 = e 1 e 2 -ξ -1 e 2 e 1 , f 3 = f 2 f 1 -ξf 1 f 2 . The Serre relations become e 1 e 3 = ξe 3 e 1 , f 3 f 1 = ξ -1 f 1 f 3 .
(2.3.9)

Furthermore

e 2 e 3 = -ξe 3 e 2 , f 3 f 2 = -ξ -1 f 2 f 3 , (2.3.10)
e 3 f 3 + f 3 e 3 = k 1 k 2 -k -1 1 k -1 2 ξ -ξ -1 , (2.3.11) e 2 3 = f 2 3 = 0.
(2.3.12)

According to [START_REF] Khoroshkin | Universal R-matrix for quantized (super)algebras[END_REF], U ξ sl(2|1) is a Hopf superalgebra with the coproduct, counit and antipode as below

∆(e i ) = e i ⊗ 1 + k -1 i ⊗ e i i = 1, 2, ∆(f i ) = f i ⊗ k i + 1 ⊗ f i i = 1, 2, ∆(k i ) = k i ⊗ k i i = 1, 2, S(e i ) = -k i e i , S(f i ) = -f i k -1 i , S(k i ) = k -1 i i = 1, 2, ε(k i ) = 1, ε(e i ) = ε(f i ) = 0 i = 1, 2.
The center and representations of U ξ sl(2|1) were studied by B. Abdesselam, D. Arnaudon and M. Bauer [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]. We focus on the case of nilpotent representations of type B with the condition odd.

Remark 2.3.2. 1. Because (e 1 ⊗ 1)(k -1 1 ⊗ e 1 ) = ξ 2 (k -1 1 ⊗ e 1 )(e 1 ⊗ 1) and
( ) ξ := 1-ξ 1-ξ = 0 then ∆(e 1 ) = m=0 m ξ (e 1 ⊗ 1) m (k -1 1 ⊗ e 1 ) -m = e 1 ⊗ 1 + k - 1 ⊗ e 1 . (2.3.13) CHAPTER 2.

INVARIANTS FROM QUANTUM GROUP

We have ∆ op (e 1 ) = 1 ⊗ e 1 + e 1 ⊗ k - 1 at the same time. It is known that e 1 , f 1 , k 1 ∈ Z where Z is the center of U ξ sl(2|1), so ∆(e 1 ) ∈ Z ⊗ Z. It follows that there exists no element R ∈ U ξ sl(2|1) ⊗ U ξ sl(2|1) such that ∆ op (x) = R∆(x)R -1 ∀x ∈ U ξ sl(2|1), i.e. the superalgebra U ξ sl(2|1) is not quasitriangular.

2. We think that the quotient superalgebra U ξ sl(2|1)/(e 1 , f 1 ) is not quasitriangular but a quotient like U ξ sl(2|1)/(e 1 , f 1 , k 1 -1, k 2 -1) could be, a proof of this might be found along the lines of [START_REF] Lentner | New R-matrices for small quantum groups[END_REF]. This is not the quotient that interests us in this chapter.

3. The unrolled version U H ξ sl(2|1) (defined in Section 2.4) seems to be quasitriangular only in a topological sense (see [START_REF] Ha | A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra[END_REF]). However, we will show in Theorem 2.4.4 and Proposition 2.4.5 that some representations (the weight modules) form a ribbon category.

It is commonly admitted that the superalgebra [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]). Nevertheless, we give in Appendix A.1 an elementary proof of this fact stated in Lemma 2.3.3. Its Borel part is a superalgebra U ξ (n + ) which has a vector space basis {e ρ 2 e σ 3 e p 1 ρ, σ ∈ {0, 1}, p ∈ {0, 1, ..., -1}}. It is well known that U ξ (n + ) is a Nichols algebra of diagonal type associated with the generalized Dynkin diagram

U ξ sl(2|1)/(e 1 , f 1 ) has a Poincaré-Birkhoff-Witt basis {e ρ 2 e σ 3 e p 1 k s 1 k t 2 f ρ 2 f σ 3 f p 1 , ρ, σ, ρ , σ ∈ {0, 1}, p, p ∈ {0, 1, ..., -1}, s, t ∈ Z} (see
-1 ξ 2 ξ -2
(see [START_REF] Heckenberger | Nichols algebras of diagonal type and arithmetic root systems[END_REF]). We now explain this point of view. We consider the group algebra B = CG in which G is an abelian group generated by k 1 , k 2 , a vector space V on C generated by e 1 , e 2 . Here B is a Hopf algebra and (V, •, δ) is a Yetter-Drinfeld module on B [START_REF] Heckenberger | Nichols algebras of diagonal type and arithmetic root systems[END_REF], where the action

• : B ⊗ V → V of B on V is determined by k 1 • e 1 = ξ 2 e 1 , k 1 • e 2 = ξ -1 e 2 , k 2 • e 1 = ξ -1 e 1 , k 2 • e 2 = -e 2 ,
the matrix determining the bicharacter is (q ij ) 2×2 , q ij = (-1) |i||j| ξ a ij where |1| = 0, |2| = 1 and the coaction δ :

V → B ⊗ V of B on V is given by δ(e i ) = k i ⊗ e i i = 1, 2. It is clear that δ(b • v) = b (1) v (-1) S(b (3) ) ⊗ b (2) • v (0) = v (-1) ⊗ b • v (0) for all b ∈ B, v ∈ V .
Here we use the Sweedler notation and write

(∆ ⊗ Id)∆(b) = b (1) ⊗ b (2) ⊗ b (3) , δ(v) = v (-1) ⊗ v (0) for b ∈ B, v ∈ V .
Using Hopf algebra B and Yetter-Drinfeld module V we can determine the Nichols algebra B(V ) = T (V )/J (V ) where T (V ) = ∞ n=0 V ⊗n is the tensor algebra of V with the braided copoduct ∆(v) = 1 ⊗ v + v ⊗ 1 and counit ε(v) = 0 for v ∈ V , J (V ) is the maximal coideal in degree ≥ 2 of T (V ). We now check that e 2 2 and the Serre relation w = e 1 e 3 -ξe 3 e 1 are in J (V ). We have ∆(e 2 2 ) = ∆(e 2 ) ∆(e 2 ) = (1 ⊗ e 2 + e 2 ⊗ 1)

(1 ⊗ e 2 + e 2 ⊗ 1) = 1 ⊗ e 2 2 + (k 2 • e 2 ) ⊗ e 2 + e 2 ⊗ e 2 + e 2 2 ⊗ 1 = 1 ⊗ e 2 2 + e 2 2 ⊗ 1, so e 2 2 ∈ J (V ). We calculate ∆(e 3 ) = ∆(e 1 ) ∆(e 2 ) -ξ -1 ∆(e 2 ) ∆(e 1 ) = (1 ⊗ e 1 + e 1 ⊗ 1)(1 ⊗ e 2 + e 2 ⊗ 1) -ξ -1 (1 ⊗ e 2 + e 2 ⊗ 1)(1 ⊗ e 1 + e 1 ⊗ 1) = 1 ⊗ e 1 e 2 + (k 1 • e 2 ) ⊗ e 1 + e 1 ⊗ e 2 + e 1 e 2 ⊗ 1 -ξ -1 (1 ⊗ e 2 e 1 + (k 2 • e 1 ) ⊗ e 2 + e 2 ⊗ e 1 + e 2 e 1 ⊗ 1) = 1 ⊗ e 3 + e 3 ⊗ 1 + (1 -ξ -2 )e 1 ⊗ e 2 .
And a similar calculation gives us ∆(e 1 ) ∆(e 3 ) = 1 ⊗ e 1 e 3 + ξe 3 ⊗ e 1 + (1 -ξ -2 )ξ 2 e 1 ⊗ e 1 e 2 + e 1 ⊗ e 3 + e 1 e 3 ⊗ 1 + (1 -ξ -2 )e 2 1 ⊗ e 2 , and By maximality of J (V ), this implies that w ∈ J (V ). The bosonization of B(V ) is then isomorphic to a Hopf subalgebra of the bosonization of the Hopf superalgebra U ξ sl(2|1).

∆(e 3 ) ∆(e 1 ) = 1 ⊗ e 3 e 1 + ξe 1 ⊗ e 3 + e 3 ⊗ e 1 + e 3 e 1 ⊗ 1 + (1 -ξ -2 )e 1 ⊗ e 2 e 1 + (1 -ξ -2 )ξ -1 e 2 1 ⊗ e 2 .

Lemma 2.3.3. The set of vectors {e

ρ 2 e σ 3 e p 1 k s 1 k t 2 f ρ 2 f σ 3 f p 1 ρ, σ, ρ , σ ∈ {0, 1}, p, p ∈ {0, 1, ..., -1}, s, t ∈ Z} is a basis of U ξ sl(2|1)/(e 1 , f 1 ). Proof. See in Appendix A.1.

Pivotal Hopf superalgebra U ξ sl(2|1)

Recall that the even category of representations of a superalgebra is the category of representations in which one restricts to the morphisms of even degree.

Proposition 2.3.4. Given φ

0 = k - 1 k -2 2 , so ∀u ∈ U ξ sl(2|1), S 2 (u) = φ 0 uφ -1 0 .
Proof. This can be verified for generator elements k i , e i , f i , i = 1, 2.

It follows that the Hopf superalgebra U ξ sl(2|1) provided with the pivotal element

φ 0 = k - 1 k -2
2 is pivotal superalgebra (see [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]). Let U ξ sl(2|1)-mod 0 be the category of finite dimensional modules over U ξ sl(2|1) with even morphisms then U ξ sl(2|1)-mod 0 is a pivotal category thanks to Proposition 2.2.4. If V is an object of U ξ sl(2|1)-mod 0 , its dual is a C-vector space V * = Hom C (V, C) provided with the action of u given by (u, ϕ) → (-1)

deg u deg ϕ ϕ • ρ V (S(u)) where ρ V : U ξ sl(2|1) → End C (V ) is the representation of U ξ sl(2|1).
The unit element of category U ξ sl(2|1)-mod 0 is the module C provided with the representation ε :

U ξ sl(2|1) → C ∼ = End C (C).
If one has a basis (e i ) i of V with dual basis (e * i ) i , recall that the dual morphisms given by

-→ ev V : e * i ⊗ e j → e * i (e j ) = δ j i , -→ coev V : 1 → i e i ⊗ e * i , ←- ev V : e j ⊗ e * i → (-1) deg e j e * i (φ 0 .e j ), ←- coev V : 1 → i (-1) deg e i e * i ⊗ (φ -1 0 .e i ).

Category of nilpotent weight modules

This section allows to define the superalgebra U H ξ sl(2|1) from U ξ sl(2|1). Then we define the even category C H of nilpotent finite dimensional weight modules over U H ξ sl(2|1) and prove that this category is G-graded and ribbon. The category C H is used to construct the topological invariants in next sections.

Typical module

where λ i = ξ µ i with i = 1, 2. For µ = (µ 1 , µ 2 ) ∈ C 2 we say that U ξ sl(2|1)-module V µ is typical if it is a simple module of dimension 4 . Other simple modules are said to be atypical. The basis of a typical module is formed by vectors w ρ,σ,p = f ρ 2 f σ 3 f p 1 w 0,0,0 where ρ, σ ∈ {0, 1}, 0 ≤ p < . The odd elements are w 0,1,p and w 1,0,p , others are even. The representation of typical U ξ sl(2|1)-module V µ 1 ,µ 2 is determined by

k 1 w ρ,σ,p = λ 1 ξ ρ-σ-2p w ρ,σ,p , (2.4.1) k 2 w ρ,σ,p = λ 2 ξ σ+p w ρ,σ,p , (2.4.2) f 1 w ρ,σ,p = ξ σ-p w ρ,σ,p+1 -ρ(1 -σ)ξ -σ w ρ-1,σ+1,p , (2.4.3) f 2 w ρ,σ,p = (1 -ρ)w ρ+1,σ,p , (2.4.4) e 1 w ρ,σ,p = -σ(1 -ρ)λ 1 ξ -2p+1 w ρ+1,σ-1,p + [p][µ 1 -p + 1]w ρ,σ,p-1 , (2.4.5) e 2 w ρ,σ,p = ρ[µ 2 + p + σ]w ρ-1,σ,p + σ(-1) ρ λ -1 2 ξ -p w ρ,σ-1,p+1 . (2.4.6)
where ρ, σ ∈ {0, 1} and p ∈ {0, 1, ..., -1}.

We also have [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]).

V µ V µ+ϑ ⇔ ϑ ∈ ( Z) 2 . Remark 2.4.1. The module V µ is typical if [µ 1 -p + 1] = 0 ∀p ∈ {1, ..., - 1} (µ 1 = p -1 + 2 Z ∀p ∈ {1, ..., -1}) and [µ 2 ][µ 1 + µ 2 + 1] = 0 (µ 2 = 2 Z, µ 1 + µ 2 = -1 + 2 Z) (see
We call U H ξ sl(2|1) the C-superalgebra generated by e i , f i , k i , k -1 i and h i for i = 1, 2 with Relations (2.3.1) -(2.3.8) plus the relations

[h i , e j ] = a ij e j , [h i , f j ] = -a ij f j , [h i , h j ] = 0, [h i , k j ] = 0 i, j = 1, 2.
The superalgebra U H ξ sl(2|1) is a Hopf superalgebra where ∆, S and ε are determined as in U ξ sl(2|1) and by

∆(h i ) = h i ⊗ 1 + 1 ⊗ h i , S(h i ) = -h i , ε(h i ) = 0 i = 1, 2.
Note that U H ξ sl(2|1) can be seen as a semidirect product of C[h 1 , h 2 ] acting on U ξ sl(2|1).

Let U H ξ sl(2|1)-mod 0 be the category of finite dimensional modules over U H ξ sl(2|1) with even morphisms then U H ξ sl(2|1)-mod 0 is a pivotal category thanks also to Proposition 2.2.4. We call nilpotent weight U H ξ sl(2|1)-module an object of U H ξ sl(2|1)-mod 0 on which e 1 = f 1 = 0 and ξ h i = k i for i = 1, 2 are diagonalizable. Let C H be the full subcategory of U H ξ sl(2|1)-mod 0 formed by all nilpotent weight modules over U H ξ sl(2|1). The category C H is pivotal similar to C (see Section 2.3.2).

We define the actions of h i , i = 1, 2 on the basis of V µ 1 ,µ 2 by

h 1 w ρ,σ,p = (µ 1 + ρ -σ -2p)w ρ,σ,p , h 2 w ρ,σ,p = (µ 2 + σ + p)w ρ,σ,p .
Thus V µ 1 ,µ 2 is a weight module of C H . A module in C H is said to be typical if, seen as a U ξ sl(2|1)-module, it is typical. For each module V we denote V the same module with the opposite parity. We set G = C/Z × C/Z and for each µ ∈ G we define C H µ as the subcategory of weight modules which have their weights in the coset µ (modulo

Z × Z). So {C H µ } µ∈G is a G-graduation (where G is an additive group): let V ∈ C H µ , V ∈ C H µ , then the weights of V ⊗ V are congruent to µ + µ (modulo Z × Z). Furthermore, if µ = µ then Hom C H (V, V ) = 0 because a morphism preserves weights.
We also define

G s = {g ∈ G such that ∃ V ∈ C H
g simple and atypical}. It follows from [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF] that

G s = 0, 1 2 × C/Z ∪ C/Z × 0, 1 2 ∪ (µ 1 , µ 2 ) : µ 1 + µ 2 ∈ 0, 1 2 .

Character of representations of U H ξ sl(2|1)

Definition 2.4.2. The character of a weight module V is

χ V = µ dim(E µ (V ))X µ 1 1 X µ 2 2
where E µ (V ) is the proper subspace of the proper value µ = (µ 1 , µ 2 ) of (h 1 , h 2 ).

Note that we do not use the concept of a super-character defined as above by replacing the dimension by the super-dimension.

A finite dimensional representation of U ξ gl(2), subalgebra generated by

e 1 , f 1 , k i is defined by V = Span C {v 0 , ..., v -1 } [1] k 1 v p = λ 1 ξ -2p v p with p ∈ {0, 1, ..., -1}, f 1 v p = v p+1 with p ∈ {0, 1, ..., -2} and f 1 v -1 = 0, e 1 v p = [p][µ 1 -p + 1]v p-1 , ξ µ 1 = λ 1 , k 2 v p = λ 2 ξ p v p with p ∈ {0, 1, ..., -1}.

It extends to the generators h 1 , h 2 by

h 1 v p = (µ 1 -2p)v p with p ∈ {0, 1, ..., -1}, h 2 v p = (µ 2 + p)v p with p ∈ {0, 1, ..., -1} so that ξ h i = k i , i = 1, 2 on V . We have the character of representation of U ξ gl(2) χ V gl(2) µ 1 ,µ 2 = X µ 1 1 X µ 2 2 1 -x 1 -x where x = X -2 1 X 2 .
In the case of a typical representation, the nilpotent representation V µ 1 ,µ 2 of U ξ sl(2|1) with highest weight (µ 1 , µ 2 ) is determined by

k 1 w ρ,σ,p = λ 1 ξ ρ-σ-2p w ρ,σ,p , k 2 w ρ,σ,p = λ 2 ξ σ+p w ρ,σ,p
with h 1 w ρ,σ,p = (µ 1 + ρ -σ -2p)w ρ,σ,p and h 2 w ρ,σ,p = (µ 2 + σ + p)w ρ,σ,p . So the nilpotent representation V µ 1 ,µ 2 has the following character

χ V sl(2|1) µ 1 ,µ 2 = χ V gl(2) µ 1 ,µ 2 ,ρ=σ=0 + χ V gl(2) µ 1 ,µ 2 ,ρ=1,σ=0 + χ V gl(2) µ 1 ,µ 2 ,ρ=0,σ=1 + χ V gl(2) µ 1 ,µ 2 ,ρ=σ=1 = X µ 1 1 X µ 2 2 1 -x 1 -x (1 + X 1 )(1 + X 1 x). (2.4.7)

Braided category C H

Let U q sl(2|1) be the C(q)-subsuperalgebra of the h-adic quantized enveloping superalgebra of sl(2|1) generated by the elements e

i , f i , k i , k -1 i for 1 ≤ i ≤ 2 where q = e h ∈ C[[h]][h -1 ]. Let A = C[q, q -1 , ( -1) q ! -1 ]
. Let U A sl(2|1) be the A-subsuperalgebra of U q sl(2|1) generated by the elements e i , f i , k i , k -1 i for 1 ≤ i ≤ 2 and the relations (2.3.1) -(2.3.12) in which ξ is replaced by q.

The C-superalgebra U ξ sl(2|1) can be seen as the specialisation at q = ξ of U A sl(2|1), i.e. U ξ sl(2|1) = U A sl(2|1)/ (q -ξ) U A sl(2|1) (see also [START_REF] Concini | Representations of quantum groups at roots of 1[END_REF]). Then U ξ sl(2|1) is a superalgebra over C with generators e i , f i , k i , k -1 i for 1 ≤ i ≤ 2 and relations (2.3.1) -(2.3.12).

In articles [START_REF] Khoroshkin | Universal R-matrix for quantized (super)algebras[END_REF][START_REF] Yamane | Quantized enveloping algebras associatied with simple Lie superalgebras and their universal R-matrices[END_REF] the authors showed that R q = Řq K q where Řq =

∞ i=0 {1} i e i 1 ⊗ f i 1 (i) q ! 1 j=0 (-{1}) j e j 3 ⊗ f j 3 (j) q ! 1 k=0 (-{1}) k e k 2 ⊗ f k 2 (k) q ! , ( 0
) q ! = 1, (n) q ! := (1) q (2) q . . . (n) q , (k) q = 1-q k 1-q and K q = q -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 is a universal R-matrix element of superalgebra U q sl(2|1).
That is, we have the following relations in the h-adic completion of these algebras

(∆ ⊗ Id)(R q ) = R q 13 R q 23 , (Id ⊗∆)(R q ) = R q 13 R q 12 , ∆ op (x)R q = R q ∆(x) for all x ∈ U q sl(2|1). The superalgebra U q sl(2|1) has a Poincaré-Birkhoff- Witt basis {e p 1 e σ 3 e ρ 2 h s 1 1 h s 2 2 f ρ 2 f σ 3 f p 1 , p, p ∈ N, ρ, σ, ρ , σ ∈ {0, 1}, s 1 , s 2 ∈ N}.
Using this basis we can write U q sl(2|1) as a direct sum U q sl(2|1) = U < ⊕ I where U < is a C(q)-module generated by the elements e p 1 e σ 3 e ρ 2 h s 1

1 h s 2 2 f σ 3 f ρ 2 f p 1
for 0 ≤ p, p < ; ρ, σ, ρ , σ ∈ {0, 1}, s 1 , s 2 ∈ N and I is generated by the other monomials. Set p : U q sl(2|1) → U < the projection with kernel I. We define

R < = p ⊗ p(R q ) = p ⊗ Id(R q ) = Id ⊗p(R q ).
The proposition below shows that the "truncated R-matrix" R < satisfies the properties of an R-matrix "modulo truncation".

Proposition 2.4.3. R < satisfies:

1. (p ⊗ p ⊗ p)(∆ ⊗ Id(R < )) = (p ⊗ p ⊗ p)R < 13 R < 23 , 2. (p ⊗ p ⊗ p)(Id ⊗∆(R < )) = (p ⊗ p ⊗ p)R < 13 R < 12 , 3. (p ⊗ p)(R < ∆ op (x)) = (p ⊗ p)(∆(x)R < ) for all x ∈ U q sl(2|1).

Proof. The above relations and p

• p = p give us (p ⊗ p ⊗ p)(∆ ⊗ Id(R q )) = (p ⊗ p ⊗ p)(∆ ⊗ Id)(Id ⊗p(R q )) = (p ⊗ p ⊗ p)(∆ ⊗ Id)(R < ). At the same time (p ⊗ p ⊗ p)(R q 13 R q 23 ) = (p ⊗ p ⊗ p)((p ⊗ Id ⊗ Id)(R q 13 )(Id ⊗p ⊗ Id)(R q 23 )) = (p ⊗ p ⊗ p)(R < 13 R < 23 ). So (p ⊗ p ⊗ p)(∆ ⊗ Id(R < )) = (p ⊗ p ⊗ p)R < 13 R < 23 .
(

Similarly we also have

(p ⊗ p ⊗ p)(Id ⊗∆)(R < ) = (p ⊗ p ⊗ p)(R < 13 R < 12 ).
(2.4.9)

For the third equality, it is enough to check on the generator elements. It is true when

x = h i because ∆(h i ) is symmetric and ∆(h i )(e j ⊗ f j ) = e j ⊗ h i f j + h i e j ⊗ f j = e j ⊗ f j (h i -a ij ) + e j (h i + a ij ) ⊗ f j = e j ⊗ f j (1 ⊗ (h i - a ij ) + (h i + a ij ) ⊗ 1) = (e j ⊗ f j )∆(h i ). For x = e i we have (p ⊗ p)(∆ op (e i )R q ) = (p ⊗ p)(1 ⊗ e i + e i ⊗ k -1 i )R q = (p ⊗ p)((1 ⊗ e i )R q ) + (p ⊗ p)((e i ⊗ k -1 i )R q ) = (p ⊗ p)((1 ⊗ e i )R < ) + (p ⊗ p)((e i ⊗k -1 i )R < ) = (p⊗p)(∆ op (e i )R < ). On the other side (p⊗p)(R q ∆(e i )) = (p ⊗ p)(R < ∆(e i )). So we have (p ⊗ p)(∆ op (e i )R < ) = (p ⊗ p)(R < ∆(e i )).
For x = f i we proceed analogously. So we deduce that

(p ⊗ p)(∆ op (x)R < ) = (p ⊗ p)(R < ∆(x)) ∀x ∈ U q sl(2|1).
Let K be the operator in C H ⊗ C H defined by

K = ξ -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 that is ∀V, W ∈ C H , K V ⊗W = exp (ρ V ⊗W ( 2iπ (-h 1 ⊗ h 2 -h 2 ⊗ h 1 -2h 2 ⊗ h 2 )))
is a linear map on the finite dimensional vector space V ⊗ W . For example, if w ρ,σ,p ⊗ w ρ ,σ ,p ∈ V µ ⊗ V µ , one has

K V ⊗W (w ρ,σ,p ⊗ w ρ ,σ ,p ) = ξ -(µ 1 +ρ-σ-2p)(µ 2 +σ +p )-(µ 2 +σ+p)(µ 1 +ρ -σ -2p )-2(µ 2 +σ+p)(µ 2 +σ +p ) w ρ,σ,p ⊗w ρ ,σ ,p .
We have

∆ ⊗ Id(K) = K 13 K 23 , Id ⊗∆(K) = K 13 K 12 .
(2.4.10)

Let Ř< be the universal truncated quasi R-matrix of U q sl(2|1),

q = e h ∈ C[[h]] given by Ř< = p ⊗ p( Řq ) = Id ⊗p( Řq ) = p ⊗ Id( Řq ), i.e. Ř< = -1 i=0 {1} i e i 1 ⊗ f i 1 (i) q ! 1 j=0 (-{1}) j e j 3 ⊗ f j 3 (j) q ! 1 k=0 (-{1}) k e k 2 ⊗ f k 2 (k) q ! . Set Ř = Ř< | q=ξ , i.e. Ř = -1 i=0 {1} i e i 1 ⊗ f i 1 (i) ξ ! 1 j=0 (-{1}) j e j 3 ⊗ f j 3 (j) ξ ! 1 k=0 (-{1}) k e k 2 ⊗ f k 2 (k) ξ ! ∈ U H ξ sl(2|1)⊗U H ξ sl(2|1).
Theorem 2.4.4. The operator R = ŘK led to a braiding {c V,W } in the category C H where c V,W :

V ⊗ W → W ⊗ V is determined by v ⊗ w → τ (R(v ⊗ w)). Here τ : V ⊗ W → W ⊗ V, v ⊗ w → (-1) deg v deg w w ⊗ v.
Proof. It is sufficient to prove that the operator R satisfies

∆ ⊗ Id(R) = R 13 R 23 , Id ⊗∆(R) = R 13 R 12 , R∆ op (x) = ∆(x)R (2.4.11)
for all x ∈ U H ξ sl(2|1). Let χ q : U q sl(2|1)⊗U q sl(2|1) → U q sl(2|1)⊗U q sl(2|1) be the automorphism determined by x ⊗ y → K q (x ⊗ y)K -1 q , this one induces an automorphism

χ ξ : U H ξ sl(2|1)⊗U H ξ sl(2|1) → U H ξ sl(2|1)⊗U H ξ sl(2|1).
We consider the element Ř< of U q sl(2|1) ⊗ U q sl(2|1), Proposition 2.4.3 implies the relations

∆ ⊗ Id( Ř) = Ř13 (χ ξ ) 13 ( Ř23 ),
(2.4.12)

Id ⊗∆( Ř) = Ř13 (χ ξ ) 13 ( Ř12 ), (2.4.13) Ř (χ ξ ) (∆ op (x)) = ∆(x) Ř for all x ∈ U H ξ sl(2|1).
(2.4.14)

We will prove the equality (2.4.12), and that the other two are similar. From the first equality of the Proposition 2.4.3, we deduce that (∆ ⊗ Id)( Ř< K q ) = Ř< 13 (K q ) 13 Ř< 23 (K q ) 23 . The term in the left of this equality is equal to (∆ ⊗ Id)( Ř< )(∆ ⊗ Id)(K q ) = ∆ ⊗ Id( Ř< )(K q ) 13 (K q ) 23 . The right one is equal to Ř< 13 (K q ) 13 Ř< 23 (K q ) 23 = Ř< 13 (χ q ) 13 ( Ř< 23 )(K q ) 13 (K q ) 23 . Now because K q is invertible, the result is ∆ ⊗ Id( Ř< ) = Ř< 13 (χ q ) 13 ( Ř< 23 ). The element Ř< has no pole when q is a root of unity of order . Hence we can specialize this relation at q = ξ and ∆ ⊗ Id( Ř) = Ř13 (χ ξ ) 13 ( Ř23 ). Finally, as operators on

V 1 ⊗ V 2 ⊗ V 3 in which V 1 , V 2 , V 3 ∈ C H , Equation (2.4.10) implies that ∆ ⊗ Id(R) = (∆ ⊗ Id)( Ř)(∆ ⊗ Id)(K) = Ř13 (χ ξ ) 13 ( Ř23 )K 13 K 23 = Ř13 K 13 Ř23 K -1 13 K 13 K 23 = Ř13 K 13 Ř23 K 23 = R 13 R 23 .
Thus the relations of equation (2.4.11) hold.

The category C H is pivotal and braided with the braiding c V,W :

V ⊗W → W ⊗ V, v ⊗ w → τ • R(v ⊗ w) where V, W ∈ C H .

Ribbon category C H

To prove the next proposition we will use the semi-simplicity of C g (g ∈ G\G s ) which is proven later in Theorem 2.4.14.

