Sebastian Posur

CONSTRUCTING EQUIVARIANT VECTOR BUNDLES VIA THE BGG CORRESPONDENCE

Keywords: computing with equivariant modules References 2010 Mathematics Subject Classification. 14F05, 16E05, 16E20 BGG correspondence, equivariant vector bundle, Grothendieck group, cohomology table

We describe a strategy for the construction of finitely generated G-equivariant Z-graded modules M over the exterior algebra for a finite group G. By an equivariant version of the BGG correspondence, M defines an object F in the bounded derived category of G-equivariant coherent sheaves on projective space. We develop a necessary condition for F being isomorphic to a vector bundle that can be simply read off from the Hilbert series of M . Combining this necessary condition with the computation of finite excerpts of the cohomology table of F makes it possible to enlist a class of equivariant vector bundles on P 4 that we call strongly determined in the case where G is the alternating group on 5 points.

Contents

1. Introduction 2. Constructing equivariant vector bundles via the BGG correspondence 2.1. The BGG correspondence 2.2. Tate sequences and hypercohomology 2.3. The construction strategy 2.4. Cohomology tables of vector bundles 2.5. An isomorphism of Grothendieck groups 2.6. Equivariant BGG correspondence 2.7. Adaption of the construction strategy to the equivariant setup 3. Strongly determined A 5 -equivariant vector bundles Appendix A. Software packages Appendix B.

Introduction

Interesting problems concerning vector bundles on projective space have remained open for many years. For example, Hartshorne conjectures in [Har74] that there are no indecomposable vector bundles of rank 2 on P n C for n ě 7, which is equivalent to the conjecture that every codimension 2 nonsingular subvariety of P n C is a complete intersection for n ě 7. Furthermore, the only known rank 2 indecomposable vector bundle on P 4

C is the famous Horrocks-Mumford bundle [START_REF] Horrocks | A rank 2 vector bundle on P 4 with 15, 000 symmetries[END_REF] (up to twists by line bundles and pullbacks by finite maps [DS87]). It is unclear whether there exist more such vector bundles.

The purpose of this paper is to provide tools for the construction of G-equivariant vector bundles on projective space for a finite group G over a splitting field k. The main idea is to start with a triple of Z-graded irreducible k-representations V, T, U of G concentrated in degrees ´1, t, u P Z (where we assume t " u), and to consider G-equivariant E-linear graded maps of the form

p φ : E b k p n`1 ľ W b k T q ÝÑ E b k p n`1 ľ W b k U q,
where dim k pV q " n`1, E " Ź V is the graded exterior algebra of V , and W :" Hom k pV, kq the graded k-dual. Due to an equivariant version of the famous BGG correspondence, the module M " ker p φ corresponds to an object F in the bounded derived category of Gequivariant coherent sheaves on PV with some hypercohomology groups prescribed by T and U . The tools described in this paper will then help us with the decision whether F is isomorphic to a vector bundle.

The paper is organized as follows. Section 2 gives the theoretical background of our construction strategy. We start with a review of the classical BGG correspondence in Subsection 2.1 and show how it can be used for the computation of hypercohomology groups (Subsection 2.2). In Subsection 2.3 we state our construction strategy of objects F in D b pCoh PV q in the non-equivariant case, where Coh PV is the category of coherent sheaves on PV . The next two subsections are concerned with the question when F is isomorphic to a vector bundle. This question can in theory be decided by finding two particularly sparse columns in the cohomology table of F (Lemma 2.7), but since we cannot in practice compute the entire cohomology table for finding (or disproving the existence) of two such columns, we also give a necessary condition in Corollary 2.11 that can be simply read off from the Hilbert series of M " ker p φ (and thus is computationally quite cheap). Subsection 2.6 and Subsection 2.7 lift the BGG correspondence and our construction strategy to the G-equivariant setup.

In Section 3 we apply our tools in the case where G is the alternating group A 5 on 5 points and V is its unique irreducible representation of dimension 5. We construct all those G-equivariant vector bundles for which the corresponding map p φ is already determined (up to non-zero scalar) by its source and range. We call such vector bundles strongly determined. Interestingly, in the case G " A 5 , all such bundles turn out to have supernatural cohomology, a notion of importance in Boij-Söderberg theory [ES09]. We summarize our findings in Theorem 3.6.

All necessary computations for obtaining Theorem 3.6 were performed with Cap [GSP17], a software project for category theory implemented in GAP [START_REF]GAP -Groups, Algorithms, and Programming[END_REF]. We address our strategy to compute in a G-equivariant E-linear setup on the computer in Appendix B.

Notation. By a module we will always mean a left module. A graded module is understood as a Z-graded module over a Z-graded ring. Given a graded module M " À iPZ M i , its twist by d P Z is denoted by M pdq " À iPZ M d`i . Similarly, the twist by d of a coherent sheaf F (or more generally of a cochain complex of coherent sheaves) on projective space is denoted by Fpdq " F b Opdq. If we consider an object T in a triangulated category T , its translation by d is denoted by T rds. If we say a property holds for F up to translation and twist, e.g., being isomorphic to a vector bundle with Chern polynomial given by 1 `2h 2 `2h 3 , we mean that this property is true for at least one of the objects in tFpdqres | d, e P Zu.

Constructing equivariant vector bundles via the BGG correspondence

2.1. The BGG correspondence. Let k be a field, n P N 0 , and V be an pn `1qdimensional Z-graded k-vector space concentrated in degree ´1. We denote by W the graded k-dual of V which is necessarily concentrated in degree 1. We write E :" Ź V for the graded exterior algebra of V , and S :" SympW q for the graded symmetric algebra of W . Let E-grmod denote the abelian category of finitely generated graded E-modules. Its corresponding stable category E-grmod is the quotient category of E-grmod modulo those morphisms that factor over a projective object. Since E-grmod is a Frobenius category, i.e., it has enough projectives and injectives, and these two classes of objects coincide, E-grmod can be equipped with a triangulated structure [GM94, Theorem 2.9]. If we regard a graded E-module M as an object in E-grmod, we shall write M . Furthermore, we denote the category of coherent sheaves on projective space PV :" ProjpSq by Coh PV , and its bounded derived category by D b pCoh PV q.

Theorem 2.1 (BGG).

There is an equivalence of triangulated categories BGG : E-grmod " Ý Ñ D b pCoh PV q that maps kpiq to O PV piqr´is for all i P Z. Here, we regard k » E{ radpEq as a graded E-module concentrated in degree 0.

The BGG correspondence was first proved in [BGG78]. A more detailed construction and proof of its correctness can be found in [OSS11, Appendix].

Tate sequences and hypercohomology.

The BGG correspondence provides a great computational tool for sheaf cohomology (Theorem 2.5) that we are now going to explain. We will use the more general language of Frobenius categories and stable categories in the sense of [GM94] (since this language also applies to the equivariant setup discussed in Subsection 2.6), but it is safe to simply think of E-grmod and E-grmod. Definition 2.2. Let A be a Frobenius category and let K ' pAq denote its category of cochain complexes with cochain homotopy classes as morphisms. The category of Tate sequences TatepAq is defined as the full subcategory of K ' pAq generated by (possibly infinite) acyclic cochain complexes only having projective objects in each cohomological degree.

Note that TatepAq inherits a triangulated structure from K ' pAq. Moreover, sending a cochain complex in TatepAq to its 0-th syzygy object yields a functor Syz : TatepAq Ñ A mapping into the stable category of A. We also get a functor the other way around using the following construction.

Construction 2.3. Let A P A. Take a projective resolution P ' A and an injective resolution P ãÑ I ' . Joining both resolutions with the differential

P 0 A ãÑ I 0
(having cohomological degree 0) yields an acyclic cochain complex of projectives, thus a Tate sequence, denoted by TatepAq.

See [GM94, Chapter 5, 2.9.1] for the following folklore theorem.

Theorem 2.4. Let A be a Frobenius category. Then the two constructions Syz and Tate yield an equivalence of triangulated categories TatepAq » A. Now, let F P D b pCoh PV q correspond to M P E-grmod via the BGG correspondence. It is easy to see that M can be chosen as a reduced E-module, i.e., it does not contain a nonzero projective submodule. If we choose a minimal projective and injective resolution in Construction 2.3, then TatepM q has the property that tensoring with the E ´E bimodule k yields zero differentials in all cohomological degrees. We call such Tate sequences minimal and denote the minimal Tate sequence of F by TatepFq.

Theorem 2.5. Let ω E be the graded k-dual of E. The minimal Tate sequence of an object F P D b pCoh PV q is of the form

. . . ω E b k `Àn i"0 H i pFp´iqq ˘ωE b k `Àn i"0 H i pFp´i `1qq ˘. . . d 0
where the hypercohomology groups H i pFpjqq are regarded as graded k-vector spaces concentrated in degree j. Thus, from this cochain complex, we can read off the hypercohomology groups of F as the socles of the objects.

Proof where we use that elements in the socle of an E-module M correspond to E-module homomorphisms k Ñ M . Substituting Fpjq for F gives the result in the general case.