Proposition 2.4.5. The family of isomorphisms θ

V : V → V determined by θ V = (Id V ⊗ ←- ev V )(c V,V ⊗ Id V * )(Id V ⊗ -→ coev V ), V ∈ C H is a twist. That is θ V = θ V ∀V ∈ C H where θ V = ( -→ ev V ⊗ Id V )(Id V * ⊗c V,V )( ←- coev V ⊗ Id V ). Proof. Firstly, if V is a typical module of highest weight µ = (µ 1 , µ 2 ), V ∈ C H g , g ∈ G\G s , we have θ V = ( -→ ev V ⊗ Id V )(Id V * ⊗c V,V )( ←- coev V ⊗ Id V ) = X 1 X 2 X 3 . We use the vector of lowest weight (µ 1 -2 + 2, µ 2 + ) of V, w 1,1, -1 := w ∞ , to calculate. X 3 (w ∞ ) = ρ,σ,p (-1) ρ+σ w * ρ,σ,p ⊗ φ -1 0 w ρ,σ,p ⊗ w ∞ = ρ,σ,p (-1) ρ+σ ξ µ 1 +2µ 2 +2σ+2p w * ρ,σ,p ⊗ w ρ,σ,p ⊗ w ∞ . X 2 X 3 (w ∞ ) = ρ,σ,p (-1) ρ+σ ξ µ 1 +2µ 2 +2σ+2p w * ρ,σ,p ⊗ (τ • R)(w ρ,σ,p ⊗ w ∞ ). K(w ρ,σ,p ⊗ w ∞ ) = ξ -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 w ρ,σ,p ⊗ w ∞ = ξ -µ 1 (µ 2 +σ+p+ )-µ 2 (µ 1 +2µ 2 +σ+ρ+2)-2(σ+p) w ρ,σ,p ⊗ w ∞ . X 2 X 3 (w ∞ ) = ρ,σ,p (-1) ρ+σ ξ µ 1 +2µ 2 ξ -µ 1 (µ 2 +σ+p+ )-µ 2 (µ 1 +2µ 2 +σ+ρ+2) w * ρ,σ,p ⊗ w ∞ ⊗ w ρ,σ,p = ρ,σ,p (-1) ρ+σ ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +2µ 2 +σ+ρ) w * ρ,σ,p ⊗ w ∞ ⊗ w ρ,σ,p . So X 1 X 2 X 3 (w ∞ ) = ρ,σ,p (-1) ρ+σ ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +2µ 2 +σ+ρ) w * ρ,σ,p (w ∞ ) ⊗ w ρ,σ,p = ξ -µ 1 (µ 2 + )-µ 2 (µ 1 +2µ 2 +2) w ∞ .
Secondly, we have

θ V = (Id V ⊗ ←- ev V )(c V,V ⊗ Id V * )(Id V ⊗ -→ coev V ) = Y 1 Y 2 Y 3 . Y 3 (w 0,0,0 ) = ρ,σ,p w 0,0,0 ⊗ w ρ,σ,p ⊗ w * ρ,σ,p , Y 2 Y 3 (w 0,0,0 ) = ρ,σ,p (τ • R)(w 0,0,0 ⊗ w ρ,σ,p ) ⊗ w * ρ,σ,p where K(w 0,0,0 ⊗ w ρ,σ,p ) = ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w 0,0,0 ⊗ w ρ,σ,p and R(w 0,0,0 ⊗ w ρ,σ,p ) = ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w 0,0,0 ⊗ w ρ,σ,p . Y 2 Y 3 (w 0,0,0 ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w ρ,σ,p ⊗w 0,0,0 ⊗w * ρ,σ,p . Y 1 Y 2 Y 3 (w 0,0,0 ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w ρ,σ,p ⊗ w * ρ,σ,p ((-1) ρ+σ φ 0 w 0,0,0 ) = ρ,σ,p (-1) ρ+σ ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w ρ,σ,p ⊗ w * ρ,σ,p (ξ -µ 1 -2µ 2 w 0,0,0 ) = ξ -2µ 1 µ 2 -2µ 2 2 -2µ 2 -µ 1 w 0,0,0 = ξ -µ 1 (µ 2 + )-µ 2 (µ 1 +2µ 2 +2) w 0,0,0 . We can deduce that θ V = θ V for every typical module V with highest weight µ = (µ 1 , µ 2 ), V ∈ C H g , g ∈ G\G s .
Note that the calculation does not change if we reverse the parity of vectors. So we have the affirmation for a semisimple module in degree g

∈ G\G s . Let a module W ∈ C H g , g ∈ G. By Theorem 2.4.14 it exists h ∈ G such that C H h , C H g+h are semi-simple. For a module V ∈ C H h we have W ⊗ V ∈ C H g+h is semi-simple. Because θ W ⊗V = (θ W ⊗ θ V )c V,W c W,V = θ W ⊗V = (θ W ⊗ θ V )c V,W c W,V and θ V = θ V , we deduce that θ W = θ W ∀W ∈ C H , i.e. the family θ V is a twist. Lemma 2.4.6. Let µ = (µ 1 , µ 2 ) ∈ C × C, then the value of the twist θ Vµ on a simple module V µ with highest weight µ is ξ -µ 1 -2µ 2 (1+µ 1 +µ 2 ) Id Vµ . That is, θ Vµ = ξ -µ 1 -2µ 2 (1+µ 1 +µ 2 ) Id Vµ = -ξ -2(α 2 2 +α 1 α 2 ) Id Vµ where α = (α 1 , α 2 ) = (µ 1 -+ 1, µ 2 + 2 ).
Proof. By the proof of Proposition 2.4.5,

θ Vµ = ξ -µ 1 -2µ 2 (1+µ 1 +µ 2 ) Id Vµ .
The category C H is a braided pivotal category with a twist, i.e. C H is a ribbon category.

Let T be the ribbon category of C H -colored oriented ribbon graphs in the sense of Turaev [START_REF] Turaev | Quantum invariants of knots and 3-manifolds[END_REF]. The set of morphisms T ((

(V 1 , ±), ..., (V n , ±)), ((W 1 , ±), ..., (W n , ±))
) is a space of linear combinations of C -colored ribbon graphs. The ribbon Reshetikhin-Turaev functor F : T → C H is defined by the Penrose graphical calculus.

Definition 2.4.7. If T ∈ T ((V µ , +), (V µ , +)) where V µ is a simple weight module of U H ξ sl(2|1), then F (T ) = x. Id Vµ ∈ End U H ξ sl(2|1) (V µ ) for x ∈ C. We define the bracket of T by T = x. For example, if V µ , V µ ∈ C H , we define S (V µ , V µ ) = Vµ V µ .
We write S (µ, µ ) for S (V µ , V µ ).

Another example is the bracket of the twist

Vµ = -ξ -2(α 2 2 +α 1 α 2 ) , (α 1 , α 2 ) = (µ 1 -+ 1, µ 2 + 2 ). Proposition 2.4.8. Let V = V µ be a typical module, V = V µ be a simple module, then S (µ, µ ) = ξ -4α 2 α 2 -2(α 2 α 1 +α 1 α 2 ) { α 1 }{α 2 }{α 2 + α 1 } {α 1 } where α = (α 1 , α 2 ) = (µ 1 -+ 1, µ 2 + 2 ), α = (α 1 , α 2 ) = (µ 1 -+ 1, µ 2 + 2 ). Proof. Let S = S(µ, µ ) ∈ End C (V µ 1 ,µ 2 ) be the endomorphism determined by the diagram Vµ V µ . We have S (µ, µ ) Id V µ = (Id V ⊗ ←- ev V )(c V,V ⊗ Id V * )(c V ,V ⊗ Id V * )(Id V ⊗ -→ coev V ) = X 1 X 2 X 3 X 4 .
The definition gives us X 4 (w 0,0,0 ) = ρ,σ,p w 0,0,0 ⊗ w ρ,σ,p ⊗ w * ρ,σ,p and

X 3 X 4 (w 0,0,0 ) = ρ,σ,p (τ • R)(w 0,0,0 ⊗ w ρ,σ,p ) ⊗ w * ρ,σ,p . K(w 0,0,0 ⊗ w ρ,σ,p ) = ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w 0,0,0 ⊗ w ρ,σ,p . R(w 0,0,0 ⊗ w ρ,σ,p ) = ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w 0,0,0 ⊗ w ρ,σ,p . So X 3 X 4 (w 0,0,0 ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) . w ρ,σ,p ⊗ w 0,0,0 ⊗ w * ρ,σ,p . X 2 X 3 X 4 (w 0,0,0 ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) . (τ • R)(w ρ,σ,p ⊗ w 0,0,0 ) ⊗ w * ρ,σ,p . Furthermore, the element ( Ř -1)(w ρ,σ,p ⊗ w 0,0,0 ) ∈ V µ 1 ,µ 2 ⊗ V µ 1 ,µ 2 is a sum of vectors of the form v ⊗ w
where w is a weight vector of V µ 1 ,µ 2 and v is a weight vector of V µ 1 ,µ 2 which has a higher weight than w ρ,σ,p .

X 2 X 3 X 4 (w 0,0,0 ) = ρ,σ,p (ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) . w 0,0,0 ⊗ w ρ,σ,p ⊗ w * ρ,σ,p + k w k ⊗ v k ⊗ z k ). X 1 X 2 X 3 X 4 (w 0,0,0 ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p) w 0,0,0 ⊗ (-1) ρ+σ w * ρ,σ,p (φ 0 w ρ,σ,p ) = ρ,σ,p ξ -µ 1 (µ 2 +σ+p)-µ 2 (µ 1 +ρ-σ-2p)-2µ 2 (µ 2 +σ+p)-µ 1 -2(µ 2 +σ+p) w 0,0,0 = ξ -(2µ 2 +µ 1 +1)(2µ 2 +µ 1 +1)+(µ 1 +1)(µ 1 +1)-(µ 1 +µ 1 +1) { (µ 1 + 1)}{µ 2 }{µ 2 + µ 1 + 1} {µ 1 + 1} w 0,0,0 = ξ -4α 2 α 2 -2(α 2 α 1 +α 1 α 2 ) { α 1 }{α 2 }{α 2 + α 1 } {α 1 } w 0,0,0 .
By the definition S(µ, µ )(w 0,0,0 ) = S (µ, µ )w 0,0,0 , we deduce the proposition.

Definition 2.4.9.

If µ = (µ 1 , µ 2 ) ∈ C\ 1 2 Z ∪ (-1 + 2 Z) × C\ 2 Z and µ 2 + µ 1 + 1 ∈ C\ 2 Z, we define d(µ) = {µ 1 + 1} { µ 1 }{µ 2 }{µ 2 + µ 1 + 1} = {α 1 } { α 1 }{α 2 }{α 1 + α 2 }
, so there is a symmetry

d(µ )S (µ, µ ) = d(µ)S (µ , µ).

Semi-simplicity of category C H

Remember that

G = C/Z × C/Z and G s = {g ∈ G such that ∃ V ∈ C H
g simple and atypical}. Recall that we denote with a bar a module with opposite parity. Then if

V ∈ C H , V V ⊗ I. Lemma 2.4.10. If C H µ is semi-simple, then a module of C H µ is

determined up to an isomorphism and parity by its character: let

V = V 1 ⊕ ... ⊕ V m
be a decomposition of V into simple modules and let V be a module with the same character then

V V 1 ⊗ ε 1 ⊕ ... ⊕ V m ⊗ ε m where ε i ∈ {I, I} for 1 ≤ i ≤ m.
The above lemma and the character of representation

V µ 1 ,µ 2 ⊗ V µ 1 ,µ 2 gives us the following theorem. Theorem 2.4.11. Let V µ , V µ be two typical modules. If µ + µ / ∈ G s then V µ 1 ,µ 2 ⊗ V µ 1 ,µ 2 = ⊕ -1 k=0 (V µ 1 +µ 1 -2k,µ 2 +µ 2 +k ⊕ V µ 1 +µ 1 -2k+1,µ 2 +µ 2 +k ⊕ V µ 1 +µ 1 -2k,µ 2 +µ 2 +k+1 ⊕ V µ 1 +µ 1 -2k-1,µ 2 +µ 2 +k+1 ) (2.4.15)
where V is the module V with opposite parity.

Proof. According to the formula (2.4.7), we have

χ V sl(2|1) µ 1 ,µ 2 ⊗V sl(2|1) µ 1 ,µ 2 = χ V sl(2|1) µ 1 ,µ 2 χ V sl(2|1) µ 1 ,µ 2 = X µ 1 +µ 1 1 X µ 2 +µ 2 2 1 -x 1 -x (1 + X 1 )(1 + X 1 x) -1 k=0 (X -2 1 X 2 ) k (1 + X 1 + X 2 + X -1 1 X 2 ) = 1 -x 1 -x (1 + X 1 )(1 + X 1 x) -1 k=0 X µ 1 +µ 1 -2k 1 X µ 2 +µ 2 +k 2 + X µ 1 +µ 1 -2k+1 1 X µ 2 +µ 2 +k 2 + X µ 1 +µ 1 -2k 1 X µ 2 +µ 2 +k+1 2 + X µ 1 +µ 1 -2k-1 1 X µ 2 +µ 2 +k+1 2 .
The analysis of parity of highest weight vectors allows to conclude.

Remark 2.4.12. Not all terms in the decomposition of the above theorem are distinct.

We defined a graduation C H = µ∈G C H µ . Let Proj be the subcategory of C H containing projective modules, Proj is an ideal (see [START_REF] Geer | Generalized trace and modified dimension functions on ribbon categories[END_REF]), i.e. Proj is closed under retracts and absorbent for the tensor product. We have the following proposition. Proposition 2.4.13. For µ ∈ G, the three conditions below are equivalent 1. All the simple U ξ sl(2|1)-modules of C µ are projective.

The category

C µ is semi-simple. 3. The C-superalgebra of finite dimension U/(k 1 -ξ µ 1 , k 2 -ξ µ 2 ) is semi- simple where U = U ξ sl(2|1)/(e 1 , f 1 ).
Proof. The equivalence is classic knowing that C µ is also a category of the

U/(k 1 -ξ µ 1 , k 2 -ξ µ 2 )-modules.
Theorem 2.4.14.

1. If µ ∈ G\G s then C H µ is semi-simple. 2. A typical U H ξ sl(2|1)-module is projective. We select and fix a µ ∈ G\G s , denote µ i = (µ 1 + i 1 , µ 2 + i 2 ) ∈ µ, i 1 , i 2 = 0, 1, ..., -1, that is µ i ∈ {(µ 1 + i 1 , µ 2 + i 2 ) : i 1 , i 2 = 0, 1, ..., -1}.
We have the two following lemmas. Lemma 2.4.15. For all µ i , µ j ∈ µ :

µ i = µ j there exists z ij ∈ Z such that χ µ i (z ij ) = χ µ j (z ij ) where χ µ i (z ij ) ∈ C is defined by ρ µ i (z ij ) = χ µ i (z ij ) Id Vµ i . Proof. We consider µ = (µ 1 , µ 2 ), µ = (µ 1 + k, µ 2 + m) k, m = 0, 1, ..., -1.
We suppose that ∀ z ∈ Z : χ µ (z) = χ µ (z). Consider the central elements C p where p ∈ Z (see [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]). We have

χ µ (C p ) = (ξ -ξ -1 ) 2 ξ (2p-1)(µ 1 +2µ 2 ) [µ 2 ][µ 2 + µ 1 + 1], χ µ (C p ) = (ξ -ξ -1 ) 2 ξ (2p-1)(µ 1 +2µ 2 +k+2m) [µ 2 + m][µ 2 + µ 1 + k + m + 1]. Because χ µ (C p ) = χ µ (C p ) and [µ 2 ][µ 2 + µ 1 + 1] = 0, we deduce that    χµ(C p+1 ) χµ(Cp) = χ µ (C p+1 ) χ µ (Cp) χ µ (C p ) = χ µ (C p ).

This is equivalent to

   ξ 2(µ 1 +2µ 2 ) = ξ 2(µ 1 +2µ 2 +k+2m) ξ (2p-1)(µ 1 +2µ 2 ) [µ 2 ][µ 2 + µ 1 + 1] = ξ (2p-1)(µ 1 +2µ 2 +k+2m) [µ 2 + m][µ 2 + µ 1 + k + m + 1], which implies 2(k + 2m) = 0 (modulo Z) (2.4.16)
and

[µ 2 ][µ 2 + µ 1 + 1] = ξ k+2m [µ 2 + m][µ 2 + µ 1 + k + m + 1]. (2.4.17)
Because odd, Equation (2.4.16) implies k + 2m = 0 (modulo Z) ⇔ k+m = -m (modulo Z). On the other hand, Equation (2.4.17

) is equivalent to [a][b] = [a + m][b -m] ⇔ -[a -b + m][m] = 0 ⇔ [-µ 1 -1 + m][m] = 0 ⇒ m = 0 where a = µ 2 , b = µ 1 + µ 2 + 1. Because m = 0, we have k = 0 (modulo Z) ⇒ k = 0.
Lemma 2.4.16. Let V be a vector space over C, I be a finite set and consider a family of C-linear functions

a i : V → C, i ∈ I. If for all i = j ∃ u ij ∈ V such that a i (u ij ) = a j (u ij ), then it exists u 0 ∈ V such that ∀ i = j a i (u 0 ) = a j (u 0 ). Proof. We set u = i =j x ij u ij ∈ V with x ij ∈ C, i, j ∈ I. We denote x = (x ij ) ∈ C N . We consider the set X = {x ∈ C N ∃i = j a i (u) = a j (u)} = {x ∈ C N : i =j (a i (u ij ) -a j (u ij ))x ij = 0}
, this is a finite reunion of hyperplanes of C N . This proves that ∃x / ∈ X and this x does not have the above property. That is, it exists u 0 ∈ V such that a i (u 0 ) = a j (u 0 ) for all i = j. Now we introduce a new basis of module V µ . This basis diagonalize the operator Ω in the proof of Theorem 2.4.14. We set

w ρ,σ,p =        w ρ,σ,p if ρ = σ = 0, 1 f p 1 w 1,0,0 if ρ = 1, σ = 0 e ( -1)-p 1 w 0,1,r-1 if ρ = 0, σ = 1
where p = 0, ..., -1. For the basis {w ρ,σ,p } we have the actions

k 1 w ρ,σ,p = ξ µ 1 +ρ-σ-2p w ρ,σ,p , k 2 w ρ,σ,p = ξ µ 2 +σ+p w ρ,σ,p , e 1 w 1,0,p = [p][µ 1 + 2 -p]w 1,0,p-1 , f 1 w 1,0,p = w 1,0,p+1 , e 1 w 0,1,p = w 0,1,p-1 , f 1 w 0,1,p = [p + 1][µ 1 -(p + 1)]w 0,1,p+1 .
Proof of Theorem 2.4.14. We begin to show that C µ is semi-simple. We set

A = U/(k 1 -ξ µ 1 , k 2 -ξ µ 2 ). The density theorem implies that the appli- cation ρ : A → µ i End(V µ i ) ∼ = By Lemma 2.4.15 and 2.4.16, it exists an element z ∈ Z such that ∀ µ i = µ j χ µ i (z) = χ µ j (z) and we set z i = χ µ i (z) i = 1, ..., 2 and we introduce the ideal J = 2 i=1 (z -z i )A.
Firstly, we consider the representation ρ : A/(z -z i ) → End C (V µ i ). We will prove that ρ is a surjection. We have End C (V µ i ) ∼ = M 4 (C). We consider the elements Ω =

k 1 ξ+k -1 1 ξ -1 {1} 2 + f 1 e 1 = k 1 ξ -1 +k -1 1 ξ {1} 2 + e 1 f 1 , c = k 1 k 2 2 , k 1 in U ξ sl(2|1).
The actions of these elements on the basis w ρ,σ,p are defined by Ωw 0,0,p = (ξ

µ 1 +1 + ξ -µ 1 -1 )w 0,0,p , Ωw 1,1,p = (ξ µ 1 +1 + ξ -µ 1 -1 )w 1,1,p , Ωw 0,1,p = ξ µ 1 + ξ -µ 1 {1} 2 w 0,1,p , Ωw 1,0,p = ξ µ 1 +2 + ξ -µ 1 -2 {1} 2 w 1,0,p , cw ρ,σ,p = ξ µ 1 +2µ 2 +ρ+σ w ρ,σ,p , k 1 w ρ,σ,p = ξ µ 1 +ρ-σ-2p w ρ,σ,p .
We now check that for all

w ρ,σ,m = w ρ ,σ ,j ∃ u ∈ {Ω, c, k 1 } such that χ ρ,σ,m µ i (u) = χ ρ ,σ ,j µ i (u) where ρ(u)w ρ,σ,m = χ ρ,σ,m µ i (u)w ρ,σ,m for ρ, σ, ρ , σ ∈ {0, 1}, m, j ∈ {0, ..., -1}. Indeed, if ρ + σ = ρ + σ then we select u = c and we have cw ρ,σ,m = cw ρ ,σ ,j . If ρ + σ = ρ + σ then we consider two cases: if (ρ, σ) = (ρ , σ ) we select u = Ω and Ωw ρ,σ,m = Ωw ρ ,σ ,j ; if (ρ, σ) = (ρ , σ ) we select u = k 1 and we have k 1 w ρ,σ,m = k 1 w ρ ,σ ,j because m = j.
By Lemma 2.4.16 it exists a vector u 0 ∈ C Ω, c, k 1 -space generated by the elements Ω, c, k 1 such that χ ρ,σ,m µ i (u 0 ) = χ ρ ,σ ,j µ i (u 0 ) for all w ρ,σ,m = w ρ ,σ ,j . The matrix B determined by the application ρ(u 0 ) is a diagonal matrix which has 4 different eigenvalues. The image of the projection on the ith eigenspace of B is the matrix E ii , i = 1, ..., 4 . Hence the matrix E ii is in the image of ρ.

For i ∈ {1, ..., 2 }, j ∈ {1, ..., 4 } we have ρ(A/(z -z i ))(v j ) ⊂ V µ i (here we denote v j the j-th vector of the basis) and

V µ i is simple. Thus we deduce ρ(A/(z -z i ))(v j ) = V µ i . This proves that it exists a 0 ∈ A/(z -z i ) such that ρ(a 0 )(v j ) = v n ∀n ∈ {1, ..., 4 }.
The endomorphism ρ(a 0 ) determines the matrix (ρ(a 0 )) where ρ(a 0 ) jn = 1. The matrix E jn is equal to E jj ρ(a 0 ) jn E nn , i.e. the matrix E jn is the image of an element in A/(z -z i ). So the application ρ is a surjection. This implies that the application

2 i=1 A/(z -z i ) → 2 i=1 M 4 (C) is surjective. Secondly, the composition 2 i=1 A/(z -z i ) → A/J → 2 i=1 A/(z -z i ) is the identity. Thus, the application A/J → 2 i=1 A/(z -z i ) is surjective. We deduce a series of surjections A A/J 2 i=1 A/(z -z i ) 2 i=1 M 4 (C), this sequence determines the surjection A 2 i=1 M 4 (C).
Furthermore, the two algebras A and 2 i=1 M 4 (C) have the same dimension 16 4 . This implies that this surjection is an isomorphism. This demonstrates that A is semi-simple. The category C µ is also semi-simple. Now we prove that C H µ is semi-simple. Let V H be a module in C H µ . Set W = Ker e 1 ∩Ker e 2 ∩Ker e 3 , it is a vector space of the highest weight vectors (the weights for (h 1 , h 2 )). We call {v j } n j=1 a basis of weight vectors of W , we have

h i v j = µ i j v j , i = 1, 2. So each v j generates a U H ξ sl(2|1)-module V j , V j = U H ξ sl(2|1).v j = U ξ sl(2|1).v j = U -.v j where U -= Alg f 1 , f 2 , f 3 ⊂ U ξ sl(2|1) and dim(U -) = 4 . Thus dim(V j ) ≤
4 and V j is simple (because there is no module in C H µ of dimension strictly between 0 and 4 ).

Set

V = n i=1 V i ⊂ V H . We can write V H = V ⊕ V as a U ξ sl(2|1)- module. However W ⊂ V which implies V = 0 (because there is no highest weight vector in V ) and V H = V = n i=1 V i . Because the V i are simple, so V H = i∈I V i where I ⊂ {1, ..., n}. Thus V H is semi-simple. For the second assertion (2), if V ∈ C H µ and C H µ is semi-simple, then V is projective. If not, (2) follows from S (V µ , V ) = 0 where V µ is any projective typical module which implies that V is a direct factor of V µ ⊗ V ⊗ V * µ ∈ Proj.
This implies that V is a projective module.

Modified traces on projective modules

In this section we recall the definition of an ambidextrous module presented by N. Geer, B. Patureau-Mirand and V. Turaev in [START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF] and of a modified trace on an ideal in a category introduced by N. Geer, J. Kujawa and B. Patureau-Mirand in [START_REF] Geer | Generalized trace and modified dimension functions on ribbon categories[END_REF]. We prove there exists a modified trace on the ideal of projective modules of the category C H . The modified trace allows us to construct an invariant of embedded graphs in Theorem 2.5.5.

Ambidextrous modules

For each object V of the category C and any endomorphism

f of V ⊗ V set ptr R (f ) = (Id V ⊗ ←- ev V ) • (f ⊗ Id V * ) • (Id V ⊗ -→ coev V ) ∈ End(V ), ptr L (f ) = ( -→ ev V ⊗ Id V ) • (Id V * ⊗f ) • ( ←- coev V ⊗ Id V ) ∈ End(V ).
In the ribbon category C H of nilpotent weight U H ξ sl(2|1)-modules, we say that a module V is ambidextrous if V simple and ptr L (f ) = ptr R (f ) for all f ∈ End(V ⊗ V ) (see [START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF]). Proof. We will prove this theorem in two steps:

Step 1. Proving the existence of two nonzero U H ξ sl(2|1)-invariant vectors x -w + and x + w -.

Step 2. Applying Theorem 3.1.3 [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF] gives us the affirmation that V µ is ambidextrous.

Call

v + , v + the highest weight vectors of V µ , V * µ and v -, v -the lowest weight vectors of V µ , V * µ . Set x -= f 2 f 3 f -1 1 , x + = e 2 e 3 e -1 1 , w + = v + ⊗ v + , w -= v -⊗ v -.
We will prove that the two vectors x -w + and x + w -are U H ξ sl(2|1)-invariant.

We consider the actions of generator elements e i , h i , f i on x -w + . The highest weight vector (resp. lowest) of V µ is v + = w 0,0,0 (resp. v -= w 1,1, -1 ). The highest weight vector (resp. lowest) of

V * µ is v + = w * 1,1, -1 (resp. v -= w * 0,0,0 ). The weight of vector w + = v + ⊗ v + is equal to the sum of the weights of v + and v + . That is weight(w + ) = (µ 1 , µ 2 ) + (-µ 1 + 2 -2, -µ 2 - ) = (2 -2, -). Furthermore, weight(x -w + ) = weight(f 2 f 3 f -1 1 w + ) = weight(f 2 f 1 f 2 f -1 1 w + ) = -weight(e 1 )-2weight(e 2 )+weight(w + ) = -(2, -1)- 2(-1, 0) + (2 -2, -) = (0, 0). It implies that h i x -w + = 0.
We also have the relations below between the generator elements in U H ξ sl(2|1) (see (B1) [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]):

f 1 f ρ 2 f σ 3 f p 1 = ξ ρ-σ f ρ 2 f σ 3 f p+1 1 -ρ(1 -σ)ξ -ρ f ρ-1 2 f σ+1 3 f p 1 , f 2 f ρ 2 f σ 3 f p 1 = (1 -ρ)f ρ+1 2 f σ 3 f p+1 1 , [e 1 , f ρ 2 f σ 3 f p 1 ] = σ(1 -ρ)(-1) σ f ρ+1 2 f σ-1 3 f p 1 ξ h 1 -2p+1 + [p]f ρ 2 f σ 3 f p-1 1 [h 1 -p + 1], e 2 f ρ 2 f σ 3 f p 1 -(-1) ρ+σ f ρ 2 f σ 3 f p 1 e 2 = ρf ρ-1 2 f σ 3 f p 1 [h 2 + p + σ] + σ(-1) ρ f ρ 2 f σ-1 3 f p+1 1 ξ -h 2 -p where (p, ρ, σ) ∈ N × {0, 1} × {0, 1}. With the above relations, it is easy to check f i x -w + = 0. The fourth relation above gives us e 2 f 2 f 3 f -1 1 -f 2 f 3 f -1 1 e 2 = f 3 f -1 1 [h 2 + ]. Because e 2 (v + ⊗ v + ) = 0 and [h 2 + ](v + ⊗ v + ) = 0, we deduce e 2 x -w + = 0. The third relation gives [e 1 , f 2 f 3 f -1 1 ] = [ -1]f 2 f 3 f -2 1 [h 1 -+2]. Because e 1 (v + ⊗ v + ) = 0 and [h 1 -+ 2](v + ⊗ v + ) = 0, we deduce e 1 x -w + = 0.
Consequently, we conclude that x -w + is an U H ξ sl(2|1)-invariant vector. The demonstration that the vector x + w -is U H ξ sl(2|1)-invariant is analogous using the relations obtained by applying the automorphism ω of superalgebra U H ξ sl(2|1) where ω(e i ) = (-1)

deg e i f i , ω(f i ) = (-1) deg f i e i , ω(k i ) = k -1 i , ω(h i ) = -h i , i = 1, 2.
Furthermore ∆x -= x -⊗ 1+ a sum of tensor products of two elements of U H ξ sl(2|1) with negative weight. Thus ∆x -w + contains the nonzero vector

x -v + ⊗ v + = f 2 f 3 f -1 1 v + ⊗ v + = w 1,1, -1 ⊗ v + .
We conclude that the vector x -w + is nonzero. Similarly, the vector x + w -is nonzero.

For step 2, we use the following results:

The decomposition of the tensor product

V ⊗ V * is a direct sum of inde- composable modules V ⊗ V * = P 1 ⊕ ... ⊕ P m . The set of invariant vectors w ∈ V ⊗ V * is in bijection with -→ coev V (C) because Hom C (C, V ⊗ V * ) ∼ = Hom C (V, V ) ∼ = C.
The vector w + (resp. w -) is the highest weight vector (resp. lowest weight vector) of V ⊗ V * . Then there exists a unique integer k (resp. l) such that w + ∈ P k (resp. w -∈ P l ). The weight of w + (resp. [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF], it gives us the affirmation that V µ ambidextrous. Remark 2.5.2. All typical modules are projective and ambidextrous.

w -) is λ + = (2 -2, -) (resp. λ -= (-2 + 2, )). Because λ -= -λ + and (V ⊗ V * ) * (V ⊗ V * ), this implies P * k P l . In addition, -→ coev V (1) ∈ P l , -→ coev V (1) ∈ P k because x + P l ⊂ P l , x -P k ⊂ P k , then P k = P l . That is P k = P * k . By Theorem 3.1.3

Modified traces on the projective modules

Definition 2.5.3. Let I be an ideal of C (see [START_REF] Geer | Generalized trace and modified dimension functions on ribbon categories[END_REF]). The family of linear applications t = (t

V : End C (V ) → k) V ∈I is a trace (modified trace) on I if it satisfies: ∀U, V ∈ I, ∀W ∈ C , ∀f ∈ Hom C (U, V ), ∀g ∈ Hom C (V, U ), t V (f • g) = t U (g • f ) ∀f ∈ End C (V ⊗ W ), t V ⊗W (f ) = t V (ptr R (f )).
We also have

∀f ∈ End C (W ⊗ V ), t W ⊗V (f ) = t V (ptr L (f )).
Given V as a typical module. The module V is ambidextrous and projective. This implies that the ideal generated by this module is I V = Proj (see [START_REF] Geer | Generalized trace and modified dimension functions on ribbon categories[END_REF]). Hence the modified trace is also defined on non simple projective modules: Theorem 2.5.4. There exists a unique modified trace t = {t P } P ∈Proj on the ideal Proj of projective modules of C H , t P : End(P ) → C, P ∈ Proj.

If P = V µ is a typical module, then t Vµ (f ) = f d(µ), f ∈ End(V µ ), d(µ) = t Vµ (Id Vµ
) is determined by Definition 2.4.9.

Invariants of embedded graphs

Recall that C H is the C-linear ribbon category of nilpotent weight modules over U H ξ sl(2|1), Proj is the ideal of projective modules and t is a trace on Proj.

We call G the set of C H -colored closed ribbon graphs, that are the C Hcolored ribbon graphs in S 3 . We have G ∼ = End T (∅).

We use the concept of a cutting presentation of C H -colored closed ribbon graph: If a diagram T represents a C H -colored ribbon graph which is an endomorphism of T , its lower and upper parts are formed by the same sequences of k vertical colored strands. It is then possible, as for a braid of k strands, to consider the closure T obtained by joining its k top vertices to its k bottom vertices by k parallel strands. This construction is actually the categorical trace in T : we have T = tr T (T ) ∈ End T (∅). We say that T is a cutting presentation with k strands of the closed graph T and that T is the closure of T (see [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]).

A closed graph T of T is said to be C H -colored admissible if there is at least one strand of T colored by P ∈ Proj. Let G a be the set of isotopy classes of C H -colored admissible ribbon graphs.

From the trace t on Proj we have the theorem below.

Theorem 2.5.5. The application

F : G a → C T → t P (F (T ))
is well defined. Here, P ∈ Proj, T ∈ End T ((P, +)) is a cutting presentation with one strand of T . That is to say the complex number t P (F (T )) does not depend on the choice of T but only of the isotopy class of the C H -colored graph T .

Proof. First, we select an edge of T and cut, we have the graph T . Then, we select and cut a second edge of T , we have the graph T . By cutting T in both these places, one obtains a graph T 2 ∈ End Ga ((P, +), (P , +)) which is a presentation with two strands of T and such that

T = T 2 , T = T 2 .
Finally we use the properties of the compatibility of trace t:

t P (F (T )) = t P (ptr R (F (T 2 ))) = t P ⊗P (F (T 2 )) = t P (ptr L (F (T 2 ))) = t P (F (T )).
Remark 2.5.6. In the case P = V µ typical, we have

F   T   = d(µ) T .
The affirmation of the above theorem gives us a link invariant in the following corollary.

Corollary 2.5.7. Let L be an oriented link with n ordered components then the application F : {admissible C 2 -coloring of L} → C determines a meromorphic function f L : C 2n → C of the 2n complex parameters defining the coloring.

Invariant of 3-manifolds

In the article [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF] the authors constructed C -decorated 3-manifold invariants where C is a ribbon category. In the previous section, it was proven that C H is a ribbon category, this suggests we construct an invariant of C Hdecorated 3-manifolds. We recall some concepts, definitions and results from [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF].

Relative G-(pre)modular categories

Let C be a k-linear ribbon category where k is a field. A set of objects of C is said to be commutative if for any pair {V, W } of these objects, we have

c V,W • c W,V = Id W ⊗V and θ V = Id V . Let (Z, +) be a commutative group. A realization of Z in C is a commutative set of objects {ε t } t∈Z such that ε 0 = I, qdim(ε t ) = 1 and ε t ⊗ ε t = ε t+t for all t, t ∈ Z.
A realization of Z in C induces an action of Z on isomorphism classes of objects of C by (t, V ) → ε t ⊗ V . We say that {ε t } t∈Z is a free realization of Z in C if this action is free. This means that ∀t ∈ Z\{0} and for any simple object V ∈ C , V ⊗ ε t V . We call simple Z-orbit the reunion of isomorphism classes of an orbit for this action.

F     Ω µ V     = ∆ -Id V , F     Ω µ V     = ∆ + Id V Figure 2.1 -V ∈ C g and
1 ∈ G is the unit), 3. there is a Z-bilinear application G × Z → k × , (g, t) → g •t such that ∀V ∈ C g , ∀t ∈ Z, c V,ε t • c ε t ,V = g •t Id ε t ⊗V , 4.
there exists X ⊂ G such that X -1 = X and G cannot be covered by a finite number of translated copies of X , in other words ∀g 1 , ..., g n ∈ G, ∪ n i=1 (g i X ) = G, 5. for all g ∈ G \ X , the category C g is semi-simple and its simple objects are in the reunion of a finite number of simple Z-orbits, [START_REF] Concini | Representations of quantum groups at roots of 1[END_REF]. there exists a nonzero trace t on ideal Proj of projective objects of C and d is the associated modified dimension, [START_REF] Concini | Quantum coadjoint action[END_REF]. there exists an element g ∈ G \ X and an object V ∈ C g such that the scalar ∆ + defined in Figure 2.1 is nonzero; similarly, there exists an element g ∈ G \ X and an object V ∈ C g such that the scalar ∆ - defined in Figure 2.1 is nonzero,

8. the morphism S(U, V ) = F (H(U, V )) = 0 ∈ End C (V ), for all simple objects U, V ∈ Proj, where H(U, V ) = U V ∈ End C ((V, +)).
The category C H of U H ξ sl(2|1)-modules is G-modular relative to X . Indeed, we have C H being G-graded by G = C/Z × C/Z. We set Z = Z × Z and {ε n } n∈Z the set of simple highest weight modules n = (n 1 , n 2 ), i.e. ε n is a U H ξ sl(2|1)-module of dimension 1 (with the basis {w}) determined by 1) and (2) of Definition 2.6.1 are satisfied. We consider a typical module V µ . We have

h 1 w = n 1 w, h 2 w = n 2 w, e i w = f i w = 0. Because c ε m ,ε n = τ and θ ε n = Id, the two conditions (
c ε n ,Vµ (w ⊗ w ρ,σ,p ) = τ • R(w ⊗ w ρ,σ,p ) = ξ -n 1 µ 2 -n 2 µ 1 -2n 2 µ 2 w ρ,σ,p ⊗ w. Next c Vµ,ε n • c ε n ,Vµ (w ⊗ w ρ,σ,p ) = c Vµ,ε n (ξ -n 1 µ 2 -n 2 µ 1 -2n 2 µ 2 w ρ,σ,p ⊗ w) = ξ -2n 1 µ 2 -2n 2 µ 1 -4n 2 µ 2 w ⊗ w ρ,σ,p = ξ -2 (µ 2 n 1 +(µ 1 +2µ 2 )n 2 ) w ⊗ w ρ,σ,p . So we can determine the Z-bilinear application G × Z → C × , (µ, n) → ξ -2 (µ 2 n 1 +(µ 1 +2µ 2 )n 2 ) which satisfies c Vµ,ε n • c ε n ,Vµ (w ⊗ w ρ,σ,p ) = ξ -2 (µ 2 n 1 +(µ 1 +2µ 2 )n 2 ) Id ε n ⊗Vµ (w ⊗ w ρ,σ,p
). This means that we have condition (3) of the definition. Condition ( 4) is also satisfied with

X = G s = 0, 1 2 × C/Z ∪ C/Z × 0, 1 2 ∪ (µ 1 , µ 2 ) : µ 1 + µ 2 ∈ 0, 1 2 . It was proven that C H g is semi-simple for g ∈ G \ G s (Theorem 2.4.14) and V µ ⊗ ε n
V µ+ n , i.e. the condition ( 5) is satisfied. Theorem 2.5.4 implies that condition ( 6) is true.

To compute ∆ -, we first use the graphical calculus

F      Ω µ Vµ      = -1 s,t=0 d(µ st )F      Vµ st Vµ      = -1 s,t=0 d(µ st ) θ -1 Vµ θ -1 V * µ st F   Vµ st Vµ   = -1 s,t=0 d(µ st ) θ -1 Vµ θ -1 V * µ st S (µ st , µ) Id Vµ .
We have

θ -1 Vµ = -ξ 2(α 2 2 +α 1 α 2 ) , θ -1 V * µ st = -ξ 2((α 2 +t) 2 +(α 1 +s)(α 2 +t))
and

S (µ st , µ) = ξ -4α 2 (α 2 +t)-2(α 2 (α 1 +s)+α 1 (α 2 +t)) 1 d(µ) . Thus F      Ω µ Vµ      = -1 s,t=0 d(µ st ) d(µ) ξ 2(t 2 +st) Id Vµ = 1 d(µ){ α 1 } -1 s,t=0 {α 1 + s} {α 2 + t}{α 1 + α 2 + s + t} ξ 2(t 2 +st) Id Vµ = 1 d(µ){ α 1 } -1 s,t=0 ξ -(α 2 +t) {α 2 + t} - ξ -(α 1 +α 2 +s+t) {α 1 + α 2 + s + t} ξ 2(t 2 +st) Id Vµ . Because -1 s,t=0 ξ -(α 2 +t) ξ 2(t 2 +st) {α 2 + t} = -1 t=0 ξ 2t 2 ξ -(α 2 +t) {α 2 + t} -1 s=0 ξ 2st = -1 t=0 ξ 2t 2 ξ -(α 2 +t) {α 2 + t} δ 0 t = ξ -α 2 {α 2 } , - 1
s,t=0 ξ -(α 1 +α 2 +s+t) ξ 2(t 2 +st) {α 1 + α 2 + s + t} = - -1 s,t=0 ξ 2(t 2 +st) 1 1 -ξ 2(α 1 +α 2 +s+t) = - -1 s,t=0 ξ 2(t 2 +st) ∞ k=0 ξ 2k(α 1 +α 2 +s+t) = - ∞ k=0 -1 t=0 ξ 2(t 2 +kα 1 +kα 2 +kt) -1 s=0 ξ 2(k+t)s = - ∞ k=0 -1 t=0 ξ 2(t 2 +kα 1 +kα 2 +kt) δ 0 t+k mod N = -   1 + -1 t=0 ξ 2t 2 ∞ j=1 ξ 2( j-t)(α 1 +α 2 +t)   = -1 + -1 t=0 ξ -2t(α 1 +α 2 ) ξ 2 (α 1 +α 2 ) 1 -ξ 2 (α 1 +α 2 ) = -+ ξ α 1 +α 2 {α 1 + α 2 } then F      Ω µ Vµ      = 1 d(µ){ α 1 } 1 ξ α 2 {α 2 } - ξ α 1 +α 2 {α 1 + α 2 } + 1 Id Vµ = 1 {α 1 } {α 1 + α 2 }ξ -α 2 -{α 2 }ξ α 1 +α 2 + {α 2 }{α 1 + α 2 } Id Vµ = 1 {α 1 } {α 1 } = Id Vµ .
This means that ∆ -= 1. By using the automorphism ω of superalgebra U ξ sl(2|1) where ω(e i ) = (-1)

deg e i f i , ω(f i ) = (-1) deg f i e i , ω(k i ) = k -1 i , ω(h i ) = -h i , i = 1, 2
and computing we also have ∆ + = 1. Condition ( 8) is obviously true.

Hence category C H is relatively G-modular.

Invariants of 3-manifolds

Definition 2.6.2. Let (M, T, ω) be a triple where M is a compact connected oriented 3-manifold, T ⊂ M is a C H -colored ribbon graph (possibly empty) and ω ∈ H 1 (M \ T, G).

The triple (M, T, ω) is compatible if each edge e of T is colored by an element of C ω(me)

where m e is an oriented meridian of the edge e.

2. Let L ∪ T ⊂ S 3 where L is an oriented link in S 3 \ T which gives a presentation of (M, T ) by surgery. The presentation

L ∪ T is computable if for each component L i of L whose meridian is denoted m i , we have ω(m i ) / ∈ X .
We suppose that (M, T, ω) is a compatible triple.

Definition 2.6.3. The formal linear combination

Ω µ = µ i ∈µ d(V µ i )V µ i is a Kirby color of degree µ ∈ G \ G s if {V µ i } is a set of representatives of simple Z-orbits of C µ .
Theorem 2.6.4. Let (M, T, ω) a compatible triple admitting a computable presentation L ∪ T ⊂ S 3 then

N (M, T, ω) = F (L ω ∪ T )
is a well defined topological invariant, i.e. depends only on the diffeomorphism class of the triple (M, T, ω) where L ω is obtained as the link L in which we have colored the i-th component L i by a Kirby color of degree ω(m i ) where m i is a meridian of L i .

Example

We consider an example in the case = 3. Let M be the lens space L(5, 2) which is given by surgery presentation on the Hopf link L (Figure 2.2). It has two oriented components L i , i = 1, 2 with framings 3, 2 and let m i be an oriented meridian of L i . The linking matrix of L with respect to the components L i is lk = 3 1 1 2 .

Let ω ∈ H 1 (M \T, G) and suppose that the triple (M, ∅, ω) is computable. We compute the values ω = (ω 1 , ω 2 ) where

µ = ω 1 = ω(m 1 ), µ = ω 2 = Ω µ Ω µ 3 2 Figure 2.2 -Surgery presentation of L(5,2)
ω(m 2 ) from the equations 3µ + µ = 0 and µ

+ 2µ = 0 (in C/Z × C/Z). Hence µ = ( k 5 , 2k 5 ), µ = ( 2k 5 , 4k 5 ), k = 1, ..., 4.
Here we set

ω k = (ω 1 k , ω 2 k ), ω 1 k = ( k 5 , 2k 5 ), ω 2 k = ( 2k 5 , 4k 5 ), k = 1, ..., 4. We have ω 4 = -ω 1 , ω 3 = -ω 2 . Using variables as in Lemma 2.4.6 we have (α 1 , α 2 ) = µ + (-+ 1, 2 ) = ( k 5 -2, 2k 5
+ 3 2 ), (α 1 , α 2 ) = µ + (-+ 1, 2 ) = ( 2k 5 -2, 4k 5 + 3 2 ). We color the i-th component L i by a Kirby color of degree ω(m i ), i.e. Ω ω(m 1 ) = Ω µ = 2 s,t=0 d(α st )V αst and Ω ω(m 2 ) = Ω µ = 2 i,j=0 d(α ij )V α ij where α st = (α 1 + s, α 2 + t), α ij = (α 1 + i, α 2 + j). By Lemma 2.4.6, Proposition 2.4.8 we have N (M, ∅, ω) = s,t i,j d(α st )d(α ij ) θ Vα st 3 θ V α ij 2 d(α st )S (α ij , α st ) in which d(α st ) = {α 1 + s} { (α 1 + s)}{α 2 + t}{α 1 + α 2 + s + t} , θ Vα st = -ξ -2((α 2 +t) 2 +(α 1 +s)(α 2 +t)) , θ V α ij = -ξ -2((α 2 +j) 2 +(α 1 +i)(α 2 +j)) , S (α ij , α st ) = 1 d(α st ) ξ -4(α 2 +j)(α 2 +t)-2((α 2 +j)(α 1 +s)+(α 1 +i)(α 2 +t)) .
Using computer algebra software Sagemath, we have (ξ

1 10 has degree 8 over Q) N (M, ∅, ±ω 1 ) = 1 15 -2ξ 7
10 -2ξ In this case, the result

N (M, ∅, ω) = N (M, ∅, -ω) is consistent with (M, ∅, ω) (M, ∅, -ω). Ω µ V i V j Figure 2.
3 -Morphism of the relative modularity condition

Relative G-modular category C H

This section proves the category C H has a relative G-modular structure. Following M. De Renzi [START_REF] Renzi | Non-semisimple extended topological quantum field theories[END_REF] this implies the invariant N in Section 2.6 extends to a family of 1 + 1 + 1-TQFTs.

Definition 2.7.1 ([42]). A pre-modular G-category C relative to X with modified dimension d and periodicity group Z is said a modular G-category relative to (G, Z) if it satisfies the modular condition, i.e. it exists a relative modularity parameter

ζ ∈ C * such that d(V i )f µ ij =    ζ( -→ coev V i • ←- ev V i ) if i = j, 0 if i = j
for all µ, ν ∈ G \ X and for all i, j ∈ ν which V i , V j are not in the same Zorbit, where f µ ij is the morphism determined by the C -colored ribbon tangle depicted in Figure 2 Proof. In Section 2.6 we proven that the category C H of nilpotent weight modules over U H ξ sl(2|1) is G-premodular category relative to (G, Z). Now we show that this category is a relative G-modular category. It is necessary to verify the relative modularity condition. We consider the morphism f which is represented by the diagram

Ω µ V k V i V j
.

By the handle-slide the circle colored by V k along the circle of f µ ij and an isotopy we have two equalities given by the diagrams as in Figure 2.4. It

Ω µ V k V i V j . = Ω µ+ν V k V i V j . = Ω µ+ν V k V i V j Figure 2.4 -Second Kirby's move on f follows that S (V k , V i )f µ ij = S (V k , V j )f µ+ν ij for all V k ∈ C ν .
It implies

f µ+ν ij = S (V k 1 , V i ) S (V k 1 , V j ) f µ ij = S (V k 2 , V i ) S (V k 2 , V j ) f µ ij for V k 1 , V k 2 ∈ C ν .
We denote the highest weights of

V i , V j , V k 1 and V k 2 by (ν 1 + i 1 , ν 2 + i 2 ), (ν 1 + j 1 , ν 2 +j 2 ), (ν 1 +s 1 , ν 2 +s 2 ) and (ν 1 +t 1 , ν 2 +t 2 ) for 0 ≤ i 1 , i 2 , j 1 , j 2 , s 1 , s 2 , t 1 , t 2 ≤ - 1
. By Proposition 2.4.8 we have

S (V k 1 , V i ) = ξ -4(ν 2 +s 2 )(ν 2 +i 2 )-2((ν 2 +s 2 )(ν 1 +i 1 )+(ν 1 +s 1 )(ν 2 +i 2 )) 1 d(V i ) , S (V k 1 , V j ) = ξ -4(ν 2 +s 2 )(ν 2 +j 2 )-2((ν 2 +s 2 )(ν 1 +j 1 )+(ν 1 +s 1 )(ν 2 +j 2 )) 1 d(V j ) , S (V k 2 , V i ) = ξ -4(ν 2 +t 2 )(ν 2 +i 2 )-2((ν 2 +t 2 )(ν 1 +i 1 )+(ν 1 +t 1 )(ν 2 +i 2 )) 1 d(V i ) , S (V k 2 , V j ) = ξ -4(ν 2 +t 2 )(ν 2 +j 2 )-2((ν 2 +t 2 )(ν 1 +j 1 )+(ν 1 +t 1 )(ν 2 +j 2 )) 1 d(V j ) .
Hence

S (V k 1 , V i ) S (V k 1 , V j ) = ξ -4(ν 2 +s 2 )(i 2 -j 2 )-2((ν 2 +s 2 )(i 1 -j 1 )+(ν 1 +s 1 )(i 2 -j 2 )) d(V j ) d(V i ) , S (V k 2 , V i ) S (V k 2 , V j ) = ξ -4(ν 2 +t 2 )(i 2 -j 2 )-2((ν 2 +t 2 )(i 1 -j 1 )+(ν 1 +t 1 )(i 2 -j 2 )) d(V j ) d(V i ) .
We see that

S (V k 1 , V i ) S (V k 1 , V j ) : S (V k 2 , V i ) S (V k 2 , V j ) = ξ -4(s 2 -t 2 )(i 2 -j 2 )-2((s 2 -t 2 )(i 1 -j 1 )+(s 1 -t 1 )(i 2 -j 2 ))
and the term -4(s

2 -t 2 )(i 2 -j 2 ) -2 ((s 2 -t 2 )(i 1 -j 1 ) + (s 1 -t 1 )(i 2 -j 2 )) is determined
i = j ∈ (Z/ Z) 2 it exists k 1 = k 2 ∈ (Z/ Z) 2 such that B(i -j, k 1 -k 2 ) = 0. Thus for all i = j ∈ ν it exists k 1 = k 2 ∈ ν such that S (V k 1 , V i ) S (V k 1 , V j ) = S (V k 2 , V i ) S (V k 2 , V j ) , it implies that f µ ij = 0 if i = j. If i = j we have f µ ii = f µ+ν ii for µ, ν ∈ G \ G s . We see that f µ ii ∈ End U H (V i ⊗V * i ) and W = V i ⊗V * i has a vector U H -invariant y. As Hom U H (V i ⊗ V * i , C) Hom U H (C, V i ⊗ V * i ) End U H (V i ) C Id V i then these imply that two morphisms f µ ii and -→ coev V i • ←- ev V i are proportional, i. e. there is a λ ∈ C * such that f µ ii = λ -→ coev V i • ←- ev V i .
First, we show the existence of vector invariant y. Let V k ∈ C ν , by Lemma 4.9 of [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF] we can do a handle-slide move on the circle component of the graph representing

f µ ii ⊗ Id V k to obtain the equality c W,V k • (f µ ii ⊗ Id V k ) = c -1 V k ,W • (f µ+ν ii ⊗ Id V k ) = c -1 V k ,W • (f µ ii ⊗ Id V k ). The braidings c W,V k , c -1 V k ,W : W ⊗ V k → V k ⊗ W are given by c W,V k = τ s • R and c -1 V k ,W = R -1 • τ s where R = ŘK with Ř = -1 i=0 {1} i e i 1 ⊗ f i 1 (i) ξ ! (1 -e 3 ⊗ f 3 )(1 -e 2 ⊗ f 2 ), ( 0
) ξ ! = 1, (i) ξ ! = (1) ξ (2) ξ • • • (i) ξ , (k) ξ = 1 -ξ k 1 -ξ and K = ξ -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 .
Let x = 0 be a weight vector of weight 0 of W and v ∈ V k be an even weight vector of weight ν = (ν 1 , ν 2 ), set y = f µ ii (x) ∈ W . Let W + be the vector space generated by {e i 1

1 e i 3 3 e i 2 2 y | i 1 + i 2 + i 3 > 1 for 0 ≤ i 1 ≤ -1, 0 ≤ i 2 , i 3 ≤ 1}, W -be the vector space generated by {f i 1 1 f i 3 3 f i 2 2 v | i 1 + i 2 + i 3 > 1 for 0 ≤ i 1 ≤ -1, 0 ≤ i 2 , i 3 ≤ 1}, V + be the vector space gener- ated by {e i 1 1 e i 3 3 e i 2 2 v | i 1 + i 2 + i 3 > 1 for 0 ≤ i 1 ≤ -1, 0 ≤ i 2 , i 3 ≤ 1} and V - be the vector space generated by {f i 1 1 f i 3 3 f i 2 2 y | i 1 + i 2 + i 3 > 1 for 0 ≤ i 1 ≤ -1, 0 ≤ i 2 , i 3 ≤ 1}. Because the weight of x is 0 then K(y ⊗ v) = y ⊗ v. Hence c W,V k (y ⊗ v) = τ s • ŘK(y ⊗ v) = v ⊗ y + (ξ -ξ -1 )f 1 v ⊗ e 1 y + f 3 v ⊗ e 3 y + f 2 v ⊗ e 2 y + W -⊗ W + and c -1 V k ,W (y ⊗ v) = R -1 • τ s (y ⊗ v) = (S ⊗ Id U H )(R)(v ⊗ y) = (S ⊗ Id U H ) v ⊗ y + (ξ -ξ -1 )e 1 v ⊗ f 1 y -e 3 v ⊗ f 3 y -e 2 v ⊗ f 2 y + V + ⊗ V - = v ⊗ y -(ξ -ξ -1 )k 1 e 1 v ⊗ f 1 y + k 1 k 2 e 3 v ⊗ f 3 y + k 2 e 2 v ⊗ f 2 y + S V + ⊗ V -.
Setting the above equations equal we have e 1 y = f 1 y = 0 and e 2 y = f 2 y = 0. By the relations e 1 f 1 -

f 1 e 1 = k 1 -k -1 1 ξ-ξ -1 , e 2 f 2 + f 2 e 2 = k 2 -k -1 2 ξ-ξ -1
, it implies that k 2 i y = y for i = 1, 2 and also since k i act as ξ h i and the weights of W are in Z × Z, we have that the eigenvalues of k i are in ξ Z which does not contain -1 (note that is odd). Thus k i y = y for i = 1, 2 and y is an invariant vector of W .

Second, we compute λ in

f µ ii = λ -→ coev V i • ←- ev V i .
We consider the value F of the braid closure of the graphs in this equality.

F       Ω µ V i V i       = k F       d(V k ) V k V i V i       = k F      d(V k ) # V k V i V k V i      = k F      V k V i      F      V k V i      = k d(V i )S (V k , V i ) d(V i )S (V * k , V i ) = k d 2 (V i )S (V k , V i ) S (V * k , V i )
where Ω µ = k∈µ d(V k )V k and the second equality by

F (L 1 # V L 2 ) = d -1 (V )F (L 1 )F (L 2 ).
Furthermore

S (V * k 1 , V i ) = ξ 4(ν 2 +s 2 )(ν 2 +i 2 )+2((ν 2 +s 2 )(ν 1 +i 1 )+(ν 1 +s 1 )(ν 2 +i 2 )) 1 d(V i ) , it implies that F       Ω µ V i V i       = -1 s 1 ,s 2 =0 1 2 = 1.
For the graph of

-→ coev V i • ←- ev V i , the value F of its closure is F      V i      = F      V i      = d(V i ).
Hence

λ = d -1 (V i ) and it proves that d(V i )f µ ii = -→ coev V i • ←- ev V i .
We see that the relative modularity parameter ζ = ∆ -∆ + = 1.

Chapter 3

Modified trace from pivotal Hopf G-coalgebra

This chapter is the content of the paper [START_REF] Ha | Modified trace from pivotal Hopf G-coalgebra[END_REF] available in https://arxiv.org/abs/1804.02416. Abstract. In a recent paper the authors A. Beliakova, C. Blanchet and A. M. Gainutdinov have shown that the modified trace on the category Hpmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra H. We generalize this fact to the context of G-graded categories and Hopf G-coalgebra studied by Turaev-Virelizier. We show that the symmetrised G-integral on a finite type pivotal Hopf G-coalgebra induces a modified trace in the associated G-graded category.

MSC: 57M27, 17B37 Key words: modified trace, G-integral, symmetrised G-integral, pivotal Hopf G-coalgebra.

Introduction

The notion of a modified trace was introduced by N. Geer, J. Kujawa and B. Patureau-Mirand in the article [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF]. This is one of the topological tools which can be used first to renormalize the Reshetikhin-Turaev invariant of links. Later F. Costantino, N. Geer and B. Patureau-Mirand used the modified trace to construct a class of invariants of 3-manifolds (CGP invariant) via link surgery presentations (see [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF]). The modified trace is also used to construct invariants of 3-manifolds of Reshetikhin-Turaev type from quantum group associated to the Lie superalgebra sl(2|1) (see Chapter 2) and for constructing the logarithmic invariant of Hennings type (see [START_REF] Beliakova | Logarithmic Hennings invariants for restricted quantum sl[END_REF]). In order to construct invariant of 3-manifolds, M. Hennings proposed a method based on the theory of integral for a finite dimensional Hopf algebra (see [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF]). The notion of integral was introduced by R. G. Larson and M. E. Sweedler in [START_REF] Larson | An associative orthogonal bilinear form for Hopf algebras[END_REF] and is studied in the book [START_REF] Radford | Hopf Algebras[END_REF] of Radford. It is known that under some assumption, both the space of modified trace and that of integral are one dimensional (see [START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF][START_REF] Radford | Hopf Algebras[END_REF]). A close relation between the modified trace and the integral has been established recently in [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]. The authors proved that a symmetrised integral for a finite dimensional pivotal Hopf algebra gives a modified trace t on H-pmod with an explicit formula. We would like to adapt these results to the unrestricted quantum groups at roots of unity. They are infinite dimensional Hopf algebra but can be understood as a Hopf G-coalgebra organized into a bundles of algebra over a Lie group. For a finite type Hopf G-coalgebra H = (H α ) α∈G there exists a family of linear forms on H α , called G-integral (see [START_REF] Virelizier | Hopf group-coalgebra[END_REF]). The aim of this chapter is to establish a correspondence between the G-integral for the finite type unimodular pivotal Hopf G-coalgebra H and the modified trace in the associated G-graded category H-mod. We introduce now these two notions.