Remark 2.6. Using the natural isomorphism ω E » E b k Ź n`1 W we can rewrite the objects of the Tate sequence in Theorem 2.5 in terms of free E-modules:

ω E b k `n à i"0 H i pFp´iqq ˘» E b k `n`1 ľ W b k n à i"0 H i pFp´iqq ˘.
2.3. The construction strategy. The BGG correspondence makes it possible to construct objects in D b pCoh PV q with prescribed hypercohomology groups. Let T, U be finite dimensional Z-graded k-vector spaces concentrated in degrees t, u, respectively. Assume t " u. Let

φ : p n`1 ľ W b k T q ÝÑ E b k p n`1 ľ W b k U q
be a monomorphism of graded vector spaces. By the universal property of free modules, φ gives rise to a morphism of graded E-modules

p φ : E b k p n`1 ľ W b k T q ÝÑ E b k p n`1 ľ W b k U q
whose kernel is a reduced E-module (it does not contain a non-zero projective submodule). The natural monomorphism

kerp p φq ãÑ E b k p n`1 ľ W b k T q is an injective hull (since t " u). Thus, E b k p Ź n`1 W b k T q is the
first object in a minimal injective resolution of kerp p φq. From Theorem 2.5 and Remark 2.6 we conclude

(1)

H i ´BGGpkerp p φqqp´i `1q
¯" " T, for p´i `1q " t, 0, else, for all i P Z. Since p φ does not have to be the next step of a minimal injective resolution, we cannot give an interpretation for U in terms of hypercohomology groups as well. However, we will improve this situation when we pass to an equivariant setup in Subsection 2.7.

In conclusion, choosing a monomorphism φ : p

Ź n`1 W b k T q ÝÑ E b k p Ź n`1 W b k U q defines an
object in D b pCoh PV q with hypercohomology as shown above. But since our ultimate goal is the construction of vector bundles, we have to learn how to detect if BGGpkerp p φqq is isomorphic to a vector bundle.

2.4. Cohomology tables of vector bundles. Let F P D b pCoh PV q. To simplify notation, we set h i j :" dim k H i pFpjqq and refer to ´hi j ¯i,j as the cohomology table of F (even though the table contains the dimensions of hypercohomology groups). We adopt the convention of [START_REF] Eisenbud | Sheaf cohomology and free resolutions over exterior algebras[END_REF] and display a cohomology table as follows:

. . . . . . . . . . . . h 2 ´3 h 2 ´2 h 2 ´1 . . . . . . h 1 ´2 h 1 ´1 h 1 0 . . . . . . h 0 ´1 h 0 0 h 0 1 . . . . . . . . . . . .
From its cohomology table, lots of information about F may be deduced. For example, the following statements are well-known: ' F is isomorphic to a coherent sheaf if and only if there exists a u 0 P N such that h i j " 0 for all j ą u 0 and i " 0. ' F is isomorphic a vector bundle if and only if it is isomorphic to a coherent sheaf and there exists an l 0 P N such that h i j " 0 for all j ă ´l0 and i " n. Unfortunately, these statements assume that we know the whole cohomology table, whereas concrete computations usually give us only excerpts bounded to the left and right. Luckily, it is true that if such an excerpt of a cohomology table of F "looks like" the cohomology table of a vector bundle, then F is already a vector bundle. The following lemma makes this statement precise. To see this, let M denote the r-th syzygy object of TatepFq. Then

Lemma 2.7 (Vector bundle criterion). Let F P D b pCoh PV q. Then F is isomorphic to a vector bundle if and only if its cohomology table has two columns with only one non-zero entry such that the non-zero entry in the left column is in the n-th row and the non-zero entry in the right column is in the

M ãÑ TatepFq r Ñ TatepFq r`1 Ñ . . .
is a minimal injective resolution from whose socles we can read off the right part of the cohomology table. Since taking socles of graded E-modules is a functorial process equivalent to applying Hom E pk, ´q, we have socpTatepFq r`i q " Ext i E pk, M q for i ě 0. But this Ext-group can also be computed as a subquotient of Hom E pP i , M q, where P ' is a minimal projective resolution of k. Now, the dark gray range simply consists of those degrees in which the objects of Hom E pP ' , M q may be supported. By dualizing M , we have the following range for the non-zero entries to the left: . . .

An isomorphism of Grothendieck groups.

In this subsection we are going to work with Chern classes and so we assume that k is an algebraically closed field. We give a necessary condition for an object M P E-grmod to correspond (up to translation and twist) to a vector bundle in D b pCoh PV q that can be simply read off from the Hilbert series of M and thus is computationally much faster than computing (an excerpt of) the cohomology table. Our main tools are the Grothendieck groups of abelian and triangulated categories.

First, let A be an abelian category. An additive invariant for A consists of an abelian group H and a map α : ObjpAq Ñ H such that αpBq " αpAq `αpCq for all short exact sequences 0 Ñ A Ñ B Ñ C Ñ 0. The Grothendieck group K 0 pAq is defined as the universal additive invariant of A. It can be constructed as the abelian group freely generated by the the isomorphism classes 1 rAs for A P A modulo the relations rAs ´rBs rCs " 0 for all short exact sequences as above. Any additive invariant α uniquely factors over the universal map A Þ Ñ rAs.

Example 2.8. In the case A " Coh PV , it follows from the Koszul complex and the existence of resolutions by line bundles that (2)

K 0 pCoh PV q » Zrx, x ´1s{x n`1 ÿ i"0 p´1q i x ´iˆn i ˙yZrx,x ´1s » xx 0 , . . . , x n y Z ,
where the first isomorphism identifies the line bundles O PV piq with the indeterminates x i for i P Z. In this case, well-known additive invariants for Coh PV are the rank and the Chern polynomial of a coherent sheaf F. From the universal property of the Grothendieck group, they can be computed as follows: If rFs " ř iPZ a i rO PV piqs for finitely many non-zero a i P Z, then rankpFq "

ÿ iPZ a i P Z
1 To avoid set theoretic issues, we suppose that the isomorphism classes form a set.

and chernpFq "

ź iPZ p1 `ihq a i P pZrhs{xh n`1 yq ŵhere pZrhs{xh n`1 yq ˆdenotes the group of units of the ring Zrhs{xh n`1 y. Next, we turn to the Grothendieck group K 0 pTq of a triangulated category T. It is defined as the abelian group freely generated by the isomorphism classes rT s for T P T modulo the relations rSs ´rT s `rU s " 0 for every distinguished triangle S Ñ T Ñ U Ñ Sr1s. Since any short exact sequence in an abelian category A induces a distinguished triangle in D b pAq, we have an inclusion K 0 pAq ãÑ K 0 pD b pAqq that actually is an isomorphism with

K 0 pD b pAqq Ñ K 0 pAq : rC ' s Þ Ñ ÿ iPZ p´1q i rH i pC ' qs
as an inverse. In particular, we have an isomorphism K 0 pCoh PV q » K 0 pD b pCoh PV qq that allows us to speak about the rank and the Chern polynomial of an arbitrary object in D b pCoh PV q. Moreover, given a triangulated functor F : T Ñ U between triangulated categories, it functorially induces a well-defined group homomorphism K 0 pF q : K 0 pTq Ñ K 0 pUq : rT s Þ Ñ rF pT qs.

In particular, the BGG equivalence BGG : E-grmod " Ý Ñ D b pCoh PV q induces an isomorphism K 0 pE-grmodq » K 0 pD b pCoh PV qq which we are now going to investigate. First, the Grothendieck group of K 0 pE-grmodq is given as the quotient K 0 pE-grmodq{xrP s | P is projectivey. Such a description is actually valid for every stabilization of an exact category (see [Hap87, Corollary in Section 9.5]). Now, since every object in E-grmod admits a finite filtration with composition factors in tkpiq | i P Zu, the group K 0 pE-grmodq is generated by the classes rkpiqs. Moreover, there are no relations between these generators because rM s " ř iPZ dim k pM i qrkp´iqs encodes the Hilbert function of M P E-grmod, which is an additive invariant. Thus, we get an isomorphism

(3) K 0 pE-grmodq » Zrt, t ´1s{x n`1 ÿ i"0 t i ˆn i ˙yZrt,t ´1s
that identifies kp´iq with the indeterminate t i .

Lemma 2.9. Under the identifications (2) and (3), the isomorphism of Grothendieck groups K 0 pE-grmodq » K 0 pD b pCoh PV qq induced by the BGG correspondence is given by

Zrt, t ´1s{x n`1 ÿ i"0 t i ˆn i ˙yZrt,t ´1s ÝÑ Zrx, x ´1s{x n`1 ÿ i"0 p´1q i x ´iˆn i ˙yZrx,x ´1s t i Þ ÝÑ p´1q i x ´i
Proof. Via the BGG correspondence the object kp´iq is sent to O PV piqr´is. Since rT riss " p´1q i rT s P K 0 pTq for all T in a triangulated category T, the claim follows.