G-integral

Let H = ({H α , m α , 1 α }, ∆, ε, S) be a Hopf G-coalgebra over a field k (see in Section 3.2). A right G-integral for the Hopf G-coalgebra H is a family of k-linear forms µ = (µ α : H α → k) α∈G satisfying (µ α ⊗ Id H β )∆ α,β (x) = µ αβ (x)1 β for any x ∈ H αβ . (3.1.1) Similarly, a left G-integral µ l α ∈ α∈G H * α satisfies (Id Hα ⊗µ l β )∆ α,β (x) = µ l αβ (x)1 α for any x ∈ H αβ .
The linear form µ 1 is an usual right integral for the Hopf algebra H 1 (see e.g [START_REF] Radford | Hopf Algebras[END_REF]). If H is a finite type Hopf G-coalgebra, i.e. a Hopf G-coalgebra in which dim(H α ) < +∞ for any α ∈ G, the space of right (resp. left) G-integral is known to be 1-dimensional (see e.g [START_REF] Virelizier | Hopf group-coalgebra[END_REF]). A pivotal Hopf G-coalgebra is a pair (H, g), where the pivot is the family

g = (g α ) α∈G ∈ α∈G H α satisfying ∆ α,β (g αβ ) = g α ⊗ g β for any α, β ∈ G, ε(g 1 ) = 1 k , and S α -1 S α (x) = g α xg -1 α for any x ∈ H α . Note that g -1 = (S α -1 (g α -1 )) α∈G , i.e. g -1 α = S α -1 (g α -1
) (see e.g [START_REF] Virelizier | Hopf group-coalgebra[END_REF]). In particular, g 1 is a pivotal element for H 1 and g 1 is invertible with g -1 1 = S 1 (g 1 ), ε(g 1 ) = 0 (see e.g [START_REF] Kassel | Quantum Groups[END_REF]). The symmetrised right G-integral on (H, g) associated with µ is the family µ = ( µ α ) α∈G ∈ α∈G H * α defined by

µ α (x) := µ α (g α x) for any x ∈ H α .
Similarly, a symmetrised left G-integral on (H, g) is

µ l α (x) := µ l α (g -1 α x) for any x ∈ H α . (3.1.2) A pivotal Hopf G-coalgebra is G-unibalanced if its symmetrised right G- integral is also symmetrised left G-integral, i.e. µ α = µ l α for any α ∈ G.
In the case (H, g) is unimodular, i.e. H 1 is unimodular, we show that the symmetrised G-integrals are symmetric linear forms on H and they are nondegenerate (see Proposition 3.2.7).

Modified trace

Let C be a pivotal k-linear category [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]. Let Proj(C) be the tensor ideal of projective objects of C. A modified trace on ideal Proj(C) is a family of k-linear forms t = {t P : End C (P ) → k} P ∈Proj(C) satisfying the cyclicity property and the partial trace property (see in Section 3.3.2).

Main results

Let (H, g) = ({H α , m α , 1 α }, ∆, ε, S, g) be a finite type unimodular pivotal Hopf G-coalgebra. If t is a right (resp. left) modified trace on H-pmod, it defines a family of linear forms

λ t = (λ t α ) α∈G ∈ α∈G H * α by λ t α (h) = t Hα (R h ) for h ∈ H α , H α is a projective object of H-mod and R h is the right multiplication of H α .
Theorem 3.1.1. The application t → λ t defined above gives a bijection between the space of right (resp. left) modified traces and the space of symmetrised right (resp. left) G-integrals. Furthermore, (H, g) is G-unibalanced if and only if the right modified trace is also left.

The chapter contains five section. In Section 3.2 we recall some definitions and results for a Hopf G-coalgebra, we also define a pivotal Hopf G-coalgebra, a symmetrised G-integral for a pivotal Hopf G-coalgebra H and prove that the symmetrised G-integrals are symmetric non-degenerate forms on H. Section 3.3 recall some results about modified traces and the proof of Reduction Lemma in the context of G-graded categories. In Section 3.4 we present the decomposition of tensor product H α ⊗ H β and the proof of the main theorem. In Section 3.5 we give an application of the main theorem in the case associated to a quantization of the Lie algebra sl(2).

Pivotal Hopf G-coalgebra

In this section, we recall some facts about Hopf G-coalgebra. For details see [START_REF] Turaev | Homotopy Quantum Field Theory[END_REF][START_REF] Virelizier | Hopf group-coalgebra[END_REF]. We then define a pivotal Hopf G-coalgebra, a symmetrised G-integral and give some of its properties.

Pivotal Hopf G-coalgebra

(∆ α,β ⊗ Id Cγ )∆ αβ,γ = (Id Cα ⊗∆ β,γ )∆ α,βγ , 2. for all α ∈ G, (Id Cα ⊗ε)∆ α,1 = Id Cα = (ε ⊗ Id Cα )∆ 1,α . A Hopf G-coalgebra is a G-coalgebra H = ({H α } α∈G , ∆, ε) endowed with a family S = {S α : H α → H α -1 } α∈G of k-linear maps (the antipode) such that 1. each H α is an algebra with product m α and unit element 1 α ∈ H α , 2. ε : H 1 → k and ∆ α,β : H αβ → H α ⊗ H β are algebra homomorphisms for all α, β ∈ G, 3. for any α ∈ G m α (S α -1 ⊗ Id Hα )∆ α -1 ,α = ε1 α = m α (Id Hα ⊗S α -1 )∆ α,α -1 .
The antipode automatically satisfies additional property:

Lemma 3.2.2. Given a Hopf G-coalgebra H = ({H α } α∈G , ∆, ε, S), then m α = α α α , ∆ α,β = β α αβ , η α = α , ε = 1 , S α = α α -1 .
1. S α (xy) = S α (y)S α (x) for any x, y ∈ H α , 2. S α (1 α ) = 1 α -1 , 3. ∆ β -1 ,α -1 S αβ = τ (S α ⊗ S β )∆ α,β where τ : H α -1 ⊗ H β -1 → H β -1 ⊗ H α -1 is the flip switching the two factors of H α -1 × H β -1 , 4. εS 1 = ε.

Graphical axioms for Hopf G-coalgebras

We will use the diagrams for the structural maps and the identities corresponding to the Hopf G-coalgebra H = (H α ) α∈G . For simplicity we write α instead of H α in the diagrams. Figure 3.1 presents the structural maps of the Hopf G-coalgebra which are the product, coproduct, unit, counit and the antipode, respectively. Note that these maps are in the category Vect k of finite dimensional vector spaces over a field k.

The identity of the coassociativity and the algebra homomorphism ∆ α,β are defined as in Figure 3.2. The antipode properties are shown in Figure 3.3. Finally, the compatibility between the antipode and the unit, counit are illustrated in Figure 3.4.

Example 3.2.3. Let H be a possibly infinite dimensional pivotal Hopf algebra

with the pivot φ. Suppose there is a commutative Hopf subalgebra C contained in the center of H (for example H can be the unrestricted quantum group in [START_REF] Concini | Quantum coadjoint action[END_REF]; an other example will be detailed in Section 3.5). Let G = Hom Alg (C, k) be the group of characters on C with multiplication given by gh = (g ⊗ h)

• ∆ α α α -1 = α α α -1 , α 1 = α 1 = α 1 , α -1 β -1 αβ = αβ α -1 β -1
.

∆(z -g(z)) = ∆(z) -(g 1 ⊗ g 2 )(∆(z)) = z (1) ⊗ z (2) -g 1 (z (1) ) ⊗ g 2 (z (2) ) = z (1) -g 1 (z (1) ) ⊗ z (2) + g 1 (z (1) ) ⊗ z (2) -g 2 (z (2) )
where we used the Sweedler's notation ∆(z) = z (1) ⊗ z [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] . This implies that ∆(I g ) ⊂ I g 1 ⊗ H + H ⊗ I g 2 . We thus have that a well defined coproduct ∆ g 1 ,g 2 given by the commutative diagram below

H ∆ -H ⊗ H H g 1 g 2 p g 1 g 2 ? ∆ g 1 ,g 2 -H g 1 ⊗ H g 2 p g 1 ⊗ p g 2 ?
where p g : H → H g is the projective morphism. The family {H g } g∈G with coproduct ∆ g,h is a G-coalgebra. It is also a Hopf G-coalgebra with the family of antipode given by the commutative diagram

H S -H H g p g ? S g -H g -1 .
p g -1

?

The family S g for g ∈ G is well defined since S(z

-g(z)) = S(z) -g(z) = S(z) -g -1 (S(z)) ∈ I g -1 .
We say a Hopf G-coalgebra H is of finite type if H α is finite dimensional over k for all α ∈ G.

Pivotal structure

We recall that a G-grouplike element of a Hopf G-coalgebra H is a family g = (g α ) α∈G ∈ α H α such that ∆ α,β (g αβ ) = g α ⊗ g β for any α, β ∈ G and ε(g 1 ) = 1 k . Note that g 1 is a grouplike element of the Hopf algebra H 1 . It follows [START_REF] Virelizier | Hopf group-coalgebra[END_REF] that the set of the G-grouplike elements of H is a group and if g = (g α ) α∈G , then g -1 = (S α -1 (g α -1 )) α∈G .

Definition 3.2.4. A G-grouplike element

g ∈ H is called a pivot if S α -1 S α (x) = g α xg -1
α for all x ∈ H α . The pair (H, g) of a Hopf G-coalgebra H and a pivot g is called a pivotal Hopf G-coalgebra.

Remark that for a pivotal Hopf G-coalgebra H = ({H α } α∈G , ∆, ε, S, g), H 1 is a pivotal Hopf algebra. Example 3.2.5. Let H be a Hopf G-coalgebra as in Example 3.2.3. Let φ g be the image of φ in the quotient H g . Then H is a pivotal Hopf G-coalgebra. Similarly, the symmetrised left G-integral is defined by µ l α (x) := µ l α (g -1 α x) for any x ∈ H α . Applying (3.1.2) for g -1 αβ x, x ∈ H αβ we get the defining relation for the symmetrised left G-integral:

Symmetrised right and left G-integrals

(g -1 α ⊗ µ l β )∆ α,β (x) = µ l αβ (x)1 α . (3.2.2)
The graphical representation for Equality (3.2.1) is given in Figure 3.5. The graphical representation of the relation for the left symmetrised G-integral is similar. Since the pivot is invertible Equation (3.2.1) for µ is equivalent to Equation (3.1.1) for µ. As the space of right G-integrals is one-dimensional, relation (3.2.1) defines µ uniquely (up to a scalar). Similarly the symmetrised left G-integral µ l defined by (3.2.2) is unique. Note also that the symmetrised G-integral for H 1 is the one in the sense of [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF].

Recall that a left (resp. right) cointegral in H 1 is an element Λ ∈ H 1 such that xΛ = ε(x)Λ (resp. Λx = ε(x)Λ) for all x ∈ H 1 ([2]). Definition 3.2.6.

A Hopf G-coalgebra H is unimodular if the Hopf algebra H 1 is unimodular, this means that the spaces of left and right cointegrals in

H 1 coincide.

A family of linear forms ϕ

α ∈ H * α for α ∈ G is symmetric non- degenerate if for any α ∈ G the associated bilinear forms (x, y) → ϕ α (xy), x, y ∈ H α is.
µ α (xy) = µ α (g α xy) = µ α (S α -1 S α (y)g α x) = µ α (g α yx) = µ α (yx)
Proposition 3.2.9. We have the relation µ α -1 (S α (x)) = µ α (a α x) for any x ∈ H α . Proof. By (3.2.4) we get

(Id Hα ⊗µ 1 (a 1 ?)) ∆ α,1 (x) = µ α (a α x)1 α for x ∈ H α .
By Lemma 3.2.8 we get

(Id Hα ⊗µ 1 • S 1 ) ∆ α,1 (x) = (µ α -1 • S α )(x)1 α for x ∈ H α .
Furthermore, Proposition 4.7 [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] gives

µ 1 (S 1 (x)) = µ 1 (a 1 x) for x ∈ H 1 . This implies that µ α (a α x)1 α = (µ α -1 • S α )(x)1 α for all x ∈ H α , i.e. µ α -1 (S α (x)) = µ α (a α x) for any x ∈ H α .
Recall that a finite type pivotal Hopf G-coalgebra (H, g) is G-unibalanced if its symmetrised right G-integral is also left.

Lemma 3.2.10. Assume (H, g) is a unimodular pivotal Hopf G-coalgebra.

Then (H, g) is G-unibalanced if and only if a α = g 2 α for any α ∈ G. Proof. First, we assume that a α = g 2 α . Applying (3.2.3) on g αβ x we have

(g -1 α ⊗ µ β )∆ α,β (x) = µ αβ (x)1 α .
This equality states that µ β is a symmetrised left G-integral, i.e. µ β = µ l β . Second, we assume that (H, g) is G-unibalanced. By applying the equality (3.2.5) on g -1 α x and the G-unibalanced condition one gets

µ l α (g -1 α x) = µ l α (x) = µ α (x) = µ α (g α x) = µ α (a α g -1 α x)
for any x ∈ H α . The last equality gives µ α (a α g -1 α -g α )x = 0 for any x ∈ H α .

By Proposition 3.2.7, µ α is non-degenerate. Therefore, the above equality holds if and only if a α = g 2 α .

Traces on finite G-graded categories

In this section we recall some notions and results from [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]. Let (H, g) be a finite type unimodular pivotal Hopf G-coalgebra. We determine the pivotal structure in pivotal G-graded category H-mod. We also prove the Reduction Lemma in the context of G-graded categories and recall the close relation between a modified trace on H 1 -pmod and a symmetrised integral for H 1 [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF].

Cyclic traces

Let C be a k-linear category. We call cyclic trace on C a family of k-linear maps t = {t P : End C (P ) → k} P ∈C (3.3.1) satisfying cyclicity property, i.e. t V (gh) = t U (hg) for g ∈ Hom C (U, V ) and h ∈ Hom C (V, U ) with U, V ∈ C. We say that a cyclic trace t is non-degenerate if the pairings

Hom C (M, P ) × Hom C (P, M ) → k, (f, g) → t P (f g) (3.3.2)
are non-degenerate for all P, M ∈ C. For a finite dimensional algebra A, let A-pmod be the category of projective A-modules. There is a bijection from the space of cyclic traces on A-pmod to the space of symmetric linear forms on A:

Lemma 3.3.1. There is an isomorphism of algebras

R : A op → End A (A) given by R(h) = R h , R -1 (f ) = f (1)
where R h denotes the right multiplication with h, i.e. R h (x) = xh for any x ∈ A.

Lemma 3.3.1 implies that if t is a cyclic trace on A-pmod then

λ(h) = t A (R h ) (3.3.3)
defines a symmetric linear form on A.

Proposition 3.3.2. [2, Proposition 2.4] A symmetric linear form λ on a finite dimensional algebra

A extends uniquely to a family of cyclic traces {t P : End A (P ) → k} P ∈A-pmod which satisfies Equality (3.3.3).

If f ∈ End A (P ), one can find a i ∈ Hom(A, P ), b i ∈ Hom(P, A) i ∈ I for some finite set I such that f = i∈I a i b i (see [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]). Then the cyclicity property of t implies that

t P (f ) = i∈I t A (b i a i ) = i∈I λ (b i a i (1)) . (3.3.4)
Furthermore, the non-degeneracy of the form linear λ is equivalent to the one of the pairings (3.3.2) determined by (t P ) P ∈A-pmod in (3.3.4) (see [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF], Theorem 2.6 where a stronger non-degeneracy condition for traces is considered).

Modified trace in pivotal category

Let C be a pivotal k-linear category [START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]. Then C is a strict monoidal klinear category, with a unit object I, equipped with the data for each object V ∈ C of its dual object V * ∈ C and of four morphisms

-→ ev V : V * ⊗ V → I, -→ coev V : I → V ⊗ V * , ←- ev V : V ⊗ V * → I, ←- coev V : I → V * ⊗ V such that ( -→ ev V , -→ coev V ) and ( ←- ev V , ←- coev V )
are dualities which induce the same functor duality which is monoidal. In the category C there is a family of isomorphisms

Φ = {Φ V = ( ←- ev V ⊗ Id V * * )(Id V ⊗ -→ coev V * ) : V → V * * } V ∈C
which is a monoidal natural isomorphism called the pivotal structure. We recall the notion of a modified trace on ideal in a pivotal category C which be introduced in [START_REF] Geer | Ambidextrous objects and trace fuctions for nonsemisimple categories[END_REF][START_REF] Geer | Traces on ideals in pivotals categories[END_REF]. Given U, V, W ∈ C and f ∈ End C (W ⊗ V ). The left partial trace (with respect to W ) is the map

tr l W : Hom C (W ⊗ U, W ⊗ V ) → Hom C (U, V ) defined for f ∈ Hom C (W ⊗ U, W ⊗ V ) by tr l W (f ) = ( -→ ev W ⊗ Id V )(Id W * ⊗f )( ←- coev W ⊗ Id U ) = f V W U ∈ Hom C (U, V ).
The right partial trace (with respect to W ) is the map Next we define the category of H-mod which is a pivotal G-graded category.

tr r W : Hom C (U ⊗ W, V ⊗ W ) → Hom C (U, V ) defined for f ∈ Hom C (U ⊗ W, V ⊗ W ) by tr r W (f ) = (Id V ⊗ ←- ev W )(f ⊗ Id W * )(Id U ⊗ -→ coev W ) = f W V U ∈ Hom C (U, V ).

Pivotal structure on H-mod G-graded category

Given a multiplicative group G, we call the category C pivotal G-graded k-linear if there exists a family of full subcategories (C α ) α∈G of C such that

1. I ∈ C 1 . 2. ∀(α, β) ∈ G 2 , ∀(V, W ) ∈ C α × C β , Hom C (V, W ) = {0} ⇒ α = β. 3. ∀V ∈ C, ∃n ∈ N, ∃(α 1 , ..., α n ) ∈ G n , ∃V i ∈ C α i for i = 1, ..., n such that V V 1 ⊕ ... ⊕ V n . 4. ∀(V, W ) ∈ C α × C β , V ⊗ W ∈ C αβ .
5. ∀α ∈ G, C α does not reduce to null object.

Pivotal structure on H-mod

Let (H, g) = ({H α } α∈G , ∆, ε, S, g) be a finite type pivotal Hopf G-coalgebra, let C be the k-linear

category α∈G C α in which C α is H α -mod the cate- gory of finite dimensional H α -modules. An object V of C is a finite di- rect sum V α 1 ⊕ ... ⊕ V αn where V α i ∈ C α i . Each object V in H α -mod has a dual V * = Hom k (V, k) in H α -1 -mod with the H α -1 action defined by (hf )(x) = f (S α -1 (h)x) for h ∈ H α -1 , f ∈ V * and x ∈ V . The category C is a G-graded tensor category, i.e. for V α ∈ C α , V β ∈ C β V α ⊗ V β ∈ C αβ and for α = β Hom C (V α , V β ) = 0.
Then C is a pivotal category with pivotal structure given by the left and right duality morphisms as follows. Assume that {v j | j ∈ J} is a basis of V ∈ H α -mod and {v j | j ∈ J} is the dual basis of V * , then

-→ ev V : V * ⊗ V → k, f ⊗ v → f (v), (3.3.6) -→ coev V : k → V ⊗ V * , 1 → j∈J v j ⊗ v j , ←- ev V : V ⊗ V * → k, v ⊗ f → f (g α v), (3.3.7) ←- coev V : k → V * ⊗ V, 1 → i∈J v i ⊗ g -1 α v i .
We call H-pmod or Proj(C) the ideal of projective H-modules. As C = α∈G C α , the projective modules of C α are in H-pmod ∩ C α = H α -pmod. Lemma 3.3.3. Let (H, g) be a finite type pivotal Hopf G-coalgebra. Let t be a cyclic trace on H-pmod. Let V ∈ H-pmod and ε W ∈ H 1 -mod be endowed with the trivial action ρ εW = ε Id εW . Then

∀f ∈ End H-mod (V ⊗ ε W ), t V ⊗ εW (f ) = t V (tr r εW (f )) (3.3.8) and ∀f ∈ End H-mod ( ε W ⊗ V ), t εW ⊗V (f ) = t V (tr l εW (f )). (3.3.9)
Proof. Consider a decomposition of Id εW

Id εW = i∈I e i ϕ i where ϕ i : ε W → k, e i : k → ε W, ϕ i (e j ) = δ ij . (3.3.10)
By setting e i = Id V ⊗e i :

V → V ⊗ ε W and ϕ i = Id V ⊗ϕ i : V ⊗ ε W → V one gets Id V ⊗ εW = i∈I e i ϕ i . (3.3.11)
For f ∈ End H-mod (V ⊗ ε W ), on the one hand we have

t V ⊗ εW (f ) = i∈I t V ⊗ εW (f e i ϕ i ) = i∈I t V ( ϕ i f e i ) = i∈I t V (f ii )
where f ii = ϕ i f e i ∈ End H-mod (V ). In the above calculations, we use Equation (3.3.11) in the first equality and the cyclicity property in the second equality.

On the other hand, each map

f ∈ End H-mod (V ⊗ ε W ) is presented by graph below f V V εW εW = i,j∈I f e i ϕ i ϕ j e j = i,j∈I f ϕ i e j e i ϕ j = i,j∈I f ij ⊗ (e i ϕ j ) Id εW
where f ij = ϕ i f e j ∈ End H-mod (V ). From this graphical representation implies

t V (tr r εW (f )) = i,j∈I t V                               f ϕ i e j e i ϕ j V εW = i∈I t V (f ii ).
Therefore Equality (3.3.8) holds.

Remark that the pivotal element acts trivially on ε W so the evaluation

←-
ev εW in H-mod is just the usual evaluation of Vect k .

For Equality (3.3.9) the proof is similar.

Reduction Lemma

We

t αβ Hα⊗H β (f ) = t β H β (tr l Hα (f )).
Proof. The proof strictly follows the line of Reduction Lemma 3.2 [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]. The necessity is obvious. We now prove the sufficiency of the condition. By Proposition 3.3.2 for each α ∈ G the symmetric linear form λ α induces the cyclic trace {t α P : End Hα (P ) → k} P ∈Hα-pmod . We then prove that the cyclic trace t α satisfies the right partial trace property. First, let P ∈ H α -pmod, P ∈ H β -pmod and f ∈ End H αβ (P ⊗ P ). Suppose that Id P and Id P have the decomposition

Id P = a i • b i , Id P = a i • b i (3.3.13)
where a i : H α → P, b i : P → H α and a i : H β → P , b i : P → H β . The modified trace of f is calculated as follows:

t αβ P ⊗P (f ) = t αβ P ⊗P             f a i b i a i b i = t αβ Hα⊗H β             f a i a i b i b i (3.3.14) = t α Hα                 f a i a i b i b i = t α P                 f = t α P (tr r P (f )) .
In this calculation, one uses (3.3.13) in the first equality, in the second equality one uses the cyclicity property of cyclic traces, the third equality thanks to (3.3.12) and finally one uses the duality morphisms to move b i around the loop then applying again (3.3.13) and the cyclicity property. Second, let

P ∈ H α -pmod, V ∈ H β -mod and f ∈ End H αβ (P ⊗ V ). Set Q = P ⊗ V , note that Q ∈ H αβ -pmod and P ⊗ P * , Q ⊗ Q * ∈ H 1 -pmod. Consider two morphisms A ∈ Hom H 1 -mod (P ⊗ P * , Q ⊗ Q * ) and B ∈ Hom H 1 -mod (Q ⊗ Q * , P ⊗ P * ) are given by A = P P Q Q , B = P P Q Q f Id Id V .
According to (3.3.14) one gets

t 1 P ⊗P * (B • A) = t α P (tr r P * (B • A)) = t α P                                 f Id Id V Q P P = t α P                 f P P V = t α P (tr r V (f )) .
In above calculation, one applies the definition of the partial trace in second equality, in the third equality one uses the properties of the pivotal structure.

Similarly we also have

t 1 Q⊗Q * (A • B) = t αβ Q tr r Q * (A • B) = t αβ Q                                 f Id Id V Q Q P = t αβ Q                 f Q Q = t αβ P ⊗V (f ).
Since the cyclicity property

t 1 P ⊗P * (B • A) = t 1 Q⊗Q * (A • B), it follows that t αβ P ⊗V (f ) = t α P (tr r V (f )
). The proof in the case of the left modified trace is similar.

Applications of Theorem 3.1.1

Theorem 3.1.1 has two immediate consequences when G = {1} or H is semi-simple. First, in degree 1 the symmetrised G-integral is also the symmetrised integral of H 1 and Theorem 3.1.1 recovers the main theorem of [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] that we recall here: Theorem 3.3.5 ([2]). Let (H, g) be a finite dimensional unimodular pivotal Hopf algebra over a field k. Then the space of right (left) modified traces on H-pmod is equal to the space of symmetrised right (left) integrals, and hence is 1-dimensional. Moreover, the right modified trace on H-pmod is non-degenerate and determined by the cyclicity property and by

t H (f ) = µ(gf (1)) for any f ∈ End H (H) .
Similarly, the left modified trace is non degenerate and determined by

t H (f ) = µ l (g -1 f (1)) for any f ∈ End H (H) .
In particular, H is unibalanced if and only if the right modified trace is also left.

Second, for a finite type unimodular pivotal Hopf G-coalgebra (H, g), if H is semi-simple, i.e. H α is semi-simple for all α ∈ G then H-pmod = C. Then the categorical trace generates the space of modified traces on H-pmod: for any f ∈ End C (V ), the right and left categorical trace are

tr C V (f ) := ←- ev V (f ⊗ Id V ) -→ coev V ∈ k, C tr V (f ) := -→ ev V (Id V * ⊗f ) ←- coev V ∈ k.
As a corollary of Theorem 3.1.1 we then have the proposition.

Proposition 3.3.6. Let (H, g) be a finite type unimodular pivotal Hopf Gcoalgebra over a field k. The right categorical trace tr C

Hα and its left version C tr Hα are non-zero if and only if H α -mod is semi-simple and in this case coincide up to a scalar with the trace maps

f → µ α (f (1 α )) and f → µ l α (f (1 α ))
respectively, where f ∈ End Hα (H α ).

Proof of the main theorem

Decomposition of tensor products of the regular representations

We denote by H α the left H α -module given by the left regular action. Let us denote by ε H β the vector space underlying H β equipped with the H 1 -module structure given by

h.m = ε(h)m for m ∈ ε H β , h ∈ H 1 .
We will use Sweedler's notation: (1) the map φ α,β :

∆ α,β (h) = h (1) ⊗ h (2) for h ∈ H αβ , h (1) ∈ H α , h (2) ∈ H β .
H αβ ⊗ ε H β → H α ⊗ H β h ⊗ m → h (1) ⊗ h (2) m is an isomorphism of H αβ -modules whose inverse is ψ α,β : H α ⊗ H β → H αβ ⊗ ε H β x ⊗ y → x (1) ⊗ S β -1 (x (2) )y.
(2) the map

φ l α,β : ε H α ⊗ H αβ → H α ⊗ H β m ⊗ h → h (1) m ⊗ h (2)
is an isomorphism of H αβ -modules whose inverse is

ψ l α,β : H α ⊗ H β → ε H α ⊗ H αβ x ⊗ y → S -1 α -1 (y (1) )x ⊗ y (2) .
We prove the theorem using graphical calculus with the graphical representations for Hopf G-coalgebras given in Section 3.2.1. The maps φ α,β and ψ α,β are presented in Figure 3.6. The graphical representations for φ l α,β and ψ l α,β are similar.

Proof. In order to prove part (1), we first check that φ α,β is left inverse to ψ α,β , by computing the composition one gets where we used the property of the algebra homomorphism ∆ α,β in the second equality and the associativity of multiplication in the third equality. The map ψ α,β is also H αβ -linear by: where we used the property of the algebra homomorphism ∆ α,β in the first equality, the coassociativity of coproduct and the antipode properties are used in the second equality, the associativity of multiplication and the antipode properties are used in the third equality, and we used the antipode properties in the last equality. The proof of the part ( 2) is similar way.

ψ α,β • φ α,β =
ψ α,β
Proposition 3.4.2. Let H = (H α ) α∈G be a finite type pivotal Hopf Gcoalgebra. Then we have the equalities of linear maps: [START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]

φ α,β (1 αβ ⊗ m) = 1 α ⊗ m for m ∈ ε H β , (2) ( µ αβ ⊗ Id β ) • ψ α,β = µ α ⊗ g β Id β where Id β : H β → ε H β is the identity map in Vect k .
Proof. The equality (1) holds by the definition of the map φ α,β . Part (2) follows from the diagrammic calculus in Vect k :

ψ α,β µ αβ α β εβ = µ αβ = µα g -1 β -1 = µα g β (3.4.1)
where in the second equality of (3.4.1) we used the relation of the right symmetrised G-integral in Figure 3.5.

Proof of Theorem 3.1.1

Let (H, g) be a finite type unimodular pivotal Hopf G-coalgebra, C be the pivotal G-graded category of H-modules. The existence of modified trace on Proj(C) follows from: 1) the existence of non-zero integral on H 1 2) the existence of modified trace in C 1 by applying the results of [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] for H 1 and 3) the existence of the extension of ambidextrous trace in [START_REF] Geer | Traces on ideals in pivotals categories[END_REF]Theorem 3.6]. Nevertheless we choose to give a direct proof of this fact following the lines of [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]. Furthermore, Theorem 3.1.1 also gives an explicit formula to compute the modified trace t from the integral and conversely.

Proof of Theorem 3.1.1. First, we show that a right symmetrised G-integral provides a modified trace. Suppose that µ = ( µ α ) α∈G is the right symmetrised G-integral for H. By Proposition 3.3.2 the family of the symmetric forms associated with µ induces the family of cyclic traces t = (t α ) α∈G of H-pmod.

Here t α = {t α P : End Hα (P ) → k} P ∈Hα-pmod is determined by Second, assume that we have a right modified trace, and hence the symmetric form t α P on End Hα (P ) for any projective module P and any α ∈ G. In particular for any α, β ∈ G the symmetric forms t α Hα on End Hα (H α ) and Lemma 3.3.1). We prove that the family ν = ( ν α ) α∈G satisfies the relation of the right symmetrised G-integral.

t α Hα (f ) = µ α (f (1 α )) for f ∈ End Hα (H α ). ( 3
t αβ Hα⊗H β                 f α α β β = t αβ Hα⊗H β                       φ α,β ψ α,β f α α β β = t αβ H αβ ⊗ εHβ                       φ α,β ψ α,β f αβ αβ εβ εβ = t αβ H αβ                           φ α,β ψ α,β f αβ αβ εβ = φ α,β ψ α,β
t αβ Hα⊗H β on End H αβ (H α ⊗ H β ) satisfy t αβ Hα⊗H β (f ) = t α Hα (tr r H β (f )) for any f ∈ End H αβ (H α ⊗ H β ). (3.4.4) Let ν α (h) = t α Hα (R h ) for R h ∈ End Hα (H α ) with h ∈ H α . Then ν α (f (1 α )) = t α Hα (f ) for f ∈ End Hα (H α ) (see
Consider the maps k = ∆ α,β • (R h ⊗ ϕ) : H αβ ⊗ ε H β → H α ⊗ H β for h ∈ H αβ and ϕ ∈ ε H * β . Then k is a morphism of H αβ -modules. The graphical representation of the map k is given in Figure 3.7. Let f = k •ψ α,β : H α ⊗H β → H α ⊗H β then f ∈ End H αβ (H α ⊗H β )
. We now calculate the values of the modified trace for f ∈ End H αβ (H α ⊗ H β ) and tr r H β ( f ) ∈ End Hα (H α ). On the one hand, we have

t αβ Hα⊗H β ( f ) = t αβ Hα⊗H β (k • ψ α,β ) = t αβ H αβ ⊗ εHβ (ψ α,β • k) = t αβ H αβ ⊗ εHβ                           R h αβ εβ ϕ αβ εβ = t αβ H αβ ⊗ εHβ                           R h ϕ = t αβ H αβ ⊗ εHβ                     R h ϕ = t αβ H αβ ⊗ εHβ                     R h ϕ = t αβ H αβ                     R h ϕ = ν αβ (h)ϕ(1 β ).
In the above calculations, we use the cyclicity property in the second equality; the coassociativity of the coproduct in the fourth equality; the antipode properties in the fifth equality and finally we use the partial trace property.

On the other hand, we have

t α Hα (tr r H β ( f )) = t α Hα (tr r H β (k • ψ α,β )) = t α Hα                           R h ϕ α β α g β = R h ϕ να g β α = να g β ϕ αβ R h = ν α (h (1) )ϕ(g β h (2) )
where we use the left evaluation ←ev with the pivot g β and the right coevaluation -→ coev in the second equality and ∆ α,β (h) = h (1) ⊗ h [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF] . By Equality (3.4.4) one has t αβ Hα⊗H β ( f ) = t α Hα (tr r H β ( f )). This equality means that

ν αβ (h)ϕ(1 β ) = ν α (h (1) )ϕ(g β h (2) ) for any ϕ ∈ ε H * β , h ∈ H αβ . This equality holds for any ϕ ∈ ε H * β implies that ν αβ (h)1 β = ν α (h (1) )g β h (2) , i.e. ( ν α ⊗ g β )∆ α,β (h) = ν αβ (h)1 β for any h ∈ H αβ . Therefore the family ν = ( ν α ) α∈G is the right symmetrised G-integral for H.
For the case of the left modified trace the proof is similar.

Modified trace for G-graded quantum sl(2)

In this section we present the symmetrised G-integral for the quantization of sl( 2) and the modified trace on ideal of projective modules of category of the weight modules over U q sl(2). It explains clearly the relation between the symmetrised G-integral for a pivotal Hopf G-coalgebra and the modified trace in associated category U q sl(2)-mod.

Unrestricted quantum U q sl(2)

Let U q sl(2) be the C-algebra given by generators E, F, K, K -1 and relations:

KK -1 = K -1 K = 1, KEK -1 = q 2 E, KF K -1 = q -2 F, [E, F ] = K -K -1 q -q -1
where q = e iπ r is a 2r th -root of unity. The algebra U q sl(2) is a Hopf algebra where the coproduct, counit and antipode are defined by

∆(E) = 1 ⊗ E + E ⊗ K, ε(E) = 0, S(E) = -EK -1 , ∆(F ) = K -1 ⊗ F + F ⊗ 1, ε(F ) = 0, S(F ) = -KF, ∆(K) = K ⊗ K ε(K) = 1, S(K) = K -1 .
Let U := U q sl(2) be the algebra U q sl(2) modulo the relations E r = F r = 0 and C = C[K ±r ] be the commutative Hopf subalgebra in the center of U q sl(2). The algebra U is a pivotal Hopf algebra with the pivot g = K 1-r . Let G = (C/2Z, +) ∼ -→ Hom Alg (C, C), α → (K r → q rα := e iπα ) and let U α be the algebra U q sl(2) modulo the relations K r = q rα for α ∈ G. By applying Example 3.2.3 it follows that U = {U α } α∈G is the Hopf G-coalgebra with the coproduct and the antipode are determined by the commutative diagrams:

U ∆ -U ⊗ U U α+β p α+β ? ∆ α,β -U α ⊗ U β p α ⊗ p β ? U S -U U α p α ? S α -U -α p -α ?
where p α : U → U α is the projective morphism from U to U α . The Hopf G-coalgebra U = {U α } α∈G has the pivotal structure given by g α = q -rα K.

For α = 0 the Hopf algebra U 0 is called the restricted quantum sl(2), i.e. the algebra U q sl(2) modulo the relations E r = F r = 0 and K r = 1. The right 0-integral is the usual right integral given by

µ 0 (E m F n K l ) = ηδ m,r-1 δ n,r-1 δ l,1
where η is a constant. By definition of right G-integral (3.1.1) we get

µ α (E m F n K l ) = q rα ηδ m,r-1 δ n,r-1 δ l,1 .
One can show that the Hopf G-coalgebra {U α } α∈G is G-unibalanced.

The symmetrised right G-integral for {U α } α∈G is determined by

µ α (E m F n K l ) = ηδ m,r-1 δ n,r-1 δ l,0 . (3.5.1)

Modified trace

Let C be the category of representations of the Hopf G-coalgebra U (see Section 3.3.3). Then C is equal to the G-graded category of finite dimensional weight modules over U q sl(2) (module in which K has a diagonalizable action). For α ∈ C let V α be a r-dimensional highest weight module of highest weight α + r -1 in C (see [START_REF] Costantino | Some remarks on the unrolled quantum group of sl(2)[END_REF]). Recall the modified dimension d(V α ) of V α for α ∈ (C \ Z) ∪ rZ was computed:

d(V α ) = t Vα (Id Vα ) = d 0 r-1 k=1 {k} {α + r -k} = d 0 r{α} {rα} (3.5.2)
where t is the modified trace on ideal Proj(C) of projective modules and d 0 is a non-zero complex number. In [START_REF] Costantino | Some remarks on the unrolled quantum group of sl(2)[END_REF] for the analogous unrolled category, it is normalized by d 0 = (-1) r-1 . We now present the way to compute the modified dimension of V α using the symmetrised G-integral.

By density theorem we have the isomorphism of algebras

U α ∼ -→ k∈Hr End(V α+2k )
where H r = {-(r -1), -(r -3), ..., r -1}. Hence we have the isomorphism of left U α -modules:

U α ∼ -→ k∈Hr End(V α+2k ) ∼ -→ k∈Hr V α+2k ⊗ ε V * α+2k .
Consider the quantum Casimir element of U defined by

Ω = F E + Kq + K -1 q -1 {1} 2 = EF + Kq -1 + K -1 q {1} 2 .
For k ∈ N, by induction one gets

k-1 i=0 Ω - q -2i-1 K + q 2i+1 K -1 {1} 2 = E k F k . ( 3
Ω k -E k F k ∈ Span C {E j F j K i | j < k, i ∈ Z}.
Proof. The proof is by induction on k. Indeed, by (3.5.3)

Ω k -E k F k ∈ Span C {Ω j K i | j < k, i ∈ Z} which by the induction hypothesis is contained in Span C {E j F j K i | j < k, i ∈ Z}.
Following (3.5.1) we have the corollary.

Corollary 3.5.2. For all k ∈ {0, ..., r -2} we have

µ α Ω k = 0. For k = r -1 then µ α (Ω r-1 ) = η. Proof. It follows from (3.5.1) that Span C {E j F j K i | j < k, i ∈ Z} is con- tained in the kernel of µ α for k ∈ {0, ..., r -2}.
For α ∈ C \ Z, Ω acts on V α by the scalar w α which is calculated as follows: Let v be a highest weight vector of V α . The action of K on v defined by Kv = q α+r-1 v. This implies that Ωv = q α+r +q -α-r {1} 2 v, i.e. w α = q α+r +q -α-r {1} 2

. The elements w α+2k , 0 ≤ k < r-1 are distinct as w α+2i -w α+2j = {i-j}{α+r+i+j} {1} 2

= 0 for i = j. We consider in U α the element

L α (Ω) = r-1 k=1 (Ω -w α+2k ) r-1 k=1 (w α -w α+2k ) . This element is the projector on V α ⊗ ε V * α r k=1 V α as L α (w α+2k ) = δ 0,k . The value of symmetrised right G-integral on L α (Ω) is µ α (L α (Ω)) = 1 r-1 k=1 (w α -w α+2k ) µ α r-1 k=1
(Ω -w α+2k ) .

Corollary 3.5.2 implies that

µ α r-1 k=1 (Ω -w α+2k ) = µ α Ω r-1 = η. The equality r-1 k=1 (w α -w α+2k ) = (-1) r-1 r-1 k=1 {k}{α+k} {1} 2 gives µ α (L α (Ω)) = (-1) r-1 η r-1 k=1 {1} 2 {k}{α + k} = η r-1 k=1 {1} 2 {k} 2 (-1) r-1 r-1 k=1 {k} {α + r -k} = {1} 2r-2 η r 3 d 0 rd(V α )
where we used the identity r-1 k=1 {k} 2 = (-1) r-1 r 2 in the last equality. It is clear that the coefficient {1} 2r-2 η r 3 d 0 does not depend on α. This proves that

µ α (L α (Ω)) = rd(V α ) with the choice d 0 = {1} 2r-2 η r 3
where η = µ α (E r-1 F r-1 ).

Chapter 4

A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra

Introduction

The notion of an unrolled quantum group is introduced in [START_REF] Geer | Topological invariants from unrestricted quantum groups[END_REF] by N. Geer and B. Patureau-Mirand. Then an unrolled quantum group is a quan-75 tum group with some additional generators which should be thought of the logarithms of some other generators, for example in U H q sl(2) the additional generator is an element H with the relation q H = K (see [START_REF] Costantino | Some remarks on the unrolled quantum group of sl(2)[END_REF][START_REF] Geer | Topological invariants from unrestricted quantum groups[END_REF][START_REF] Geer | Modified quantum dimensions and re-normalized links invariants[END_REF]). This element H is a tool to construct a ribbon structure on representations of U H q sl(2). The category of weight modules of U H q sl(2) is ribbon and not semisimple but the Hopf algebra is not ribbon. With this category U H q sl(2)-mod one constructed the invariants of links and of 3-manifolds (see [START_REF] Costantino | Some remarks on the unrolled quantum group of sl(2)[END_REF][START_REF] Patureau-Mirand | Invariants topologiques quantiques non semisimples[END_REF]). For the Lie superalgebra sl(2|1), the associated unrolled quantum group is denoted by U H ξ sl(2|1) with two additional generators h 1 , h 2 from the quantum group U ξ sl(2|1). Using this unrolled quantum group in Chapter 2 one has shown that the category C H of nilpotent weight modules over U H ξ sl(2|1) is ribbon and relative G-(pre)modular and leads to an invariant of links and of 3-manifolds. The category C H is ribbon thanks to the role of the additional elements h 1 , h 2 which should be thought as the logarithms of k 1 , k 2 , i.e. ξ h i = k i for i = 1, 2. They help to construct quasitriangular ribbon structure in C H . The relations ξ h i = k i for i = 1, 2 also suggest that k 1 , k 2 can be consider as holomorphic functions of h 1 and h 2 on C 2 . Following this idea we extend the superalgebra U H = U H ξ sl(2|1)/(e 1 , f 1 ) to a ribbon superalgebra U H in a topological sense, the topology determined by the norm of uniform convergence on compact sets. Its bosonization U H σ is a ribbon algebra (see in Section 4.2.3).

It is known that for each ribbon Hopf algebra one can construct a universal link invariant (all links are framed and oriented) (see [START_REF] Habiro | Bottom tangles and universal invariants[END_REF], [START_REF] Ohtsuki | Quantum invariants[END_REF]). In fact, one can show (see [START_REF] Beliakova | Logarithmic Hennings invariants for restricted quantum sl[END_REF]) that a double braiding in a Hopf algebra is enough to construct a universal invariant for string links or bottom tangles. From some universal link invariants one could construct a 3-manifold invariant. There are many ways to do this. In [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF], M. Hennings introduced a method of building an invariant of 3-manifolds by using a universal link invariant and a right integral. He worked with a finite dimensional ribbon algebra and this condition guarantees the existence of a right integral. In other way, A. Virelizier and V. Turaev constructed the invariants which called invariants of π-links and invariants of π-manifolds, the invariants of equivalence class of π-bundles or equivalently of manifolds equipped with a map from the fundamental group to π (see [START_REF] Virelizier | Algèbres de Hopf graduées et fibrés plats sur les 3-variétés[END_REF]). They began with a ribbon Hopf π-coalgebra of finite type to construct the invariants of π-links, after that they renormalized the invariant to invariant of π-manifolds by using π-integrals (see Chapter 3). Note that the π-integrals exist if and only if the group-coalgebra is of finite type (see [START_REF] Virelizier | Hopf group-coalgebra[END_REF]). From a Hopf algebra one can construct a Hopf group-coalgebra (see Chapter 3). In our case π = G = (C/Z × C/Z, +) is commutative, therefor a G-structure on a manifold M is a cohomology class ω ∈ H 1 (M, G) and a G-link is a G-structure on the complement of a link. The Hopf G-coalgebra associated with the ribbon algebra U H σ consists of quotients of U H α by the ideal (k i -ξ α i , i = 1, 2) is ribbon but is not of finite type, i.e. the method of construction the invariant of 3-manifolds in [START_REF] Virelizier | Algèbres de Hopf graduées et fibrés plats sur les 3-variétés[END_REF] does not work here. We show that for U H σ there is an associated Hopf G-coalgebra U σ (see in Section 4.4) which is of finite type but is not ribbon. We will present an another approach to construct an invariant of 3-manifolds from U H σ . We will use first the topological ribbon structure of U H σ to construct a universal invariant of links. The value of this invariant is represented by a product of a part which is a holomorphic function of variables h 1 , h 2 and a part of elements in copies of U σ . Assume the link is a surgery link in S 3 that produces a closed 3-manifold M . Next we use a cohomology class ω ∈ H 1 (M, G) and a discrete Fourier transform to reduce this element. This universal invariant of links allows to construct an invariant of 3-manifolds (M, ω) of Hennings type.

The chapter contains four sections. In Section 4.2 we construct the topological ribbon structure of U H whose bosonization is a topological ribbon algebra. Section 4.3 builds the universal invariant of links from the topological ribbon superalgebra U H and a factorization of the invariant. Finally, in Section 4.4 we define discrete Fourier transforms from the topological ribbon superalgebra to a finite type Hopf G-coalgebra. This leads to definition in Theorem 4.4.15 of an invariant of pair (M, ω) as above.

Topological ribbon Hopf superalgebra U H

In this section we recall the definition of Hopf superalgebra U H ξ sl(2|1) and we construct a topological ribbon Hopf superalgebra U H which is a completion of U H . The topology used in the present chapter is the one of uniform convergence on compact sets for the vector space of holomorphic functions on C 2 . This topology defined for q a root of unity is very different from the widely studied h-adic topology used with q = e h ∈ C[[h]]. For example, a topological completion of the U q sl(2) over C(q) can be seen in [START_REF] Habiro | Unified quantum invariants for integral homology spheres associated with simple Lie algebras[END_REF][START_REF] Ohtsuki | Quantum invariants[END_REF].

Hopf superalgebra U H

Hopf superalgebra U H ξ sl(2|1)

We recall here Definition 2.3.1 of the Hopf superalgebra U H ξ sl(2|1).

Definition 4.2.1. Let ≥ 3 be an odd integer and ξ = exp( 2πi ). The superalgebra U ξ sl(2|1) is an associative superalgebra on C generated by the elements

k 1 , k 2 , k -1 1 , k -1
2 , e 1 , e 2 , f 1 , f 2 and the relations

k 1 k 2 = k 2 k 1 , (4.2.1) k i k -1 i = 1, i = 1, 2, (4.2.2) k i e j k -1 i = ξ a ij e j , k i f j k -1 i = ξ -a ij f j i, j = 1, 2, (4.2.3
)

e 1 f 1 -f 1 e 1 = k 1 -k -1 1 ξ -ξ -1 , e 2 f 2 + f 2 e 2 = k 2 -k -1 2 ξ -ξ -1 , (4.2.4) [e 1 , f 2 ] = 0, [e 2 , f 1 ] = 0, (4.2.5) e 2 2 = f 2 2 = 0, (4.2.6) e 2 1 e 2 -(ξ + ξ -1
)e 1 e 2 e 1 + e 2 e 2 1 = 0, (4.2.7)

f 2 1 f 2 -(ξ + ξ -1 )f 1 f 2 f 1 + f 2 f 2 1 = 0. (4.2.8)
The last two relations are called the Serre relations. The matrix (a ij ) is given by a 11 = 2, a 12 = a 21 = -1, a 22 = 0. The odd generators are e 2 , f 2 .

We define ξ x := exp( 2πix ), afterwards we will use the notation

{x} = ξ x -ξ -x .
According to [START_REF] Khoroshkin | Universal R-matrix for quantized (super)algebras[END_REF], U ξ sl(2|1) is a Hopf superalgebra with the coproduct, counit and the antipode as below

∆(e i ) = e i ⊗ 1 + k -1 i ⊗ e i i = 1, 2, ∆(f i ) = f i ⊗ k i + 1 ⊗ f i i = 1, 2, ∆(k i ) = k i ⊗ k i i = 1, 2, S(e i ) = -k i e i , S(f i ) = -f i k -1 i , S(k i ) = k -1 i i = 1, 2, ε(k i ) = 1, ε(e i ) = ε(f i ) = 0 i = 1, 2. We call U H ξ sl(2|1) the C-superalgebra generated by e i , f i , k i , k -1 i and h i for i = 1, 2 with Relations (4.2.1) -(4.2.8) plus the relations [h i , e j ] = a ij e j , [h i , f j ] = -a ij f j , [h i , h j ] = 0, [h i , k j ] = 0 i, j = 1, 2.
The superalgebra U H ξ sl(2|1) is a Hopf superalgebra where ∆, S and ε are determined as in U ξ sl(2|1) and by

∆(h i ) = h i ⊗ 1 + 1 ⊗ h i , S(h i ) = -h i , ε(h i ) = 0 i = 1, 2.
Note that U H ξ sl(2|1) can be seen as a semidirect product of C[h 1 , h 2 ] acting on U ξ sl(2|1).

Define the odd elements e

3 = e 1 e 2 -ξ -1 e 2 e 1 , f 3 = f 2 f 1 -ξf 1 f 2 . Denote by B + = {e p 1 e ρ 3 e σ 2 , p ∈ {0, 1, ..., -1}, ρ, σ ∈ {0, 1}}, B -= {f p 1 f ρ 3 f σ 2 , p ∈ {0, 1, ..., -1}, ρ , σ ∈ {0, 1}}, B 0 = {k s 1 1 k s 2 2 , s 1 , s 2 ∈ Z} and B h = {h t 1 1 h t 2 2 , t 1 , t 2 ∈ N}.
Let U H = U H ξ sl(2|1)/(e 1 , f 1 ), this is a Hopf superalgebra.

Lemma 4.2.2. The set of vectors

B + B 0 B h B -is a Poincaré-Birkhoff-Witt basis of U H .
Proof. A proof of this lemma similar to that of Lemma 2.3.3 can be obtained by replacing

C[k ±1 1 , k ±1 2 ] with C[k ±1 1 , k ±1 2 , h 1 , h 2 ].

Topological Hopf superalgebra U H

We recall some notions of topological tensor product and nuclear spaces in [START_REF] Pflauma | Holomorphic deformation of Hopf algebras and applications to quantum groups[END_REF][START_REF] Grothendieck | Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires[END_REF]. A locally convex space E is called nuclear, if all the compatible topologies on E ⊗ F agree for all locally convex spaces F , i.e. the topology on E ⊗ F compatible with ⊗ is unique. A topology is compatible with ⊗ if: 1) ⊗ : E × F → E ⊗ F is continuous and 2) for all (e, f ) ∈ E × F the linear form e ⊗ f : E ⊗ F → C, x ⊗ y → e(x)f (y) is continuous [START_REF] Pflauma | Holomorphic deformation of Hopf algebras and applications to quantum groups[END_REF]. For two nuclear spaces E and F the completion of the tensor product E ⊗ F endowed with its compatible topology is denoted E ⊗F . A finite dimensional space is nuclear, the tensor product of two nuclear spaces is nuclear space and a space is nuclear if only if its completion is nuclear [START_REF] Grothendieck | Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires[END_REF]. The complete nuclear spaces form a symmetric monoidal category Nuc with the product ⊗ (see [START_REF] Pflauma | Holomorphic deformation of Hopf algebras and applications to quantum groups[END_REF]).

A super nuclear space E is a Z/2Z-graded nuclear E = E 0 ⊕E 1 where both E 0 and E 1 are closed in E. As for SVect 0 one can form the monoidal category SNuc 0 of super complete nuclear spaces with even morphisms. We call topological Hopf superalgebra a Hopf algebra object in the monoidal category SNuc 0 . That is a super complete nuclear C-space H endowed with the even C-linear continuous maps called the product, unit, coproduct, counit and antipode

m : H ⊗H → H, η : C → H, ∆ : H → H ⊗H, ε : H → C and S : H → H
satisfy the axioms:

1. the product m is associative on H admitting 1 H = η(1) as unity.

the coproduct ∆ is coassociative, i.e. (∆ ⊗ Id

H ) • ∆ = (Id H ⊗∆) • ∆ and (ε ⊗ Id H ) • ∆ = (Id H ⊗ε) • ∆ = Id H . 3.
∆ and ε are algebra morphisms where the associative product in H ⊗H determined by (m ⊗m)

• (Id H ⊗τ ⊗ Id H ). 4. m • (S ⊗ Id H ) • ∆ = m • (Id H ⊗S) • ∆ = η • ε.
The notion of a topological Hopf algebra is defined similarly.

If V is a finite dimensional C-vector space we denote by H(V ) the space of holomorphic functions on V endowed with the topology of uniform convergence on compact sets, it is nuclear space. We will also use the notation [START_REF] Trèves | Topological vector spaces, Distributions and Kernels[END_REF]) where V 1 , V 2 are finite dimensional C-vector spaces. For a quantum group, if H is generated by Cartan generators and W is a finite dimensional vector space generated by other generators then elements of W ⊗ H(H * ) can be seen as W -valued holomorphic functions. We have the proposition. Proposition 4.2.3. Let H i be C-vector spaces of dimension n i and let W i be finite dimensional vector spaces on C for i = 1, 2. Then

H(h 1 , . . . , h n ) := H(V ) if h 1 , . . . , h n are coordinate functions on V . Re- mark that we have H(V 1 ) ⊗ H(V 2 ) H(V 1 × V 2 ) (Theorem 51.6
(W 1 ⊗ H(H * 1 )) ⊗(W 2 ⊗ H(H * 2 )) (W 1 ⊗ W 2 ) ⊗ H(H * 1 × H * 2 ).
Proof. By the symmetric and associative properties of ⊗ we have

(W 1 ⊗ H(H * 1 )) ⊗(W 2 ⊗ H(H * 2 )) (W 1 ⊗W 2 ) ⊗ H(H * 1 ) ⊗ H(H * 2 ).
Furthermore, by Theorem 51.6 [START_REF] Trèves | Topological vector spaces, Distributions and Kernels[END_REF]

H(H * 1 ) ⊗ H(H * 2 ) H(H * 1 × H * 2 ). It implies (W 1 ⊗ H(H * 1 )) ⊗(W 2 ⊗ H(H * 2 )) (W 1 ⊗W 2 ) ⊗ H(H * 1 × H * 2 ). Since the spaces W i , H(H * i ) for i = 1, 2 are complete then W i ⊗ H(H * i ) W i ⊗ H(H * i ). Thus we get (W 1 ⊗ H(H * 1 )) ⊗(W 2 ⊗ H(H * 2 )) (W 1 ⊗ W 2 ) ⊗ H(H * 1 × H * 2 ).
The space of entire functions is a nuclear space obtained as the completion of polynomial functions for the topology of uniform convergence on compact sets. We use a similar completion to define a topological ribbon Hopf superalgebra from U H . That is a topological ribbon Hopf superalgebra U H where the topology is constructed as follow. We consider

U H W ⊗ C C[h 1 , h 2 , k ±1 1 , k ±1 2
] as a vector space on C where W is a finite dimensional vector space on C with the basis

B = B + B -.
Let H be C-vector space with basis {h 1 , h 2 } and H * be its dual, let H(h 1 , h 2 ) be the vector space of holomorphic functions on

C 2 H * . Now C[h 1 , h 2 , k ±1 1 , k ±1 2 ] embeds in H(h 1 , h 2 ) by sending k i to ξ h i = exp ( 2iπ h i ). Furthermore C[h 1 , h 2 , k ±1 1 , k ±1 2 ] is dense in H(h 1 , h 2 )
equipped with the topology of uniform convergence on compact sets. Thus U H is embedded in

W ⊗ C H(h 1 , h 2 ) W ⊗ C H(h 1 , h 2 ), in particular k i = 1⊗ξ h i ∈ W ⊗ C H(h 1 , h 2 ) for i = 1, 2. This space is nuclear. As W ⊗ H(h 1 , h 2 ) is complete and U H is dense in it then the completion U H of U H is isomorphic to W ⊗ C H(h 1 , h 2 ), i.e. U H W ⊗ C H(h 1 , h 2 ).
In the following, we show that the completion U H has the topological Hopf algebraic structure continuously extended from U H with the coproduct ∆ :

U H → U H ⊗ U H . Remark 4.2.4. For each w i ∈ B there exists |w i | = (|w i | 1 , |w i | 2 ) ∈ Z 2 such that h k w i = w i (h k + |w i | k ) and ∀w i , w j ∈ B w i w j = m w m c m ij (h 1 , h 2 ),
here |w i | k ∈ Z is the weight of w i for h k with k = 1, 2. Remark 4.2.5. As U H W ⊗ C H(h 1 , h 2 ) then each u ∈ U H can be written uniquely u = 1≤i,j≤4 u i Q ij (h 1 , h 2 )v j where u i ∈ B -, v j ∈ B + , Q ij (h 1 , h 2 ) ∈ H(h 1 , h 2 ) for 1 ≤ i, j ≤ 4 .
Furthermore, by Remark 4.2.4 each u ∈ U H can be also written

u = 1≤i,j≤4 u i v j P ij (h 1 , h 2 ) = i,j Q ij (h 1 , h 2 )u i v j (4.2.9)
where

u i ∈ B -, v j ∈ B + , P ij (h 1 , h 2 ), Q ij (h 1 , h 2 ) ∈ H(h 1 , h 2 ) for 1 ≤ i, j ≤ 4 .
Let K be a compact set in