Corollary 2.10. Let M P E-grmod. Then rankpBGGpM qq " ÿ iPZ p´1q i dim k pM i q P Z and chernpBGGpM qq "

ź iPZ p1 `ihq p´1q i dim k pM ´iq P pZrhs{xh n`1 yq ˆ.
Proof. Immediate from Example 2.8 and Lemma 2.9.

We can now state and prove our necessary condition for being a vector bundle up to translation and twist that can be read off from the Hilbert series.

Corollary 2.11 (Necessary condition). Let M P E-grmod such that BGGpM q is isomorphic to a vector bundle up to translation and twist. Define r :" ř iPZ p´1q i dim k pM i q. We set e :" 1 if r ă 0, and e :" 0 otherwise. Then the degree of ź iPZ p1 `ihq p´1q i`e dim k pM ´iq " 1 `c1 h `c2 h 2 `¨¨¨`c n h n P pZrhs{xh n`1 yq îs lower or equal to |r|, where c 1 , . . . , c n P Z.

Proof. Let E be a vector bundle and j P Z such that BGGpM q -Erjs. Then rankpEq " |r| and chernpEq " chernpErj `esq " ś iPZ p1 `ihq p´1q i`e dim k pM ´iq by Corollary 2.10. Now, the claim follows from the fact that the degree of the Chern polynomial of a vector bundle is lower or equal to the rank. 2.6. Equivariant BGG correspondence. Let G be a finite group and k be a splitting field for G. Assume G acts linearly on V . Then this action extends to an action on E by Z-graded algebra automorphisms. Now, the BGG correspondence can be lifted to a Gequivariant setup as follows: let pE ¸Gq-grmod denote the category of finitely generated Gequivariant Z-graded E-modules, i.e., the objects are Z-graded k-vector spaces M equipped with a graded E-module structure and a G-action compatible with the grading such that the multiplication map E bM Ñ M is G-equivariant, and the morphisms are graded linear maps compatible with both the E-action and G-action. Then this category is a Frobenius category by the following lemma. Lemma 2.12.

(1) Every projective object in pE ¸Gq-grmod is isomorphic to an object of the form E b U for some Z-graded finite dimensional G-module U and conversely, every object of this form is projective. (2) The classes of projective and injective objects in pE ¸Gq-grmod coincide.

(3) Every object in pE ¸Gq-grmod has a projective and an injective resolution.

Proof. As a graded E-module, E b U is projective. Since Hom pE¸Gq-grmod pE b U, ´q can be computed as the composition of the exact functor Hom E-grmod pE b U, ´q and the exact functor of invariants, E b U is also projective as an object in pE ¸Gq-grmod.

Given any M P pE ¸Gq-grmod, the epimorphism π : M Ñ M { radpM q, where radpM q denotes the graded radical of M as an E-module, is a G-equivariant morphism. Since k is a splitting field of G, π admits a section on the level of graded G-modules, i.e., a graded G-module morphism σ : M { radpM q Ñ M such that π ˝σ " id. Then, the morphism E b impσq Ñ M : e b m Þ Ñ em is a projective cover of M in pE ¸Gq-grmod, since it is also a projective cover in E-grmod. If M is projective, then it is its own projective cover and therefore isomorphic to an object of the form E b U by the uniqueness of projective covers. Using the duality Hom k p´, kq, we get all the corresponding statements for injective modules. Since Hom k p´, kq respects the class of objects of the form E b U , projectives and injectives coincide.

By modding out the class of projective-injective objects, we can form the stable category E ¸G-grmod that equivalently may be described as TateppE ¸Gq-grmodq (see Theorem Remark 2.13. It is hard to find an explicit form of the equivariant BGG correspondence in the literature. The reason for this might be that it seems very likely that one can state and prove the BGG correspondence internal to an appropriate class of tensor categories without using any new argument, but simply by using the language of tensor categories. 2.7. Adaption of the construction strategy to the equivariant setup. Let T , U be finite dimensional Z-graded irreducible G-modules concentrated in degrees t, u, respectively. If t " u, then any monomorphism φ : p

Ź n`1 W b k T q ÝÑ E b k p Ź n`1 W b k U q of graded G-modules defines a morphism of G-equivariant graded E-modules p φ : E b k p n`1 ľ W b k T q ÝÑ E b k p n`1 ľ W b k U q
that fits into a minimal Tate sequence. In fact, we have already seen in Subsection 2.3 that the natural monomorphism kerp p φq ãÑ E b k p Ź n`1 W b k T q is the injective hull of a reduced E-module. To see that p φ fits into a minimal Tate sequence it suffices to show that cokerp p φq is a reduced E-module as well. But this follows from the fact that socpE b k p

Ź n`1 W b k U qq Ď imp p
φq which holds due to the irreducibility of U . So, in addition to equality (1), we also get (4)

H i ´BGGpkerp p φqqp´i `2q
¯" " U, for p´i `2q " u, 0, else, for all i P Z.