H * . If φ ∈ K and x(h 1 , h 2 ) ∈ H(h 1 , h 2 ) then φ * x(h 1 , h 2 ) is the evaluation of x at φ, that is φ * x(h 1 , h 2 ) = x(φ(h 1 ), φ(h 2 )) ∈ C. For x = k w k x k (h 1 , h 2 ) ∈ U H , define a norm associated to K on U H as follow x K = k w k x k (h 1 , h 2 ) K = sup k sup φ∈K |φ * (x k (h 1 , h 2 ))| (4.2.10) = sup k sup φ∈K |x k (φ(h 1 ), φ(h 2 ))| = sup φ∈K φ * x B ∞ where φ * x = k w k φ * x k ∈ W .
Remark that all norms on W are equivalent so the choice above of the norm . B ∞ in the basis B does not matter. In particular, x K = 1 when x ∈ B. The set { . K } K compact induce the topology of uniform convergence on compact sets. If E is a nuclear space, it is a locally convex space and its topology is generated by the open balls of the continuous semi-norms. A linear map f : E → F between nuclear spaces is continuous if and only if for any continuous semi-norm . F on F there exists a continuous semi-norm . E on E and a constant η ∈ R + such that

∀x ∈ E f (x) F ≤ η x E .
The following three propositions show that the Hopf algebra maps on U H are continuous. This implies that these maps induce a topological Hopf algebra structure on U H . Proposition 4.2.6. For each compact set K ⊂ H * , there exists a compact set K and a λ K ∈ R such that ∀ x, y ∈ U H , we have W ⊗2 ⊗ H(h i,j ) where h i,1 = h i ⊗ 1, h i,2 = 1 ⊗ h i for i = 1, 2 and the h i,j are seen as coordinates functions on H * × H * . Thus we can write each x ∈ U H ⊗U H form x = k w k x k (h i,j ) where w k ∈ B ⊗ B and x k (h i,j ) ∈ H(h i,j ). We can define a norm of x ∈ U H ⊗U H associated to a compact set

xy K ≤ λ K x K y K . Proof. Given x = i w i x i (h 1 , h 2 ), y = j w j y j (h 1 , h 2 ) then xy = i w i x i (h 1 , h 2 ) j w j y j (h 1 , h 2 ) = i,j w i w j x i (h 1 + |w j | 1 , h 2 + |w j | 2 )y j (h 1 , h 2 ) = i,j,k w k c k i,j (h 1 , h 2 )x i (h 1 + |w j | 1 , h 2 + |w j | 2 )y j (h 1 , h 2 ). xy K = sup k sup φ∈K | i,j c k i,j (φ(h 1 ), φ(h 2 ))x i (φ(h 1 ) + |w j | 1 , φ(h 2 ) + |w j | 2 )y j (φ(h 1 ), φ(h 2 ))| ≤ sup k sup φ∈K | i,j c k i,j (φ(h 1 ), φ(h 2 ))| sup i sup φ∈K |x i (φ(h 1 ) + |w j | 1 , φ(h 2 ) + |w j | 2 )| sup j sup φ∈K |y j (φ(h 1 ), φ(h 2 ))| = λ K x K+C y K where λ K = sup k sup φ∈K | i,j c k i,j (φ(h 1 ), φ(h 2 ))|
K 2 ⊂ H * × H * by x K 2 = sup k sup φ∈K 2 |φ * x k (h i,j )| = sup k sup φ∈K 2 |x k (φ(h i,j ))|. ( 4
∆x K 2 ≤ λ K 2 x K . Proof. Let U be a compact set, U = U 1 × U 2 ⊂ H * × H * C 4 . First there exists λ U ∈ R such that for any a, a , b, b ∈ U H we have (a ⊗ b)(a ⊗ b ) U 1 ×U 2 = aa ⊗ bb U 1 ×U 2 = aa U 1 bb U 2 (4.2.12) ≤ λ U 1 a U 1 +C 1 a U 1 λ U 2 b U 2 +C 2 b U 2 = λ U 1 λ U 2 a U 1 +C 1 b U 2 +C 2 a U 1 b U 2 = λ U a ⊗ b U +C 1 ×C 2 a ⊗ b U = λ U a ⊗ b U a ⊗ b U where λ U = λ U 1 λ U 2 and U = U + C 1 × C 2 . Second let a compact set K 2 ⊂ H * × H * and let K ⊂ H * be the compact set {ϕ + ψ| (ϕ, ψ) ∈ K 2 }. For x ∈ U H , x = j w j x j (h 1 , h 2 ), we have ∆x K 2 = j ∆w j ∆x j (h 1 , h 2 ) K 2 ≤ j ∆w j ∆x j (h 1 , h 2 ) K 2 = j s w 1,s j ⊗ w 2,s j x j (h 1,1 + h 1,2 , h 2,1 + h 2,2 ) K 2 ≤ j s w 1,s j ⊗ w 2,s j x j (h 1,1 + h 1,2 , h 2,1 + h 2,2 ) K 2 ≤ j s λ K 2 ,j,s w 1,s j ⊗ w 2,s j K 2 +(C 1 ,C 2 ) x j (h 1,1 + h 1,2 , h 2,1 + h 2,2 ) K 2 ≤ j λ K 2 ,j x j (h 1,1 + h 1,2 , h 2,1 + h 2,2 ) K 2
where the sums are finite and λ K 2 ,j,s , λ K 2 ,j are constants and in the fifth inequality one used Inequality (4.2.12). Furthermore, let H be vector space on C with basis {h 1 , h 2 }. The symmetric algebra S(H × H) S(H ⊕ H) SH ⊗ SH (see [START_REF] Kassel | Quantum Groups[END_REF]), it is a commutative algebra on C generated by

h 1 ⊗ 1, h 2 ⊗ 1, 1 ⊗ h 1 ,
⊗ h i ) = ψ(h i ) for i = 1, 2. It implies that x j (h 1,1 +h 1,2 , h 2,1 + h 2,2 ) K 2 = sup (ϕ,ψ)∈K 2 |(ϕ, ψ) * x j (h 1,1 + h 1,2 , h 2,1 + h 2,2 ) | = sup (ϕ,ψ)∈K 2 |(ϕ + ψ) * x j (h 1 , h 2 ) | = x j (h 1 , h 2 ) K . Hence ∆x K 2 ≤ j λ K 2 ,j x j (h 1 , h 2 ) K ≤ λ K 2 x K
where λ K 2 is a constant.

This proposition implies that the coproduct is continuous. The antipode S is also continuous by proposition below. Proposition 4.2.8. For each compact set K ⊂ H * there exists a compact set K ⊂ H * and a constant λ K such that

S(x) K ≤ λ K x K for x ∈ U H . Proof. For x = j w j x j (h 1 , h 2 ) ∈ U H we have S(x) K = j S(x j (h 1 , h 2 ))S(w j ) K = j x j (-h 1 , -h 2 )S(w j ) K ≤ j x j (-h 1 , -h 2 )S(w j ) K ≤ j λ K,j x j (-h 1 , -h 2 ) K S(w j ) K ≤ j λ K,j x j (h 1 , h 2 ) -K ≤ λ K x -K
where λ K,j , λ K,j and λ K are constants.

It is clear that the unit and counit are continuous. Hence the maps product, coproduct, unit, counit and the antipode of U H are continuous (with the topology of uniform convergence on compact sets). Thus the topology of uniform convergence on compact sets of U H is compatible with its algebraic structure. The maps product, coproduct, unit, counit and the antipode of U H continuously extend to the completion U H . Note that the coproduct U H → U H ⊗ U H extends to U H → U H ⊗U H . The space U H endows with these continuous maps is a topological Hopf superalgebra.

Similarly, for n ≥ 2 denote

h i,j = 1 ⊗ ... ⊗ h i ⊗ ... ⊗ 1 (4.2.13)
where h i is in j-th position for 1 ≤ i ≤ 2 and 1 ≤ j ≤ n. Then the completion of U H⊗n is topological vector space U H ⊗n W ⊗n ⊗H(h i,j ) with the topology of uniform convergence on compact sets. Here W ⊗n is the tensor product of n copies of W and H(h i,j ) is the vector space of holomorphic functions of 2n variables {h i,j } j=1, ...,n i=1,2 in C 2n . Note also that the maps ∆

[n]

i : U H⊗n → U H⊗(n+1) and ε

[n] i : U H⊗n → U H⊗(n-1) continuously extend to U H ⊗n , here ∆

[n] i and ε

[n] i determined by ∆ [n] i = Id ⊗... ⊗ Id i-1 ⊗∆ ⊗ Id ⊗... ⊗ Id n-i and ε [n] i = Id ⊗... ⊗ Id i-1 ⊗ε ⊗ Id ⊗... ⊗ Id n-i
where ∆, ε are in i-th position. It follows that Id ⊗∆

[n] i = ∆ [n+1] i+1 , ∆ [n] i ⊗ Id = ∆ [n+1] i , ∆ [n+1] i • ∆ [n] i = ∆ [n+1] i+1 • ∆ [n] i , ∆ [n+1] j • ∆ [n] i = ∆ [n+1] i • ∆ [n] j i = j
and we denote ∆ [n] (x) = x (1) ⊗ ... ⊗ x (n) for x ∈ U H . Hence, each element x of U H ⊗n can be written x = k w k x k (h i,j ) where w k ∈ B ⊗n , x k (h i,j ) ∈ H(h i,j ) := H(H * n ). In particular, the element k i,j := 1 ⊗ ... ⊗ k i ⊗ ... ⊗ 1 where k i is in j-th position is equal to ξ h i,j = 1 ⊗ ... ⊗ ξ h i ⊗ ... ⊗ 1 for i = 1, 2 j = 1, ..., n. Let K be a compact set in C 2n Span C {h i,j } * . As in Definition (4.2.10) we define

x K = sup k sup φ∈K |φ * (x k (h i,j ))| = sup k sup φ∈K |x k (φ(h i,j ))|.
Recall that C H is the even category of finite dimensional nilpotent modules over U H (see in Chapter 2). Proposition 4.2.9. For any

V 1 , ..., V n ∈ C H the representation ρ V 1 ⊗...⊗Vn : U H⊗n → End C (V 1 ⊗...⊗V n ) continuously extends to a representation U H ⊗n → End C (V 1 ⊗ ... ⊗ V n ).
Proof. Let K be the compact set containing the weights of V = V 1 ⊗ ... ⊗ V n . We have ρ V : U H⊗n → End(V ) be continuous on compact set K. Indeed, let x ∈ U H⊗n and write x = k w k x k (h i,j ). On the subspace of weights φ ∈

K, ρ V ( k w k x k (h i,j )) acts as ρ V ( k w k x k (φ(h i,j ))) ≤ k w k K x k K ≤ λ K x K with λ K is a constant. It implies that ρ V is continuous. This prove that it exists a continuous representation ρ V : U H ⊗n → End C (V ).

Topological ribbon superalgebra U H

It is known in Chapter 2 that the operator R = ŘK on C H where

Ř = -1 i=0 {1} i e i 1 ⊗ f i 1 (i) ξ ! 1 ρ=0 (-{1}) ρ e ρ 3 ⊗ f ρ 3 (ρ) ξ ! 1 δ=0 (-{1}) δ e δ 2 ⊗ f δ 2 (δ) ξ ! ∈ U H ⊗ U H , ( 0
) ξ ! = 1, (i) ξ ! = (1) ξ (2) ξ • • • (i) ξ , (k) ξ = 1 -ξ k 1 -ξ and K = ξ -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 ∈ U H ⊗2 (4.2.14)
satisfies these conditions below

∆ ⊗ Id(R) = R 13 R 23 , Id ⊗∆(R) = R 13 R 12 , R∆ op (x) = ∆(x)R for all x ∈ U H .
This operator is given by action of an element R is in the completion U H ⊗2 , so the proof of the lemma below follows the line of Theorem VIII.2.4 [START_REF] Kassel | Quantum Groups[END_REF]. The element R satisfies the properties

R 12 R 13 R 23 = R 23 R 13 R 12 , (ε ⊗ Id U H )(R) = 1 = (Id U H ⊗ε)(R), (S ⊗ Id U H )(R) = R -1 = (Id U H ⊗S -1 )(R), (S ⊗ S)(R) = R.
The completion U H of U H is a Hopf C-superalgebra which has a pivotal element φ 0 = k - 1 k -2 2 (see Proposition 2.3.4). We define an even element θ, invertible and in the center of U H by

θ = φ 0 .(m • τ s • (Id ⊗S)(R)) -
v ∈ V µ zv = ϕ(z)(µ)v where ϕ(z) is in H(H * ) and ϕ(z)(µ) is its value at µ = (µ 1 , µ 2 ).
Proof. Let w 0,0,0 be a highest weight vector generating V µ and z a central element of U H . Following the lemmas above, z can be written

z = ϕ(z) + (ρ,σ,p) =(0,0,0) y ρ,σ,p Q ρ,σ,p (h 1 , h 2 )e ρ 2 e σ 3 e p 1 .
Since e ρ 2 e σ 3 e p 1 w 0,0,0 = 0 for (ρ, σ, p) = (0, 0, 0) and h i w 0,0,0 = µ i w 0,0,0 i = 1, 2, we get zw 0,0,0 = ϕ(z)(µ 1 , µ 2 )w 0,0,0 . If v is an arbitrary vector of V µ , we have v = xw 0,0,0 for some x in U H . It implies that zv = zxw 0,0,0 = xzw 0,0,0 = ϕ(z)(µ 1 , µ 2 )xw 0,0,0 = ϕ(z)(µ 1 , µ 2 )v. Proof. Let u be a central element of U H such that ϕ(u) = 0. Assume u is non-zero can be written as u = (ρ,σ,p) =(0,0,0)

By using this proposition, we have

y ρ,σ,p Q ρ,σ,p (h 1 , h 2 )e ρ 2 e σ 3 e p 1
where Q ρ,σ,p (h 1 , h 2 ) are non-zero functions in H(h 1 , h 2 ), 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1 and (ρ, σ, p) = (0, 0, 0). Consider a typical highest weight U H -module V µ generated by highest weight vector w 0,0,0 . It is known that the set of 4r vectors B * = {S -1 (e ρ 2 e σ 3 e p 1 )w * 0,0,0 } forms a basis of V * µ where 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1, {w * ρ,σ,p } is the dual basis of {w ρ,σ,p } of V µ . In fact, the elements S -1 (e ρ 2 e σ 3 e p 1 ) form up to multiplication by k a 1 k b 2 a, b ∈ Z a basis of the subalgebra U + of U H generated by e ρ 2 e σ 3 e p 1 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1. Since U -w * 0,0,0 = Cw * 0,0,0 where U -is subalgebra of U H generated by

f ρ 2 f σ 3 f p 1 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1, we have Span C (B * ) U + w * 0,0,0 U + U 0 U -w * 0,0,0 U H w * 0,0,0 V * µ where U 0 is subalgebra of U H topologically generated by h 1 , h 2 . Furthermore card(B * ) = dim V * µ , hence B * is a basis of V * µ . It exists in V µ a dual basis B = { w ρ,σ,p 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1} of B * in V * µ , i.e.
given (ρ, σ, p), for any e ρ 2 e σ 3 e p 1 , w * 0,0,0 (e ρ 2 e σ 3 e p 1 w ρ,σ,p ) = δ ρ ρ δ σ σ δ p p . On the one hand, Proposition 4.2.14 implies that u w ρ,σ,p = 0 for all 0 ≤ ρ, σ ≤ 1, 0 ≤ p ≤ -1. On the other hand, we have that e ρ 0 2 e σ 0 3 e p 0 1 is an element having minimal weight of ones in the items of sum (ρ,σ,p) =(0,0,0)

y ρ,σ,p Q ρ,σ,p (h 1 , h 2 )e ρ 2 e σ 3 e p 1 such that Q ρ 0 ,σ 0 ,p 0 (h 1 , h 2 ) = 0.
It is clear that e ρ 2 e σ 3 e p 1 w ρ 0 ,σ 0 ,p 0 = 0 for e ρ 2 e σ 3 e p 1 having the weight higher than one of e ρ 0 2 e σ 0 3 e p 0 1 and e ρ 2 e σ 3 e p 1 w ρ 0 ,σ 0 ,p 0 = δ ρ 0 ρ δ σ 0 σ δ p 0 p w 0,0,0 for e ρ 2 e σ 3 e p 1 having the weight equal one of e ρ 0 2 e σ 0 3 e p 0 1 . Hence we have (ρ,σ,p) =(0,0,0)

y ρ,σ,p Q ρ,σ,p (h 1 , h 2 )e ρ
y ρ,σ,p Q ρ,σ,p (h 1 , h 2 )δ ρ 0 ρ δ σ 0 σ δ p 0 p w 0,0,0
= y ρ 0 ,σ 0 ,p 0 Q ρ 0 ,σ 0 ,p 0 (h 1 , h 2 )w 0,0,0 = Q ρ 0 ,σ 0 ,p 0 (µ)w ρ 0 ,σ 0 ,p 0 = 0.

This result prove that Q ρ 0 ,σ 0 ,p 0 (h 1 , h 2 ) = 0. Thus u = 0.

Lemma 4.2.16. Let ρ V : U H → End(V ) be a nilpotent finite dimensional representation of U H . We have

ρ V (S(θ)) = ρ V (θ).
Proof. Recall that the category C H of nilpotent representations of U H ξ sl(2|1) is a ribbon category having the twist is the family of isomorphisms θ

V : V → V, ∀V ∈ C H , θ V = ρ V (θ) where ρ V : U H → End(V ) is a representation of U H (see Chapter 2). It follows that (θ V ) * = θ V * ∀V ∈ C H . In fact (θ V ) * = (ρ V (θ)) * = ( -→ ev V ⊗ Id V * )(Id V * ⊗θ V ⊗ Id V * )(Id V * ⊗ -→ coev V ) : V * → V * has matrix (ρ V (θ)) t where (ρ V (θ)) is the matrix of the endomorphism ρ V (θ). Furthermore θ V * = ρ V * (θ) has matrix (ρ V (S(θ))) t , so we have ρ V (θ) = ρ V (S(θ)). ( 4

Bosonization of U H

It is known that each ribbon superalgebra has an associated ribbon algebra, namely its bosonization (see [START_REF] Majid | Cross products by braided groups and bosonization[END_REF]). For the ribbon superalgebra U H , its bosonization denoted by U H σ , is a topological ribbon algebra by adding an element σ from U H , i.e. as an algebra, U H σ is the semi-direct product of U H with Z/2Z = {1, σ} where the action of σ is given by

σx = (-1) deg x xσ for x ∈ U H . ( 4.2.18)
The coproduct ∆ σ , the counit ε σ and the antipode S σ on U H σ given by

1. ∆ σ σ = σ ⊗ σ, ∆ σ (x) = i x i σ deg x i ⊗ x i where ∆(x) = i x i ⊗ x i for x ∈ U H , 2. ε σ (σ) = 1, ε σ (x) = ε(x) for x ∈ U H and 3. S σ (σ) = σ, S σ (x) = σ deg x S(x) for x ∈ U H . The universal R-matrix R σ in U H σ determined by R σ = R 1 i R 1 i σ deg R 2 i ⊗ R 2 i where R 1 = 1 2 (1 ⊗ 1 + σ ⊗ 1 + 1 ⊗ σ -σ ⊗ σ) and R = i R 1 i ⊗ R 2 i is the universal R-matrix in U H . Note that the universal R-matrix R σ can be written by R σ = i a i ⊗ b i j K 1 j ⊗ K 2 j (4.2.19)
where the terms a i , b i do not contain h 1 , h 2 for all i and K = j K 1 j ⊗ K

(R σ ) -1 = j K 1 j ⊗ K 2 j i a i ⊗ b i . (4.2.20)
The pivotal element of the ribbon algebra U H σ is φ σ 0 = σφ 0 . We denote U σ the Hopf subalgebra of U H σ generated by elements e i , f i , k i , k -1 i for i = 1, 2 and σ. It is a pivotal Hopf algebra with a pivotal element φ σ 0 .

Universal invariant of link diagrams

It is well known that from a ribbon algebra one can construct a universal invariant of oriented framed links, for example one can see these constructions presented by K. Habiro (see [START_REF] Habiro | Bottom tangles and universal invariants[END_REF]), M. Hennings (see [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF]), L. Kauffman and D. E. Radford (see [START_REF] Kauffman | Oriented quantum algebras, categories and invariants of knots and links[END_REF]), T. Ohtsuki (see [START_REF] Ohtsuki | Quantum invariants[END_REF]), ... In previous section we proved that U H is a ribbon superalgebra in the topological sense so its bosonization is a topological ribbon algebra. This topological ribbon algebra allows to construct a universal invariant of oriented framed links. In this section we apply the methods above to reconstruct a universal invariant of oriented framed links associated with the unrolled quantum group U H . Then we will use this invariant to construct an invariant of 3-manifolds in the next section.

Category of tangles

We recall the category T of framed, oriented tangles (see [START_REF] Habiro | Bottom tangles and universal invariants[END_REF], [START_REF] Kassel | Quantum Groups[END_REF]). The objets are the tensor words of symbols ↓ and ↑, i.e. each word forms x 1 ⊗ ... ⊗ x n with x 1 , ..., x n ∈ {↓, ↑}, n ≥ 0. The tensor word of length 0 is denoted by 1 = 1 T . The morphisms T : w → w between w, w ∈ Ob(T ) are the isotopy classes of framed, oriented tangles in a cube [0, 1] 3 such that the endpoints at the bottom are descriped by w and those at the top by w . The composition gf of a composable pair (f, g) of morphisms in T is obtained by placing g above f , and the tensor product f ⊗ g of two morphisms f and g is obtained by placing g on the right of f . The braiding c w,w : w ⊗ w → w ⊗ w for w, w ∈ Ob(T ) is the positive braiding of parallel of strings. The dual w * ∈ Ob(T ) of w ∈ Ob(T ) is defined by

1 * = 1, ↓ * = ↑, ↑ * = ↓ and (x 1 ⊗ ... ⊗ x n ) * = x * n ⊗ ... ⊗ x * 1 for x 1 , ..., x n ∈ {↓, ↑}, n ≥ 2.
For w ∈ Ob(T ), let A string link is a tangle without closed component whose arcs end at the same order as they start, with downwards orientation.

Universal invariant of link diagrams

We recall the notion of the 0 th -Hochschild homology for an algebra A, that is HH 0 (A) := A/[A, A] where [A, A] = Span{xy -yx : x, y ∈ A}. Let L = L 1 ∪ ... ∪ L n be a (framed, oriented) link diagram consisting of n ordered circle components L 1 , ..., L n with n ≥ 0. We use the method in Ohtsuki's book [START_REF] Ohtsuki | Quantum invariants[END_REF] to construct the universal invariant. It can be described by using the generators of T (see Figure 4.1).

We can put elements of U H σ on the strings of L according to the rule depicted in Figure 4.2 or in two Figures 4.2 and 4.3. For each j = 1, ..., n, we define J L j by first obtaining a word J b L j to be the product of the elements put on the component L j where these elements are read along the orientation of L j starting from any point (base point) in L j . Then set

J L j = tr u (J b L j )
where tr u : U H σ → HH 0 U H σ is the universal trace and HH 0 U H σ is the 0 th -Hochschild homology for the algebra U H σ . We define 1. There is a similar way to define the universal invariant by using the quantum trace tr q : A → A/N where N = Span C {xy -yS 2 (x)| x, y ∈ A} (see [START_REF] Habiro | Bottom tangles and universal invariants[END_REF]).

S(α) β α S(β) S(α) S(β) S(α) β α S(β) S(α) S(β)
J L = J L 1 ⊗ ... ⊗ J Ln ∈ HH 0 U H σ⊗n . ( 4

Product by g -1 (g is pivotal element in A) induces an isomorphism HH 0 (A)

∼ -→ A/N which gives a correspondence between Ohtsuki ([38]) and Habiro ([23]) universal invariant. Theorem 4.3.2 (see also Theorem 4.5 [START_REF] Ohtsuki | Quantum invariants[END_REF]). J L is a topological invariant of framed links.

Proof. The proof in the finite dimensional setting apply without change. One can show that J L j does not depend on where we start reading the element on the closed components, and J L is invariant under the Reidemeister moves for oriented links. This proves J L is an invariant of framed links.

We can similarly define the invariant of the string links by

J T = J T 1 ⊗ ... ⊗ J Tn ∈ U H σ⊗n (4.3.2)
where T is a string link consisting of n components T i and J T i is determined by reading the elements along the orientation of T i for 1 ≤ i ≤ n. The relation between the invariant of tangles and of its closure is similar as Proposition 7.3 in [START_REF] Habiro | Bottom tangles and universal invariants[END_REF]:

Proposition 4.3.3.
If T is a string link, then we have

J cl(T ) = tr ⊗n u (φ 0 ⊗ ... ⊗ φ 0 )( J T ) = tr ⊗n u (φ -1 0 ⊗ ... ⊗ φ -1 0 )( J T )
where cl(T ) is the closure of T .

Value of universal invariant of link diagrams

For x, y ∈ C 2 × C 2 , call Q(x, y) the polarization of the quadratic form determined by the matrix B = (b ij ) which is given by b

11 = 0, b 12 = b 21 = -1, b 22 = -2. Recall that h i,j = 1 ⊗ ... ⊗ h i ⊗ ... ⊗ 1
where h i is in j-th position for i = 1, 2 and j = 1, ..., n. Let H (n) = Span C {h i,j } ⊂ U H⊗n and Q ij be the quadratic form on H (n) * defined by

Q ij (h) = Q(h [i] , h [j] ) = h t [i] Bh [j]
where

(h) = 1≤i,j≤n lk ij Q ij (h). We consider the algebraic automorphisms ϕ ij , ϕ Q L of U H σ⊗n
given by

ϕ ij (x) = ξ -Q ij (h) xξ Q ij (h) , ϕ Q L (x) = ξ -Q L (h) xξ Q L (h) for x ∈ U H σ⊗n . (4.3.3)
Remark that ϕ ij and ϕ Q L restrict to an automorphism of U σ⊗n . Indeed, we denote the weight of an element x ∈ U H for h i by |x| i , i = 1, 2, we have that |x| i ∈ Z. We also recall that

h i x = x(h i + |x| i ), xh i = (h i -|x| i )x for x ∈ U H .
These equalities imply that for x = n k=1 x k ∈ U σ⊗n we have

n k=1 x k ξ h 1,i h 2,j = ξ 1⊗...⊗(h 1 -|x i | 1 )⊗...⊗(h 2 -|x j | 2 )⊗...⊗1 n k=1 x k . (4.3.4) Then ξ h i = k i ∈ U σ implies that xξ h 1,i h 2,j = ξ h 1,i h 2,j x with x ∈ U σ⊗n . This deduces that ϕ ij (U σ⊗n ) = U σ⊗n for 1 ≤ i, j ≤ n and ϕ Q L (U σ⊗n ) = U σ⊗n . Recall that J L = tr ⊗n u (J b L ) = J b L + [ U H σ⊗n , U H σ⊗n
] where J b L depends on the choice of the base points. We have the theorem.

Theorem 4.3.4. We have ξ -Q L (h) J b L ∈ U σ⊗n and if b is an other choice of base points then ξ -Q L (h) J b L -ξ -Q L (h) J b L ∈ N Q L where N Q L = ξ -Q L (h) [ U H σ⊗n , U H σ⊗n ] ∩ U σ⊗n .
Proof. We fix the base points and represent the value of J b L by the product of two parts, the first one is in H(H * ) and the second one is in the tensor Note also that J L belongs in U H σ⊗n U H σ where

U H σ⊗n U H σ = u ∈ U H σ⊗n |u∆ [n] (x) = ∆ [n] (x)u for all x ∈ U H σ . A proof
of this assertion can be seen in Lemma 6 [START_REF] Beliakova | Logarithmic Hennings invariants for restricted quantum sl[END_REF].

Invariant of 3-manifolds of Hennings type

In the article [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF], Hennings proposed a method to construct an invariant of 3-manifolds from a universal invariant of links by using a finite dimensional ribbon algebra with its right integral. The invariant of 3-manifolds is computed from the universal invariant of links. The key point of the construction is the role of a right integral of the Hopf algebra [START_REF] Hennings | Invariants of links and 3-manifolds obtained from Hopf algebras[END_REF]. It is well known that it always exists a right integral on a finite dimensional Hopf algebra. Virelizier generalised this fact by using the notions of a finite type unimodular ribbon Hopf π-coalgebra and the right π-integral to construct an invariant of 3-manifolds with π-structure. Here π is a group and the structure is given by representation of the fundamental group in π (see [START_REF] Virelizier | Algèbres de Hopf graduées et fibrés plats sur les 3-variétés[END_REF]). When π = G is commutative a G-structure reduces to a G-valued cohomology class. In the case of the unrolled quantum algebra U H , the associated Hopf G-coalgebra can be ribbon but not finite type. However, we show that the associated Hopf G-coalgebra induces a finite type Hopf G-coalgebra by forgetting h 1 , h 2 . We show that we can still construct an invariant of 3-manifolds of Hennings type by working on the pairs (M, ω) in which M is a 3-manifold and ω is a cohomology class in H 1 (M, G). The construction of the invariant uses the discrete Fourier transform and the G-integral for the finite type Hopf G-coalgebra associated with U σ (see in Section 4.4.1). This invariant is a generalisation of the one in [START_REF] Virelizier | Hopf group-coalgebra[END_REF] that apply to U H . We recall some definitions from [START_REF] Ohtsuki | Colored ribbon Hopf algebras and universal invariants of framed links[END_REF][START_REF] Virelizier | Hopf group-coalgebra[END_REF].

Hopf G-coalgebra from pivotal Hopf algebra

U σ Definition 4.4.1. Let π be a group. A π-coalgebra over C is a family C = {C α } α∈π of C-spaces endowed with a family ∆ = {∆ α,β : C αβ → C α ⊗ C β } α,β∈π of C-linear maps (the coproduct) and a C-linear map ε : C 1 → C (the counit) such that 1. ∆ is coassociative, i.e. for any α, β, γ ∈ π, (∆ α,β ⊗ Id Cγ )∆ αβ,γ = (Id Cα ⊗∆ β,γ )∆ α,βγ , 2. for all α ∈ π, (Id Cα ⊗ε)∆ α,1 = Id Cα = (ε ⊗ Id Cα )∆ 1,α . A Hopf π-coalgebra is a π-coalgebra H = ({H α } α∈π , ∆, ε) endowed with a family S = {S α : H α → H α -1 } α∈π of C-
(S α -1 ⊗ Id Hα )∆ α -1 ,α = ε1 α = m α (Id Hα ⊗S α -1 )∆ α,α -1 . A Hopf π-coalgebra is of finite type if H α is finite dimensional algebra for any α ∈ π. Recall that C = C[k ± 1 , k ± 2 ] is the commutative Hopf subalgebra in the center of U σ . Let G = (C/Z × C/Z, +) ∼ -→ Hom Alg (C, C), (α 1 , α 2 ) → k i → ξ α i for i = 1, 2 and let U α be the algebra U σ modulo the relations k i = ξ α i for α = (α 1 , α 2 ) ∈ G, i = 1, 2. Proposition 4.4.2. The family U σ = {U α } α∈G is a finite type Hopf G- coalgebra.
Proof. By applying Example 3.2.3 it follows that {U α } α∈G is the Hopf Gcoalgebra with the coproduct and the antipode determined by the commutative diagrams:

U σ ∆ σ -U σ ⊗ U σ U α+β p α+β ? ∆ α,β -U α ⊗ U β p α ⊗ p β ? U σ S σ -U σ U α p α ? S α -U -α p -α ?
where p α : U σ → U α , x → [x] is the projection from U σ to U α . For α = 0 the Hopf algebra U 0 is called the restricted quantum sl(2|1), i.e. the algebra U σ modulo the relations k i = 1 for i = 1, 2. Furthermore dim(U α ) = 32 4 for α ∈ G. This finished the proof.

Proposition 4.4.3. The small quantum group U 0 is unimodular.

Proof. Call C the even category of finite dimensional nipotent representations of U ξ sl(2|1). We claim that the projective cover P C of the trivial module is self dual: P C P * C . The proof is analogous to Theorem 2.5.1. Furthermore P C ∈ U 0 -mod so the category U 0 -mod is unimodular. By [13, Lemma 4.2.1] confirms that U 0 is unimodular.

A consequence of the proposition above is that the Hopf G-coalgebra U σ is unimodular finite type.

= (λ α ) α∈G ∈ α∈G U * α satisfies (λ α ⊗ Id U β )∆ α,β = λ α+β 1 β (4.4.1)
for all α, β ∈ G (see in Section 3 [START_REF] Virelizier | Hopf group-coalgebra[END_REF]). Note that λ 0 is an usual right integral for the Hopf algebra U 0 . We define a family of C-linear forms {tr α } α∈G on U σ determined by tr Proof. As U σ = {U α } α∈G is a unimodular finite type Hopf G-coalgebra, by Theorem 4.2 and Lemma 6.8 [START_REF] Virelizier | Hopf group-coalgebra[END_REF] for U σ one gets

α (x) := λ α (G α x) for x ∈ U α where G α = σφ 0 | k i =ξ α i for i = 1, 2, i.e. G α = ξ -α 1 σk -2 2 mod k i -ξ α i for i = 1, 2.
λ α (xy) = λ α (S -α S α (y)x) , λ -α (S α (x)) = λ α G 2 α x and S -α S α (x) = G α xG -1 α for x, y ∈ U α .
By the definition of {tr α } α∈G we have

tr α (yx) = λ α (G α yx) = λ α (S -α S α (x)G α y) = λ α (G α xy) = tr α (xy). Furthermore, for x ∈ U α λ -α (S α (x)) = λ -α (S α (x)S α (G α )G -α ) = λ -α (S α (G α x)G -α ) = λ -α (S α S -α (G -α )S α (G α x)) = λ -α (G -α S α (G α x)) = tr -α (S α (G α x))
and λ α (G 2 α x) = tr α (G α x) so tr -α (S α (x)) = tr α (x). This implies that the family tr = (tr α ) α∈G is a G-trace for U σ .

Note that, since S

-α S α (G α ) = G α for x ∈ U α then λ α (G α x) = λ α (S -α S α (G α )x) = λ α (xG α ).
Thus we also have tr α (x) = λ α (xG α ) for x ∈ U α .

Discrete Fourier transform

For a (partial) map f :

C n → C we define t i (f ) by t i (f )(h 1 , ..., h n ) = f (h 1 , ..., h i + 1, ..., h n ) for 1 ≤ i ≤ n. Let L α = {(α 1 , ..., α n ) + Z n } be the lattice of C n corresponding to - → α = α = (α 1 , ..., α n ) ∈ (C/Z) n . A function f (h 1 , ..., h n ) ∈ H(h 1 , ..., h n ) is called -periodic in h i on the lattice L α if it satisfies f | α = t i f | α where f | α := f | L α . A function f (h 1 , ..., h n ) ∈ H(h 1 , ..., h n ) is -periodic on L α if it
is in all variables on L α . The functions {ξ mh i } i=1,...,n m∈Z are -periodic and ξ h i -ξ α i are zero on α. Let I be the ideal in the ring R = C[ξ ±h 1 , ..., ξ ±hn ] generated by ξ h i -ξ α i for 1 ≤ i ≤ n. Then an element of R/I defines a -periodic map in all variables on L α . Proposition 4.4.6 (Discrete Fourier transform). Let f = f (h 1 , ..., h n ) ∈ H(h 1 , ..., h n ) be a -periodic function on L α . Then there is a unique element F -→ α (f ) ∈ R/I which coincides with f on L α and is given by

F -→ α (f ) = -1 m 1 ,...,mn=0 a m 1 ...mn ξ m 1 h 1 +...+mnhn . (4.4.2)
The coefficients a m 1 ...mn (Fourier coefficients) are determined by

a m 1 ...mn = 1 n -1 i 1 ,...,in=0 ξ -m 1 (α 1 +i 1 )-...-mn(αn+in) f (α 1 + i 1 , ..., α n + i n ).
Proof. We consider first the function

f (h 1 ) ∈ H(h 1 ) is -periodic on L α 1 for α 1 ∈ C/Z which is denoted by f | α 1 .
The set of such functions is adimensional vector space. The family {ξ m 1 h 1 } -1 m 1 =0 of linearly independent -periodic functions on L α 1 is a basis of this space, so we can write

f | α 1 = -1 m 1 =0 a m 1 ξ m 1 h 1 .
To determine (a m 1 ) m 1 we evaluate the function at α 1 + i 1 for i 1 = 0, ..., -1, we have a linear system of variables a m 1 with m 1 = 0, .., -1

-1 m 1 =0 a m 1 ξ m 1 (α 1 +i 1 ) = f (α 1 + i 1 ) for i 1 = 0, ..., -1.
The matrix of this linear system is

A =           1 ξ α 1 • • • ξ ( -1)α 1 1 ξ α 1 +1 • • • ξ ( -1)(α 1 +1) • • • • • • • • • 1 ξ α 1 +k • • • ξ ( -1)(α 1 +k) • • • • • • • • • 1 ξ α 1 + -1 • • • ξ ( -1)(α 1 + -1)           . Note that -1 k=0 ξ k(i-j) = δ i j , so we have A -1 = 1       1 1 • • • 1 ξ -α 1 ξ -(α 1 +1) • • • ξ -(α 1 + -1) . . . . . . • • • . . . ξ -( -1)α 1 ξ -( -1)(α 1 +1) • • • ξ -( -1)(α 1 + -1)       . This implies that a m 1 = 1 -1 i 1 =0 ξ -m 1 (α 1 +i 1 ) f (α 1 + k 1 )
for m 1 = 0, .., -1. Then by induction on i for 1 ≤ i ≤ n we have a similar affirmation for the -periodic functions on L α with α ∈ (C/Z) n .

Denote U

⊗ -→ α := U α 1 ⊗ ... ⊗ U αn for - → α ∈ ((C/Z) 2 )
n in which α j = (α 1j , α 2j ) ∈ (C/Z) 2 and U 0 ⊗ -→ α the subalgebra of U ⊗ -→ α generated by k ±1 i,j = ξ ±h i,j for i = 1, 2 and j = 1, ..., n (see Equation (4.2.13)). Corollary 4.4.7. Let f = f (h i,j ) ∈ H(h i,j ) be a -periodic function on L α . Then there is a unique element of U 0 ⊗ -→ α which coincides with f on L α and it is given by Since ξ h i,j = k i,j for i = 1, 2 and j = 1, ..., n then

F -→ α (f ) = -1 i 1 ,...,in, j 1 ,...,jn=0 a i 1 ...inj 1 ...jn n s=1 k is 1,s k js 2,s ∈ U 0 ⊗ -→
F -→ α (f ) = -1 i 1 ,...,in, j 1 ,...,jn=0 a i 1 ...inj 1 ...jn n s=1 k is 1,s k js 2,s ∈ U 0 ⊗ -→ α .
Proposition 4.4.6 gives the formula determining the coefficients a i 1 ...inj 1 ...jn .

Example 4.4.8. The function K = ξ -h 1 ⊗h 2 -h 2 ⊗h 1 -2h 2 ⊗h 2 is -periodic on L 0 and we have

F -→ 0 (K) = 1 2 -1 i 1 ,i 2 ,j 1 ,j 2 =0 ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 k i 1 1 k j 1 2 ⊗ k i 2 1 k j 2 2 ∈ U 0 ⊗ U 0 . (4.4.3)
Indeed, by Corollary 4.4.7 one has

F -→ 0 (K) = -1 i 1 ,i 2 ,j 1 ,j 2 =0 a i 1 i 2 j 1 j 2 k i 1 1 k j 1 2 ⊗ k i 2 1 k j 2 2 .
The coefficients a i 1 i 2 j 1 j 2 are computed as below

a i 1 i 2 j 1 j 2 = 1 4 -1 s 1 ,s 2 ,t 1 ,t 2 =0 ξ -i 1 s 1 -j 1 t 1 -i 2 s 2 -j 2 t 2 ξ -s 1 t 2 -t 1 s 2 -2t 1 t 2 = 1 4 -1 t 1 ,t 2 =0 ξ -j 1 t 1 -j 2 t 2 -2t 1 t 2 -1 s 1 ,s 2 =0 ξ -i 1 s 1 -i 2 s 2 -s 1 t 2 -t 1 s 2 = 1 4 -1 t 1 ,t 2 =0 ξ -j 1 t 1 -j 2 t 2 -2t 1 t 2 -1 s 1 ,s 2 =0 ξ -(i 1 +t 2 )s 1 -(i 2 +t 1 )s 2 = 1 4 -1 t 1 ,t 2 =0 ξ -j 1 t 1 -j 2 t 2 -2t 1 t 2 -1 s 1 =0 ξ -(i 1 +t 2 )s 1 -1 s 2 =0 ξ -(i 2 +t 1 )s 2 = 1 4 -1 t 1 ,t 2 =0 ξ -j 1 t 1 -j 2 t 2 -2t 1 t 2 δ 0 i 1 +t 2 mod Z δ 0 i 2 +t 1 mod Z = 1 2 -1 t 1 =0 ξ -j 1 t 1 δ 0 i 2 +t 1 mod Z -1 t 2 =0 ξ -j 2 t 2 -2t 1 t 2 δ 0 i 1 +t 2 mod Z = 1 2 -1 t 1 =0 ξ -j 1 t 1 δ 0 i 2 +t 1 mod Z ξ -j 2 (-i 1 )-2t 1 (-i 1 ) = 1 2 ξ -j 1 (-i 2 ) ξ -j 2 (-i 1 )-2(-i 2 )(-i 1 ) = 1 2 ξ j 1 i 2 +j 2 i 1 -2i 1 i 2 .
For α i ∈ (C/Z) 2 we call U H per α i the subalgebra of U H σ generated by elements forms u = j f ij (h 1 , h 2 )w j where w j ∈ σ m B + B -for m = 0, 1 and

f ij (h 1 , h 2 ) ∈ H(h 1 , h 2 ) are -periodic on L α i . Denote U H per ⊗ -→ α = U H per α 1 ⊗ ... ⊗ U H per αn . We extend linearly F -→ α to a map U H per ⊗ -→ α → U ⊗ -→ α by the rule m f m (h 1,i , h 2,j )w m → F -→ α (f m (h 1,i , h 2,j ))w m . Lemma 4.4.9. The map F -→ α : U H per ⊗ -→ α → U ⊗ -→ α is an algebra map.
Proof. By the unicity in Proposition 4.4.6, as f g| L α = F -→ α (f )F -→ α (g) we have

F -→ α (f g) = F -→ α (f )F -→ α (g)
for the -periodic functions f, g on L α . Consider the elements f (h 1 , h 2 )w 1 , g(h 1 , h 2 )w 2 ∈ U H per α i where f, g areperiodic on L α i and w 1 , w 2 ∈ σ m B + B -for m = 0, 1. By Remark 4.2.4 one has (f (h 1 , h 2 )w 1 )(g(h 1 , h 2 )w 2 ) = f (h 1 , h 2 )(w 1 g(h 1 , h 2 ))w 2 = f (h 1 , h 2 )g(h

1 + |w 1 | 1 , h 2 + |w 1 | 2 )w 2
where (|w 1 | 1 , |w 1 | 2 ) is the weight of w 1 for (h 1 , h 2 ). So we have

F -→ α (f w 1 gw 2 ) = F -→ α (f (h 1 , h 2 )g(h 1 + |w 1 | 1 , h 2 + |w 1 | 2 )w 1 w 2 ) = F -→ α (f g(h 1 + |w 1 | 1 , h 2 + |w 1 | 2 ))w 1 w 2 = F -→ α (f )F -→ α (g(h 1 + |w 1 | 1 , h 2 + |w 1 | 2 ))w 1 w 2 = F -→ α (f )w 1 F -→ α (g(h 1 + |w 1 | 1 -|w 1 | 1 , h 2 + |w 1 | 2 -|w 1 | 2 ))w 2 = F -→ α (f )w 1 F -→ α (g)w 2 = F -→ α (f w 1 )F -→ α (gw 2 ).
Lemma 4.4.10. Assume x ∈ U H per ⊗ -→ α is a commutator in U H σ⊗n then F -→ α (x) is a commutator in U ⊗ -→ α .

Proof. We consider an extension of the discrete Fourier transform on the lattice L α which denoted by F . The extension will depend on (α i ) i ∈ C 2n and coincide with F -→ α on elements -periodic on L α . The transformation is defined as follow: let f = f (h 1,i , h 2,j ) be a holomorphic function of H(h 1,i , h 2,j ) then if f is -periodic on L α we define F -→ α (f ) = F -→ α (f ); if f is not -periodic on L α we define F (f ) = F -→ α (f | Rec ) given by the right hand of Equation (4.4.2) where Rec = {(α 1 + i 1 , . . . , α n + i n ), 0 ≤ i 1 , . . . , i n < }. Then F (f ) is the unique element of R/I (see Section 4.4.2) which coincides with f on Rec. The map F is also an algebra map. A proof is similar as the one of Lemma 4.4.9. As x is a commutator in U H σ⊗n we write

x = f x 1 gx 2 -gx 2 f x 1
where f, g ∈ H(h 1,i , h 2,j ) and x 1 , x 2 ∈ U ⊗ -→ α . Applying F to the above equality one gets

F (x) = F (f )x 1 F (g)x 2 -F (g)x 2 F (f )x 1 ∈ [U ⊗ -→ α , U ⊗ -→ α ].
Furthermore x ∈ U H per ⊗ -→ α then F -→ α (x) = F (x). Thus F -→ α (x) is a commutator in U ⊗ -→ α . Proof. First, by Proposition 4.4.6 we have

F -→ α (f ) = -1 m 1 ,m 2 =0 a m 1 m 2 ξ m 1 h 1 +m 2 h 2
where a m 1 m 2 = 1

2

-1 i 1 ,i 2 =0 ξ -m 1 (α 1 +i 1 )-m 2 (α 2 +i 2 ) f (α 1 + i 1 , α 2 + i 2 ). Then ∆ β,γ F -→ α (f ) = -1 m 1 ,m 2 =0
a m 1 m 2 ξ m 1 (h 1 ⊗1+1⊗h 1 )+m 2 (h 2 ⊗1+1⊗h 2 ) .

Second, the algebra homomorphism ∆ gives us ∆f (h 1 , h 2 ) = f (h 1 ⊗ 1 + 1 ⊗ h 1 , h 2 ⊗ 1 + 1 ⊗ h 2 ). Applying the discrete Fourier transform one gets

F (β,γ) (∆(f )) = -1 n 1 ,n 2 ,n 3 ,n 4 =0
b n 1 n 2 n 3 n 4 ξ n 1 (h 1 ⊗1)+n 2 (h 2 ⊗1)+n 3 (1⊗h 1 )+n 4 (1⊗h 2 )

where the Fourier coefficient

b n 1 n 2 n 3 n 4 = 1 4 -1 i 1 ,i 2 ,j 1 ,j 2 =0
ξ -n 1 (β 1 +i 1 )-n 2 (β 2 +i 2 )-n 3 (γ 1 +j 1 )-n 4 (γ 2 +j 2 ) . f (β 1 + i 1 + γ 1 + j 1 , β 2 + i 2 + γ 2 + j 2 ).

By α = β + γ, one has

b n 1 n 2 n 3 n 4 = 1 4 ξ -n 1 β 1 -n 2 β 2 -n 3 γ 1 -n 4 γ 2 -1 i 1 ,i 2 ,j 1 ,j 2 =0
ξ -n 1 i 1 -n 2 i 2 -n 3 j 1 -n 4 j 2 . f (α 1 + i 1 + j 1 , α 2 + i 2 + j 2 ).

Since f (h 1 , h 2 ) is -periodic on L α , setting s = i 1 + j 1 and t = i 2 + j 2 then

b n 1 n 2 n 3 n 4 = 1 4 ξ -n 1 β 1 -n 2 β 2 -n 3 γ 1 -n 4 γ 2 -1 i 1 ,i 2 ,s,t=0 f (α 1 + s, α 2 + t). ξ -n 1 i 1 -n 2 i 2 -n 3 (s-i 1 )-n 4 (t-i 2 ) . b n 1 n 2 n 3 n 4 = 1 4 ξ -n 1 β 1 -n 2 β 2 -n 3 γ 1 -n 4 γ 2 -1 s,t=0
f (α 1 + s, α 2 + t)ξ -n 3 s-n 4 t .

-1

i 1 ,i 2 =0
ξ (n 3 -n 1 )i 1 +(n 4 -n 2 )i 2 .

Since

-1 i 1 ,i 2 =0
ξ (n 3 -n 1 )i 1 +(n 4 -n 2 )i 2 = -1

i 1 =0 ξ (n 3 -n 1 )i 1 -1 i 2 =0 ξ (n 4 -n 2 )i 2 = 2 δ n 1 n 3 δ n 2 n 4
then b n 1 n 2 n 3 n 4 = 0 if (n 1 , n 2 ) = (n 3 , n 4 ) and when (n 1 , n 2 ) = (n 3 , n 4 ) then

b n 1 n 2 n 1 n 2 is computed b n 1 n 2 n 1 n 2 = 1 2 ξ -n 1 (β 1 +γ 1 )-n 2 (β 2 +γ 2 ) -1 s,t=0 f (α 1 + s, α 2 + t)ξ -n 1 s-n 2 t = 1 2 ξ -n 1 α 1 -n 2 α 2 -1 s,t=0 f (α 1 + s, α 2 + t)ξ -n 1 s-n 2 t = 1 2 -1 s,t=0
ξ -n 1 (α 1 +s)-n 2 (α 2 +t) f (α 1 + s, α 2 + t)

= a n 1 n 2 .
Hence

F (β,γ) (∆(f )) = -1 n 1 ,n 2 =0
b n 1 n 2 n 1 n 2 ξ n 1 (h 1 ⊗1)+n 2 (h 2 ⊗1)+n 1 (1⊗h 1 )+n 2 (1⊗h 2 ) = -1

n 1 ,n 2 =0
a n 1 n 2 ξ n 1 (h 1 ⊗1+1⊗h 1 )+n 2 (h 2 ⊗1+1⊗h 2 )

= ∆ β,γ F -→ α (f ).

Remark 4.4.12. As S(h i ) = -h i for i = 1, 2, by the similar calculations as in Lemma 4.4.11 then F --→ α S(f ) = S α F -→ α (f ).

A consequence of Lemma 4.4.11 is that R 0 = R 1 ŘF -→ 0 (K) is the universal R-matrix of U 0 with R 0 = F -→ 0 (R q ) is given by

R 0 = 1 2 R 1 -1 i,i 1 ,i 2 ,j 1 ,j 2 =0 1 ρ,δ=0 {1} i (-{1}) ρ+δ (i) ξ !(ρ) ξ !(δ) ξ ! ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 . e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ⊗ f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 (4.4.4)
where R 1 = 1 2 (1 ⊗ 1 + σ ⊗ 1 + 1 ⊗ σ -σ ⊗ σ) (see Section 4.2.3). Indeed the relations satisfied by the R-matrix R q (see [START_REF] Khoroshkin | Universal R-matrix for quantized (super)algebras[END_REF], [START_REF] Yamane | Quantized enveloping algebras associatied with simple Lie superalgebras and their universal R-matrices[END_REF]) translate to the relations for R 0 .

Invariant of 3-manifolds of Hennings type

Let L be a framed link in S 3 consisting of n components (still denote by L its link diagram), M be a 3-manifold obtained by surgery along the link L. Let ω be an element of the cohomology group H 1 (M, G) (see Section 2 [START_REF] Costantino | Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories[END_REF]). The value of the invariant of link J b L is in ξ Q L (h) U σ⊗n . Let α j = ω(m j ) = (α 1j , α 2j ) here m j is a meridian of the j-th component of L. Denote α = (α 1 , ..., α n ). Since ω is an element of the cohomology group H 1 (M, G) it vanishes on longitudes of L, this implies the relation n j=1 lk ij α j = 0, ∀i = 1, ..., n. We have Proposition 4.4.13. The function f (h 1,i , h 2,j ) = ξ Q L (h) is -periodic on L α . i.e. J (M, ω) does not change under the second Kirby's move. Changing the orientation of a component changes J L by applying an antipode (see [START_REF] Ohtsuki | Quantum invariants[END_REF]), Proposition 4.4.5 and Remark 4.4.12 imply J (M, ω) does not depend on the orientation. For the first Kirby's move, the blowing up and blowing down, it is easy to see that ω(m) = 0 for m the meridian of ±1-framed loops and the two ±1-framed loops evaluate as λ 0 (θ 0 ) and λ 0 (θ -1 0 ), respectively.

Recall that the Hopf algebra U 0 has a PBW basis {f i 1 f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k j 1 1 k j 2 2 σ m : 0 ≤ ρ, δ, ρ , δ , m ≤ 1, 0 ≤ i, i , j 1 , j 2 ≤ -1}. To prove Lemma 4.4.14 we need the proposition below.

λ 0 (f i 1 f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k j 1 1 k j 2 2 σ m ) = ηδ i -1 δ ρ 1 δ σ 1 δ i -1 δ ρ 1 δ σ 1 δ j 1 0 δ j 2 -2 δ m 0 (4.4.7)
is a right integral of U 0 where η ∈ C * is a constant and δ i j is Kronecker symbol.

Proof. See in Appendix A. 1. e max f max = 0.

2. The set {e max k s 1 k t 2 f max s, t ∈ Z} is free over C. Proof. First, let V µ be a typical module with the highest weight µ = (µ 1 , µ 2 ). We show e max f max = 0 by considering its action on V µ . We have e max f max w 0,0,0 = e 2 e 3 e -1 1 f 2 f 3 f -1 1 w 0,0,0 = e 2 e 3 e -1 1 w 1,1, -1 .

Using the representation of V µ determined in (2.

= ξ µ 1 +µ 2 +2 -1 i=1 [i][µ 1 + 1 -i] -ξ µ 1 +1 [µ 2 + 1][µ 2 ] -ξ -µ 2 [µ 1 ][µ 2 ] w 0,0,0 = -ξ µ 1 +µ 2 +2 -1 i=1 [i][µ 1 + 1 -i][µ 2 ] ξ µ 1 +1 [µ 2 + 1] + ξ -µ 2 [µ 1 ] w 0,0,0 = -ξ µ 1 +µ 2 +3 -1 i=1 [i][µ 1 + 1 -i] [µ 2 ] ξ -ξ -1 ξ µ 1 +µ 2 +1 -ξ -µ 1 -µ 2 -1 w 0,0,0 109 We have m • τ s • (Id ⊗S 0 )(σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ⊗ σ n f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 ) = (-1) δ+ρ+i ξ i(i-1)-2ρ (-1) ρ+δ k -j 2 2 k -i 2 1 σ δ f δ 2 k -δ 2 σ ρ f ρ 3 k -ρ 1 k -ρ 2 f i 1 k -i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ + (-1) δ+i+δ+ρ (ξ -2ρ -1)ξ i(i-1) k -j 2 2 k -i 2 1 σ δ f δ 2 k -δ 2 σ ρ f ρ 2 f ρ 1 k -ρ 1 k -ρ 2 f i 1 k -i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ = (-1) i ξ i(i-1)-2ρ k -j 2 2 k -i 2 1 σ δ f δ 2 k -δ 2 σ ρ f ρ 3 k -ρ 1 k -ρ 2 f i 1 k -i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ + (-1) i+ρ (ξ -2ρ -1)ξ i(i-1) k -j 2 2 k -i 2 1 σ δ f δ 2 k -δ 2 σ ρ f ρ 2 f ρ 1 k -ρ 1 k -ρ 2 f i 1 k -i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ = X 1 + X 2 .
Since k i f j = ξ -a ij f j k i , k i e j = ξ a ij e j k i , k i f 3 = ξ -(a i1 +a i2 ) f 3 k i , k i e 3 = ξ a i1 +a i2 e 3 k i for i, j = 0, 1 and σx = (-1) deg x xσ then X 1 = (-1) i ξ i(i-1)-2ρ ξ -(j 2 +δ)ρ-(j 2 +δ+ρ)i+(j 2 +δ+ρ)i+(j 2 +δ+ρ)ρ . ξ -i 2 δ+i 2 ρ+2(i 2 +ρ)i-2(i+i 2 +ρ)i-(i+i 2 +ρ)ρ+(i+i 2 +ρ)δ .

σ δ f δ 2 σ ρ f ρ 3 f i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ 1 k j 1 -j 2 -δ-ρ 2 σ ρ+δ = (-1) i ξ -i-i 2 -iρ+iδ+ρδ-2ρ σ δ f δ 2 σ ρ f ρ 3 f i 1 σ m+n e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ 1 k j 1 -j 2 -δ-ρ 2 σ ρ+δ = (-1) i+δδ+(ρ+δ)ρ+(ρ+δ+m+n)(ρ+δ) ξ -i-i 2 -iρ+iδ+ρδ-2ρ . f δ 2 f ρ 3 f i 1 e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ 1 k j 1 -j 2 -δ-ρ 2 σ 2(ρ+δ)+m+n
= (-1) i+δρ+(m+n)(δ+ρ) ξ -i-i 2 -iρ+iδ+ρδ-2ρ f δ 2 f ρ 3 f i 1 e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ 1 k j 1 -j 2 -δ-ρ 2 σ m+n and X 2 = (-1) i+ρ+ρδ (ξ -2ρ -1)ξ i(i-1)+iρ k -j 2 2 k -i 2 1 σ δ k -δ 2 σ ρ f ρ+δ

2 f ρ+i 1 k -ρ 2 k -i-ρ 1 .
σ m+n e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ = (-1) i+ρ+ρδ+(m+n)(ρ+σ) (ξ -2ρ -1)ξ -i-i 2 -iρ+ρδ+iδ .

f ρ+δ 2 f ρ+i 1 e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ 1 k j 1 -j 2 -ρ-δ 2 σ m+n .
Thus we have

θ -1 0 = m • τ s • (Id ⊗S 0 )(R 0 )(φ σ 0 ) -1 = 1 2 2 -1 i,i 1 ,i 2 ,j 1 ,j 2 =0 1 m,n,ρ,δ=0 (-1) mn {1} i (-{1}) ρ+δ (i) ξ !(ρ) ξ !(δ) ξ ! ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 (X 1 +X 2 )k 2 2 σ.
Since X 2 k 2 2 σ = (-1) i+ρ+ρδ+(m+n)(ρ+σ) (ξ -2ρ -1)ξ -i-i 2 -iρ+ρδ+iδ . f ρ+δ 2 f ρ+i 1 e i 1 e ρ 3 e δ 2 k i 1 -i 2 -i-ρ

1 k j 1 -j 2 -ρ-δ+2 2 σ m+n+1
then by Proposition 4.4.18 implies that λ 0 (X 2 k 2 2 σ) = 0. Hence we have

λ 0 (θ -1 0 ) = - 1 2 2 -1 i 1 ,i 2 ,j 1 ,j 2 =0 1 m,n=0
{1} -1 (-{1}) 1+1 ( -1) ξ !(1) ξ !(1) ξ ! (-1) mn ξ -1+i 1 j 2 +i 2 j 1 -2i 1 i 2 λ 0 (f 2 f 3 f -1 1 e -1 1 e 3 e 2 k i 1 -i 2 -( -1)-1 1

k j 1 -j 2 -1-1+2 2 σ m+n+1 ) = - 1 2 2 {1} +1 ξ -1 ( -1) ξ ! η -1 i 1 ,i 2 ,j 1 ,j 2 =0 1 m,n=0
ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 . δ 0 i 1 -i 2 mod Z δ -2 j 1 -j 2 mod Z δ 0 m+n+1 mod 2Z

where η = λ 0 (f 2 f 3 f -1 1 e -1 1 e 3 e 2 k -2 2 ), i.e.