We summarize the context of our construction strategy in the following definition.

Definition 2.14. Let G be a finite group, k a splitting field for G, and let V, T, U be finite dimensional Z-graded irreducible G-modules concentrated in degrees ´1, t, u, respectively, such that t " u. We define the associated BGG space as the subset of the projective space

P `Hom kG-grmod pp n`1 ľ W b k T q, E b k p n`1 ľ W b k U qq consisting
of those points rφs such that φ is a monomorphism, where E :" Ź V and W :" Hom k pV, kq. We denote this subset by BGG V pT, U q.

Points in BGG V pT, U q give rise to objects in D b pCoh pPV ¸Gqq with hypercohomology groups given by the equations ( 1) and (4). Definition 2.15. Let BGG V pT, U q be a BGG space consisting of exactly one point rφs. Then we call the G-equivariant object BGGpkerp p φqq P D b pCoh pPV ¸Gqq strongly determined.

Remark 2.16. It is easy to see that BGG V pT, U q is a singleton if and only if

Hom kG-grmod `pn`1 ľ W b k T q, E b k p n`1 ľ W b k U q ȋs
of dimension 1, and this dimension can be computed as the scalar product of the dual character associated to p Ź n`1 W b k T q and the character associated to the t `n `1 degree part of

E b k p Ź n`1 W b k U q.

Strongly determined A 5 -equivariant vector bundles

In this section we work with k " C. Let A 5 " xp1 2 3 4 5q, p3 4 5qy ď S 5 denote the alternating group on 5 points. The goal of this section is to find all strongly determined A 5 -equivariant vector bundles on PV , where V is the unique irreducible A 5 -representation of degree 5 (and thus PV » P 4 ). We start by recalling the character table of A 5 : pq p1 2qp3 4q p1 2 3q p1 2 3 4 5q p1 2 3 5 4q

χ 1 1 1 1 1 1 χ 3 3 ´1 ¨σΦ Φ σχ 3 3 ´1 ¨Φ σΦ χ 4 4 ¨1 ´1 ´1 χ 5 5 1 ´1 ¨Ẅe have Φ " 1`?5
2 with its Galois conjugate σΦ " 1´?5 2 , where σ is the non-trivial element in the Galois group of QpΦq over Q. By abuse of notation we will use the same symbol for a character χ and a representation affording χ. If we want to consider a representation χ as a Z-graded object concentrated in (internal) degree d P Z, we decorate its symbol with an index: χ d . Furthermore, we will highlight the individual steps of our construction procedure by putting a box around them.

Step 1: Set V :" χ 5 ´1.

Note that the k-dual W :" HompV, kq " χ 5 1 . Furthermore, Ź 5 W " χ 1 5 . In particular, tensoring a graded representation with Ź 5 W simply shifts its degree by 5. We denote the set of irreducible characters of A 5 by X :" tχ 1 , χ 3 , σχ 3 , χ 4 , χ 5 u.

Step 2: Find all i P t´1, ´2, ´3, ´4u and ϕ, ψ P X such that BGG V pϕ i , ψ 0 q is a singleton. This step can be performed by using character theory (see Remark 2.16). The cases i ą 0 and i ă ´5 lead to empty BGG spaces. In the case i " ´5, every point φ P rBGG V pϕ i , ψ 0 qs gives us (see Subsection 2.7 for the definition of p φ)

kerp p φq -`E b k p 5 ľ W b k ψ ´5q ˘ă0
which is the first syzygy object in a resolution of Ź 5 W b k ψ ´5 -ψ 0 (regarded as a trivial E-module). Thus, the case i " ´5 will always give us the A 5 -equivariant vector bundle O PV b ψ 0 p´1q.

Step 3: For pi, ϕ, ψq found in Step 2, let rφs P BGG V pϕ i , ψ 0 q. Compute the morphism

p φ : E b k p Ź 5 W b k φ i q ÝÑ E b k p Ź 5 W b k ψ 0 q (see Subsection 2.7).
Determine the Hilbert series of kerp p φq and check the necessary condition described in Corollary 2.11.

If the necessary condition does not hold, then repeat Step 3 with the next triple pi, ϕ, ψq found in Step 2 (if there is any such triple left).

Remark 3.1. It is tempting to think that the computation of the Hilbert series of kerp p φq might work without the computation of the morphism p φ itself, but simply by looking at the characters of E b k p Ź 5 W b k φ i q and E b k p Ź 5 W b k ψ 0 q. For example, let φ i " χ 4

´2

and ψ 0 " χ 1 0 . We enlist the (internal) degree-wise decompositions into characters in the following table:

deg kerp p φq E b k p Ź 5 W b k φ i q E b k p Ź 5 W b k ψ 0 q ´2 χ 4 χ 4 ´1 χ 3 `σχ 3 `χ4 `2χ 5 χ 3 `σχ 3 `χ4 `2χ 5 0 2χ 3 `2σχ 3 `3χ 4 `3χ 5 χ 1 `2χ 3 `2σχ 3 `3χ 4 `3χ 5 χ 1 1 χ 1 `2χ 3 `2σχ 3 `3χ 4 `2χ 5 χ 1 `2χ 3 `2σχ 3 `3χ 4 `3χ 5 χ 5 2 2χ 5 χ 3 `σχ 3 `χ4 `2χ 5 χ 3 `σχ 3 `χ4 3 χ 4 χ 3 `σχ 3 `χ4 4 χ 5 5 χ 1
We can see that the second column is given by subtracting the forth column from the third column and keeping only those terms with positive coefficient. Interestingly, in the case that we are interested in (our group equals A 5 and V " χ 5 ´1), the decomposition of kerp p φq into characters can always be computed by such a subtraction (a fact that we could only check a posteriori, i.e., after having already computed kerp p φq). However, this is only a coincidence and not true in general. Counterexamples exist in the case where our group equals the symmetric group on 5 points, e.g., if χ denotes the irreducible character of degree 5 that maps the conjugacy class consisting of order 6 elements to 1, and if sgn denotes the signum, then V " φ i " χ ´1 and ψ 0 " psgn χq 0 provides a counterexample.

We give an example where the necessary condition checked in Step 3 does not hold.

Example 3.2. Let rφs P BGG V pχ 4 ´2, χ 1 0 q be the unique point and F :" BGGpkerp p φqq. Using the methods described in Subsection 2.5, we can compute rankpFq " 2, chernpFq " 1 ´h `3h 2 `5h 3 `10h 4 up to translation and twist. The necessary condition described in Corollary 2.11 does not hold and thus, F cannot correspond to a vector bundle up to translation and twist. A finite excerpt of its cohomology table looks as follows: If the necessary condition does hold, then we can proceed further.

Step 4: Find a finite excerpt of the cohomology table of BGGpkerp p φqq by computing a few steps of a minimal injective and projective resolution of kerp p φq. Check if the criterion described in Lemma 2.7 holds up to translation and twist.

If kerp p φq passed

Step 3 but did not correspond to a vector bundle up to translation and twist, then we would get stuck in Step 4 since we cannot disprove the existence of the two sparse columns required by Lemma 2.7 by only computing finite excerpts of the cohomology table. On the other hand, if kerp p φq does correspond to a vector bundle, it is furthermore unclear how many steps are actually needed in the resolutions for finding two sparse columns. Nevertheless, if Step 4 succeeds, then we have found a strongly determined A 5 -equivariant vector bundle on PV . In order to find all such bundles, we have to go back to Step 3 and proceed with the next triple pi, φ, ψq (if there is any such triple left).

Luckily, in the case that we are examining (our group equals A 5 and V " χ 5 ´1), every kerp p φq that passes Step 3 turns out to be a vector bundle.

Example 3.3. We discuss a case where the vector bundle criterion does hold. Let rφs P BGG V pχ 3 ´2, χ 1 0 q be the unique point and F :" BGGpkerp p φqq. We compute rankpFq " 3, chernpFq " 1 `2h 2 `2h 3 up to translation and twist. The necessary condition described in Corollary 2.11 holds, and so we compute a finite excerpt of the cohomology table: The criterion of Lemma 2.7 also holds, thus, F is (up to translation and twist) a strongly determined A 5 -equivariant vector bundle.

For stating the main theorem we need the following definitions from Boij-Söderberg theory, in which vector bundles with supernatural cohomology play a crucial role (for a survey of Boij-Söderberg theory, see [Flø12]). Definition 3.4. For s P N 0 , a root sequence of length s is a sequence of strictly decreasing integers z : z 1 ą z 2 ą ¨¨¨ą z s .