λ 0 (θ -1 0 ) = - 1 2 2 {1} +1 ξ -1 ( -1) ξ ! η -1 i 2 ,j 2 =0
ξ i 2 j 2 +i 2 (j 2 + -2)-2i 2 i 2 1 m,n=0

(-1) mn δ 0

m+n+1 mod 2Z = -2 1 2 2 {1} +1 ξ -1 ( -1) ξ ! η -1 i 2 =0 ξ -2i 2 2 -2i 2 -1 j 2 =0 ξ 2i 2 j 2 = - 1 2 {1} +1 ξ -1 ( -1) ξ ! η -1 i 2 =0 ξ -2i 2 2 -2i 2 δ 0 i 2 mod Z = - {1} +1 ξ -1 ( -1) ξ ! η .
To computer λ 0 (θ 0 ) we use the equality

θ 0 = φ σ 0 .(m • τ s • (S 2 0 ⊗ Id)(R 0 )).
Since S 2 0 (e i 1 ) = φ σ 0 e i 1 (φ σ 0 ) -1 = σk -2 2 e i 1 k 2 2 σ = ξ 2i e i 1 , S 2 0 (e ρ 3 ) = (-1) ρ ξ 2ρ e ρ 3 , S 2 0 (e δ 2 ) = (-1) δ e δ 2 , S 2 0 (k i 1 1 ) = k i 1 1 , S 2 0 (k j 1 2 ) = k j 1 2 , S 2 0 (σ ρ+δ ) = σ ρ+δ then S 2 0 (σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ) = S 2 0 (σ m )S 2 0 (e i 1 )S 2 0 (e ρ 3 )S 2 0 (e δ 2 )S 2 0 (k i 1 1 )S 2 0 (k j 1 2 )S 2 0 (σ ρ+δ ) = (-1) ρ+δ ξ 2(i+ρ) σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ .

It implies that (S 2 0 ⊗ Id)(σ m e i 1 e ρ 3 e δ 2 k i 1

1 k j 1 2 σ ρ+δ ⊗ σ n f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 ) = (-1) ρ+δ ξ 2(i+ρ) σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ⊗ σ n f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 then m • τ s • (S 2 0 ⊗ Id)(σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ⊗ σ n f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 ) = (-1) ρ+δ+ρ+δ ξ 2(i+ρ) σ n f i 1 f ρ 3 f δ 2 k i 2 1 k j 2 2 σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ = ξ 2(i+ρ) ξ i 2 (2i+ρ-δ) ξ -j 2 (i+ρ) σ n f i 1 f ρ 3 f δ 2 σ m e i 1 e ρ 3 e δ 2 k i 1 +i 2 1 k j 1 +j 2 2
σ ρ+δ = (-1) (2n+m)(ρ+δ) ξ 2(i+ρ)+i 2 (2i+ρ-δ)-j 2 (i+ρ) f i 1 f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k i 1 +i 2 1 k j 1 +j 2 2 σ m+m+ρ+δ = (-1) m(ρ+δ) ξ 2(i+ρ)+i 2 (2i+ρ-δ)-j 2 (i+ρ) f i

1 f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k i 1 +i 2 1 k j 1 +j 2 2 σ m+m+ρ+δ .
So we have

θ 0 = σk -2 2 1 2 2 -1 i,i 1 ,i 2 ,j 1 ,j 2 =0 1 m,n,ρ,δ=0
(-1) mn {1} i (-{1}) ρ+δ (i) ξ !(ρ) ξ !(δ) ξ ! ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 .

(-1) m(ρ+δ) ξ 2(i+ρ)+i 2 (2i+ρ-δ)-j 2 (i+ρ) f i

1 f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k i 1 +i 2 1 k j 1 +j 2 2 σ m+m+ρ+δ = 1 2 2 -1 i,i 1 ,i 2 ,j 1 ,j 2 =0
1 m,n,ρ,δ=0

{1} i (-{1}) ρ+δ (i) ξ !(ρ) ξ !(δ) ξ ! ξ 2(i+ρ)+i 2 (2i+ρ-δ)-j 2 (i+ρ) .

(-1) m(ρ+δ)+mn ξ i 1 j 2 +i 2 j 1 -2i {1} -1 (-{1}) 1+1 ( -1) ξ ! (-1) mn ξ -2i 2 +i 1 j 2 +i 2 j 1 -2i 1 i 2 η. δ 0 i 1 +i 2 mod Z δ 0 j 1 +j 2 -2 mod Z δ 0

m+n+1 mod 2Z = 1 2 2 {1} +1 ( -1) ξ ! η -1 i 1 ,i 2 ,j 1 ,j 2 =0
ξ -2i 2 +i 1 j 2 +i 2 j 1 -2i 1 i 2 δ 0 i 1 +i 2 mod Z δ 0 j 1 +j 2 -2 mod Z . where η = λ 0 (f -1 1 f 3 f 2 e -1 1 e 3 e 2 k -2 2 ), i.e.

λ 0 (θ 0 ) = 1 2 {1} +1 ( -1) ξ ! η -1 i 1 ,j 1 =0
ξ -2( -i 1 )+i 1 ( -j 1 +2)+( -i 1 )j 1 -2i 1 ( -i 1 )

= 1

2 {1} +1 ( -1) ξ ! η -1 i 1 =0 ξ 2i 2 1 +4i 1 -1 j 1 =0 ξ -2i 1 j 1 = 1 2 {1} +1 ( -1) ξ ! η -1 i 1 =0 ξ 2i 2 1 +4i 1 δ 0 i 1 mod Z = {1} +1 ( -1) ξ ! η.
On the other hand

f 2 f 3 f 1 = ξ -1 f 2 f 1 f 3 = ξ -1 (f 3 + ξf 1 f 2 )f 3 = f 1 f 2 f 3 .
Since f 2 f 3 = -ξf 3 f 2 we get

η = λ 0 (f 2 f 3 f -1 1 e -1 1 e 3 e 2 k -2 2 ) = -ξλ 0 (f -1 1 f 3 f 2 e -1 1 e 3 e 2 k - 2
2 ) = -ξη.

Thus we have

λ 0 (θ -1 0 ) = - {1} +1 ξ -1 ( -1) ξ ! η = {1} +1 ( -1) ξ ! η = λ 0 (θ 0 ). Since ( -1) ξ ! = -1 i=1 1-ξ i 1-ξ = -1
(1-ξ) -1 then λ 0 (θ 0 ) = {1} +1 (1-ξ) -1

( -1)

η.

  Théorie topologique des champs quantiques pour la superalgèbre de Lie sl(2|1) Ngoc Phu HA LMBA CNRS UMR 6205 Université de Bretagne Sud Remerciement L'apprentissage et la recherche à Vannes sont mes meilleures expériences. Dans cette très jolie ville, j'ai rencontré des professeurs, des amis qui sont très sympatiques. Je tiens tout d'abord à remercier au fond du coeur mon directeur de thèse Monsieur Bertrand Patureau-Mirand. Il a fait partager sa motivation et ses connaissances pour ce domaine de recherche. Je tiens à remercier encore sa famille pour son accueil très familier et sympatiques. Je suis très reconnaissant envers les professeurs Anna Beliakova et Emmanuel Wagner d'avoir pris le temps d'évaluer mon travail et de leurs commentaires très utiles. Je remercie les professeurs Christian Blanchet, Azat M. Gainutdinov, Gaël Meigniez et Gwénaël Massuyeau d'avoir d'accepté de faire partie de mon jury.

  Thus we have ∆(w) = ∆(e 1 ) ∆(e 3 ) -ξ ∆(e 3 ) ∆(e 1 ) = 1 ⊗ w + w ⊗ 1 + (ξ 2 -1)e 1 ⊗ e 1 e 2 + e 1 ⊗ e 3 -ξ 2 e 1 ⊗ e 3 -(ξ -ξ -1 )e 1 ⊗ e 2 e 1 = 1 ⊗ w + w ⊗ 1.

Theorem 2 . 5 . 1 .

 251 Each typical module V µ of category C H is an ambidextrous module.

Proposition 2 . 7 . 2 .

 272 .3 under Reshetikhin-Turaev functor F . The category C H of nilpotent weight modules over U H ξ sl(2|1) is modular G-category relative to (G, Z) where G = C/Z × C/Z and Z = Z × Z × Z/2Z.

  by a symmetric bilinear non-degenerate B from (Z/ Z) 2 × (Z/ Z) 2 to Z/ Z which has the matrix B = (b ij ) 2×2 where b 11 = 0, b 12 = b 21 = -2 and b 22 = -4. It deduces that for all

Résumé.

  Dans un article récent, les auteurs A. Beliakova, C. Blanchet et A. M. Gainutdinov ont montré que la trace modifiée sur la catégorie H-pmod des modules projectifs correspond à l'intégrale symétrisée sur l'algèbre de Hopf pivotale de dimension finie H. Nous généralisons ce théorème au contexte des catégories G-graduées et G-cogèbre de Hopf étudiée par Turaev-Virelizier. Nous montrons que la G-intégrale symétrisée sur une G-cogèbre de Hopf pivotale de type fini induit une trace modifiée dans la catégorie G-graduée associée.

Hopf G-coalgebra Definition 3 . 2 . 1 .

 321 Let G be a multiplicative group. A G-coalgebra over a field k is a family C = {C α } α∈G of k-spaces endowed with a family ∆ = {∆ α,β :C αβ → C α ⊗ C β } α,β∈G of k-linear maps (the coproduct) and a k-linear map ε : C 1 → k (the counit) such that 1. ∆ is coassociative, i.e. for any α, β, γ ∈ G,

Figure 3 Figure 3 . 2 -

 332 Figure 3.1 -The structural maps

Figure 3 1 = 1 .Figure 3 . 4 -

 31134 Figure 3.3 -The antipode properties

Figure 3 . 5 -

 35 Figure 3.5 -The graphical representation of the relation for the right symmetrised G-integral

Proposition 3 . 2 . 7 .

 327 Assume (H, g) is unimodular, then the symmetrised right (resp. left) G-integral for (H, g) is symmetric and non-degenerate.Proof. For any α ∈ G, x, y ∈ H α , by[START_REF] Virelizier | Hopf group-coalgebra[END_REF] Lemma 7.1] we have

( 3 . 3 . 5 )

 335 Let Proj(C) be the tensor ideal of projective objects of C. A left modified trace on Proj(C) is a cyclic trace t on Proj(C) satisfying t W ⊗P (f ) = t P (tr l W (f )) for any f ∈ End C (W ⊗ P ) with P ∈ Proj(C) and W ∈ C. A right modified trace on Proj(C) is a cyclic trace t on Proj(C) satisfying t P ⊗W (f ) = t P (tr r W (f )) for any f ∈ End C (P ⊗ W ) with P ∈ Proj(C) and W ∈ C. A modified trace on ideal Proj(C) is a cyclic trace t on Proj(C) which is both a left and right trace on Proj(C).

Theorem 3 . 4 . 1 .Figure 3 . 6 -

 34136 Figure 3.6 -The graphical representations of φ α,β and ψ α,β

1 =

 1 Id H αβ ⊗ εHβwhere the associativity of the product m β is used in the first equality, then we use the coassociativity of the coproduct in the second equality, and finally we use the antipode properties in the last equality. Similarly we have φ α,β •ψ α,β = Id Hα⊗H β . Next we prove the map φ α,β is H αβ -linear by diagrammic calculus:

Figure 3 . 7 -

 37 Figure 3.7 -The graphical representation of the map k

.5. 3 ) 3 . 5 . 1 .

 3351 Lemma For k ∈ N then

Lemma 4 . 2 . 10 .

 4210 The element R = ŘK is a topological universal R-matrix of the topological Hopf superalgebra U H .

Proposition 4 . 2 . 15 .

 4215 Let u be a central element of U H . If ϕ(u) = 0 then u = 0 where ϕ is Harish-Chandra homomorphism.

2 e σ 3 e p 1 w ρ 0 ,σ 0 ,p 0 =weight(e ρ 2 e σ 3 e p 1 )

 01

  w * ⊗ w → 1, -→ coev w : 1 → w ⊗ w *denote the duality morphisms. For each object w in T , let t w : w → w denote the positive full twist defined byt w = (w⊗ -→ ev w * )(c w,w ⊗ w * )(w⊗ -→ coev w ).

Figure 4 . 1 - 1 Figure 4 . 2 -

 41142 Figure 4.1 -The morphisms c↓,↓ , c -1 ↓,↓ , -→ ev ↓ , -→ coev ↓ , -→ ev ↑ , -→ coev ↑

Figure 4 . 3 -

 43 Figure 4.3 -The cases of crossings with upwards strings where R σ = α ⊗ β and (R σ ) -1 = α ⊗ β.

.3. 1 )

 1 Remark 4.3.1.

Definition 4 . 4 . 4 .

 444 A π-trace for a Hopf π-coalgebra H = {H α } α∈π is a family of C-linear forms tr = {tr α : H α → C} α∈π which verifiestr α (xy) = tr α (yx), tr α -1 (S α (x)) = tr α (x)for all α ∈ π and x, y ∈ H α .It is known that for each finite type Hopf π-coalgebra, there exists a family of linear forms called a family of the right π-integrals ([START_REF] Virelizier | Hopf group-coalgebra[END_REF]). Call (λ α ) α∈G the family of right G-integral for the finite type Hopf G-coalgebra U σ = {U α } α∈G . This means that the family of C-linear forms λ

Proposition 4 . 4 . 5 .

 445 This family determines a G-trace by proposition below. The family {tr α } α∈G above is a G-trace for the unimodular finite type Hopf G-coalgebra U σ = {U α } α∈G .

Lemma 4 . 4 . 11 .

 4411 Let β, γ ∈ (C/Z) 2 and let -→ α = α = β+γ. Assume f (h 1 , h 2 )is a -periodic entire function on L α . Then ∆(f ) is -periodic on L (β,γ) and∆ β,γ F -→ α (f ) = F (β,γ) (∆(f )) .

Figure 4 . 5 -

 45 Figure 4.5 -Second Kirby's move

Proposition 4 . 4 . 18 .

 4418 The linear form λ 0 : U 0 → C determined by

2 .Remark 4 . 4 . 19 .f ρ 3 f δ 2 e i 1 e ρ 3 e δ 2 k j 1 1 k j 2 2δ ρ 1 δ σ 1 δ i -1 δ ρ 1 δ σ 1 δ j 1 0 δ j 2 -2 δ m 0 Lemma 4 . 4 . 20 . 1 ( 3

 24419220442013 By Equation (4.4.1) we have the remark.For α = (α 1 , α 2 ) ∈ C/Z × C/Z then λ α (f i 1 σ m ) = ηξ (α 1 +α 2 ) δ i -1is a right G-integral for the Hopf G-coalgebra {U α } α∈G .By using Proposition 4.4.18 one gets the lemma. We haveλ 0 (θ 0 ) = λ 0 (θ -1 0 ) = {1} +1 (1 -ξ) -To prove Lemma 2.3.3 we need the lemma below. Set e max = e 2 e 3 e -1 1 , f max = f 2 f 3 f -1 1 . Lemma A.1.1.

e 2 e 3 e -1 1 w 1 , 1 , - 1 = ξ µ 1 +µ 2 +2 - 1 i=1[i][µ 1 + 1 -

 111111 4.5) one getse -1 1 w 1,1, -1 = ξ µ 1 +µ 2 +2 -1 i=1 [i][µ 1 + 1 -i]w 1,1,0and by(2.4.6) i] e 2 e 1 e 2 w 1,1,0

  Ω µ is a Kirby color of degree µ. Let (G, ×) and (Z, +) be two commutative groups. A k-linear ribbon category C is G-modular relative to X with modified dimension d and periodicity group Z if 1. the category C has a G-grading {C g } g∈G , 2. the group Z has a free realization {ε t } t∈Z in C 1 (where

	Definition 2.6.1 ([8]).

  α∈G be the associated cyclic traces. Then t is a right modified trace on H-pmod if and only if for all α, β ∈ G and for all f ∈ End H αβ (H α ⊗ H β )

	t αβ Hα⊗H β (f ) = t α Hα (tr r H β (f )).	(3.3.12)

have a graded version of Reduction Lemma [2, Lemma 3.2] Lemma 3.3.4. Let (H, g) be a finite type unimodular pivotal Hopf G-coalgebra and λ = (λ α ) α∈G ∈ α∈G H * α be a family of symmetric linear forms and t = (t α ) Similarly, t is a left modified trace on H-pmod if and only if for all f ∈ End H αβ (H α ⊗ H β )

  and C ⊂ H * is the convex hull of weights of elements of B. Proposition 4.2.6 implies that the product on U H is continuous but there does not seem to exist multiplicative seminorms on U H . By Proposition 4.2.3 we have U H ⊗U H

  .2.11) For each compact set K 2 ⊂ C 4 , there exists a compact set K ⊂ C 2 and a λ K 2 ∈ R such that ∀ x ∈ U H , we have

	Proposition 4.2.7.

  1 ⊗ h 2 and Hom Alg (SH ⊗ SH, C) Hom Alg (S(H × H), C)Hom V ect (H × H, C) (H × H) * H * × H * .This isomorphism allows that for (ϕ, ψ) ∈ H * × H * one has (ϕ, ψ)(h i ⊗ 1) = ϕ(h i ) and (ϕ, ψ)[START_REF] Abdesselam | Centre and representations of U q sl(2|1) at roots of unity[END_REF]

  1 (4.2.15) Let I + be a left ideal of U H generated by e 1 , e 2 and e 3 , set I = I + ∩ U h . We haveI = I + ∩ U h = I -∩ U h and U h = H(h 1 , h 2 ) I where I -is right ideal generated by f 1 , f 2 and f 3 .Hence, I is a two-sided ideal and the projection ϕ : U h → H(h 1 , h 2 ) is a homomorphism of algebras called the Harish-Chandra homomorphism. Let V µ be a simple highest weight U H -module with highest weight µ = (µ 1 , µ 2 ). Then for any z ∈ Z( U H ) and any

	Lemma 4.2.13. Proposition 4.2.14.

  .2.17)Proof ofProposition 4.2.11. Set z = S(θ) -θ, z is in the center of U H . Let a weight module V µ in C H of weight µ and v is a weight vector of V µ . The completion U H of U H is a topological ribbon superalgebra.

	By
	Proposition 4.2.14 and Equality (4.2.17) we have ϕ(z)(µ)v = zv = 0. It
	implies that ϕ(z)(µ) = 0. Furthermore ϕ(z) ∈ H(H * ), this deduces that
	ϕ(z) = 0, so z = 0 by Proposition 4.2.15, i.e. S(θ) = θ.
	Hence the results above give us the theorem.
	Theorem 4.2.17.

  h [i] is the column matrix h 1,i h 2,i for i = 1, ..., n. Recall also the formula for the universal R-matrix and its inverse in Equations (4.2.19) and (4.2.20). Let L be a link diagram consisting of n ordered circle components L 1 , ..., L n . Denote by lk = (lk ij ) the linking matrix of the link diagram L and set Q L

  linear maps (the antipode) such that 1. each H α is an algebra with product m α and unit element 1 α ∈ H α , 2. ε : H 1 → C and ∆ α,β : H αβ → H α ⊗ H β are algebra homomorphisms for all α, β ∈ π, 3. for any α ∈ π, m α

i=1 M 4 (C) is surjective. We give here an elementary proof.

and by [START_REF] Virelizier | Hopf group-coalgebra[END_REF]Corollary 3.7] H * α is free left module rank one over H α with basis {µ α } when the action is defined by (h µ α )(x) := µ α (xh) for h, x ∈ H α .

If µ α (xy) = µ α (g α xy) = µ α (xyg α ) = (yg α µ α )(x) = 0 for all x ∈ H α , then yg α µ α = 0. It follows thus y = 0. For the symmetrised left G-integral the proof is similar. Also note that the spaces of left and right G-integrals are not equal in general. We have a lemma. Lemma 3.2.8. The left G-integral for H can be chosen as µ l α (x) = µ α -1 (S α (x)) for any x ∈ H α .

Proof. By (3.1.1) we have

β to both sides of the last equality and S -1 β (1 β -1 ) = 1 β , we obtain that (Id H β ⊗µ α -1 • S α )∆ β,α (x) = (µ (βα) -1 • S βα )(x)1 β , i.e. µ α -1 • S α satisfies the definition of the left G-integral.

G-unibalanced Hopf algebras

Let H = ({H α } α∈G , ∆, ε, S) be a finite type Hopf G-coalgebra with right G-integral µ. We call a distinguished G-grouplike of H (see e.g [START_REF] Virelizier | Hopf group-coalgebra[END_REF]) or G-

Note that a 1 is the comodulus element of the Hopf algebra H 1 (see [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]). By multiplying (3.2.3) with a -1 and replacing x by a αβ x we have

where denote by µ β (a β ?) the linear map x → µ β (a β x) for x ∈ H β . This equality implies that µ β (a β ?) is a left G-integral for H, i.e.

This is another choice for left G-integral from right G-integral. This choice of the left G-integral is the same with the one in Lemma 3.2.8 by following proposition.

where

We now show that the completion U H with the element θ will be a ribbon Hopf superalgebra.

Proposition 4.2.11. The θ is a twist, i.e. the element θ satisfies

Equalities ( 1) and ( 2) follow from the definition of θ. To prove (3), we need the following lemmas. Let U h be the sub-superalgebra of U H of all elements commuting with h 1 , h

For each h k for k = 1, 2 we have

It implies that u commutes with h i if and only if the sum of the weights of u i and v j is zero.

we can separate the elements coming from the Cartan part from the rest. Second, we fix the Cartan parts of the elements at the cross points and then push the rest of the elements to the base point of strand (along the orientation of L j ), see illustration in The product of this part gives an element w j ∈ U σ for j = 1, ..., n. At each point of crossing (i, j) between the i-strand and j-strand of L, its Cartan part gives us the element

where ε ij = ±1 is the sign of the crossing (i, j). Hence the value of J b L can be written as a product of ξ Q L (h) and an element of U σ⊗n . This means that ξ -Q L (h) J b L ∈ U σ⊗n . By the definition of the J L one has

It implies that

We have thus

We denote by J L any elements

thanks to Theorem 4.3.4 and Lemma 4.4.10. Let θ 0 be the ribbon element of the small quantum group U 0 .

Lemma 4.4.14.

There exists a normalization of (λ α ) α∈G such that

Proof. The proof is thanks to Lemma 4.4.20.

Theorem 4.4.15.

is a topological invariant of the pairs (M, ω) where n is the number of components of the surgery link L.

Remark 4.4.16. Usual quantum surgery invariants are renormalized thanks to the signature. There is no need of renormalisation here thanks to Lemma 4.4.14.

We use a result on the equivalence of 3-manifolds obtained by surgery along a link to prove Theorem 4.4.15, that is the theorem below. Theorem 4.4.17 ([32]). Let M 1 and M 2 be oriented 3-manifolds and f : M 1 → M 2 be an orientation preserving diffeomorphism. Any two surgery presentations L 1 and L 2 of M 1 and M 2 , respectively can be connected by a sequence of handle-slides, blow-up moves and blow-down moves such that the induced diffeomorphism between

). This implies that e max f max w 0,0,0 = 0, i.e. e max f max = 0.

Second, one has e 2 e 3 e -1

where (a ij ) 1≤i,j≤2 is the Cartan matrix in Definition 2.3.1, then one can write

As the set of the functions {f st : s, t ∈ Z} is linearly independent then the set {k s 1 k t 2 e max f max : s, t ∈ Z} is free over C. Thus we have the second affirmation.

Proof of Lemma 2.3.3. We consider the superalgebra U = U ξ sl(2|1)/(e 1 , f 1 ) as the one generated by generators e i , f i , k i , k -1 i and the relations as in Definition 2.3.1 with additional relations

2 , e 3 f 2 + f 2 e 3 = ξ -1 e 1 k 2 . Define the length on generators by l(e i ) = l(f i ) = 1, l(k i ) = 0 for i = 1, 2 then the above relations imply that one can reorder the monomials in U up to elements of smaller length. This implies by induction on length that the set {e ρ 2 e σ 3 e p 1 k s 1 k t 2 f ρ 2 f σ 3 f p 1 ρ, σ, ρ , σ ∈ {0, 1}, p, p ∈ {0, 1, ..., -1}, s, t ∈ Z} is a generating set for U (see [START_REF] Concini | Representations of quantum groups at roots of 1[END_REF]).

To prove the linear independence of the vectors we consider the relation

where ρ, σ, ρ , σ ∈ {0, 1}, p, p ∈ {0, 1, ..., -1}, s, t ∈ Z. The sum in Equation (A.1.1) contains four blocs associated with (ρ, σ) and can rewrite LHS of (A.1.1) = x 0,0,p,s,t,ρ ,σ ,p e p 1 k s

As e 2 2 = e 2 3 = 0 then the three last blocs (A.1.3) -(A.1.5) are zero after the left multiplication at Equation (A.1.1) by e 2 e 3 and one gets x 0,0,p,s,t,ρ ,σ ,p e 2 e 3 e p 1 k s

By (e 2 e 3 )e 1 = e 1 (e 2 e 3 ) and e 1 = 0, using the left multiplication at Equation (A. 1.6) by e -1

we get

x 0,0,0,s,t,ρ ,σ ,p e 2 e 3 e -1

Using the right multiplication Equation (A.1.7) by f -1 1 one gets

x 0,0,0,s,t,ρ ,σ ,0 e 2 e 3 e -1

Now we write the left hand side of Equation (A.1.8) as the sum of four blocs.

LHS of (A.1.8) = x 0,0,0,s,t,0,0,0 e 2 e 3 e -1

x 0,0,0,s,t,0,0,0 e 2 e 3 e -1

By second statement of Lemma A.1.1 one deduces that x 0,0,0,s,t,0,0,0 = 0 for s, t ∈ Z. Now the left hand side of Equation (A.1.8) remains three blocs (A.1.10) -(A.1.12). Similarly, we deduce that x 0,0,0,s,t,1,0,0 = x 0,0,0,s,t,0,1,0 = x 0,0,0,s,t,1,1,0 = 0. Thus we see that from Equation (A.1.7) we get x 0,0,0,s,t,ρ ,σ ,0 = 0 for 0 ≤ ρ , σ ≤ 1, s, t ∈ Z. Repeating the calculations gives x 0,0,0,s,t,ρ ,σ ,p = 0 for 0

Applying similar calculations we get

A.2 Proof of Proposition 4.4.18

It is necessary to check λ 0 satisfies the condition (4.4.1), i.e.

for all x ∈ U 0 . We check Equation (A.2.1) for the elements in PBW basis. This equation holds true for all elements

is equal to η1. The left hand side of Equation (A.2.1) at w is computed as follows. First, one has

where c uv , c u v are the coefficients in C and the powers of e 1 , f 1 and k 1 are less then -1.

Then we have the decomposition

where the terms in the sum satisfy (i, ρ, δ, j, ρ , δ , j 1 , j 2 , m) = ( -1, 1, 1, -1, 1, 1, 0, -2, 0). By Equation (4.4.7) and k i = 1 for i = 1, 2 the decomposition above implies that

Thus the linear form λ 0 is a right integral of U 0 .

A.3 Proof of Lemma 4.4.20

Firstly, we represent the decomposition of θ -1 0 in a PBW basis of U 0 . By Equation (4.2.15) the ribbon element θ 0 of U 0 is determined by

In Equation (A.3.1) the terms are determined by (-1) mn {1} i (-{1}) ρ+δ (i) ξ !(ρ) ξ !(δ) ξ ! ξ i 1 j 2 +i 2 j 1 -2i 1 i 2 .

σ m e i 1 e ρ 3 e δ 2 k i 1 1 k j 1 2 σ ρ+δ ⊗ σ n f i

Since

2 )S 0 (k i 2 1 )S 0 (f δ 2 )S 0 (f ρ 3 )S 0 (f i 1 )S 0 (σ

where in the second equality we used