For our convenience, we will expand every such sequence by z i :" 8 for i ă 1 and z i :" ´8 for i ą s. We set its associated Hilbert polynomial as

HP z ptq :" 1 s! s ź i"1
pt ´zi q and its associated cohomology table as γ z pp, qq :" " |HP z ppq|, for z q ą p ą z q`1 , 0, else.

Definition 3.5. Let z be a root sequence. A vector bundle F P Coh P n has supernatural cohomology of type z if the cohomology table of F is given by H q pFppqq " rankpFq ¨γz pp, qq. Now, we summarize the findings of our computations that were performed using the software packages described in the Appendix A in the following theorem. Theorem 3.6. Let V be the unique irreducible A 5 -representation of dimension 5. Then all strongly determined A 5 -equivariant vector bundles on PV are supernatural. All i P t´1, ´2, ´3, ´4u, and φ, ψ P tχ 1 , χ3 , σχ 3 , χ 4 , χ 5 u such that BGG V pφ i , ψ 0 q is a singleton whose element corresponds to a vector bundle F (up to translation and twist) can be found in the following table, together with the type z of F:

ψ i φ z rank χ 1 ´4 χ 5 r0, ´2, ´3, ´4s 4
´2 χ 3 r0, ´1, ´3, ´6s 3 ´2 σχ 3 r0, ´1, ´3, ´6s 3 ´1 χ 5 r0, ´1, ´2, ´3s 1 χ 3 ´3 χ 3 r0, ´2, ´3, ´5s 9 ´3 σχ 3 r0, ´2, ´3, ´5s 9 ´2 χ 1 r0, ´3, ´5, ´6s 3 ´1 χ 3 r0, ´1, ´4, ´5s 6 ´1 σχ 3 r0, ´1, ´4, ´5s 6 ´1 χ 5 r0, ´1, ´3, ´6s 3 σχ 3 ´3 χ 3 r0, ´2, ´3, ´5s 9 ´3 σχ 3 r0, ´2, ´3, ´5s 9 ´2 χ 1 r0, ´3, ´5, ´6s 3 ´1 χ 3 r0, ´1, ´4, ´5s 6 ´1 σχ 3 r0, ´1, ´4, ´5s 6 ´1 χ 5 r0, ´1, ´3, ´6s 3 χ 4 ´1 χ 4 r0, ´1, ´4, ´5s 8 χ 5 ´4 χ 1 r0, ´1, ´2, ´4s 4 ´1 χ 1 r0, ´1, ´2, ´3s 1 ´1 χ 3 r0, ´3, ´5, ´6s 3 ´1 σχ 3 r0, ´3, ´5, ´6s 3

Remark 3.7. The Horrocks-Mumford bundle [START_REF] Horrocks | A rank 2 vector bundle on P 4 with 15, 000 symmetries[END_REF] is also a strongly determined equivariant vector bundle for its group of symmetries of order 15000, but it does not have supernatural cohomology.

Appendix A. Software packages

The main tool for our computations is the Cap project (Categories, Algorithms, Programming), which is a collection of software packages2 written in GAP. The Cap project supports the programmer in the implementation of specific instances of categories, like in our case the category pE ¸Gq-grmod of finitely generated G-equivariant Z-graded modules over the exterior algebra E.

The GAP packages which enable us to work with the category pE ¸Gq-grmod are called InternalExteriorAlgebraForCAP and GroupRepresentationsForCAP, and they are freely available on GitHub. commute. If C is the category of sets, then M is a classical monoid with multiplication map µ and unit given by the unique element in the image of η. The strength of this definition lies in the fact that we may vary C. For example, an (internal) monoid in the category of topological spaces is a topological monoid. An (internal) monoid in the category of groups is an abelian group.

The diagrammatic definition of an (internal) monoid even makes sense in contexts where we do not necessarily have products, but simply a bifunctor b : CˆC Ñ C, a distinguished object 1 P C, associator isomorphisms AbpB bCq » pAbBqbC and unitor isomorphisms 1 b A » A b 1 » A natural in A, B, C P C satisfying certain relations. Such categories are called monoidal categories (see [ML98] for a proper definition). For example, Rep k pGq together with the usual tensor product b k of representations is a monoidal category. An (internal) monoid in pRep k pGq, b k q is the same as a finite dimensional G-equivariant algebra (like our exterior algebra E " Ź V in Subsection 2.6). Similar to the concept of a monoid, we can also internalize the concept of a left action of a monoid pM, µ, ηq in any monoidal category pC, bq: an (internal) left action is simply an object V P C equipped with a morphism µ V : M b V Ñ M (encoding the action of M on V from the left) satisfying the obvious conditions. Again, interpreted in the category of sets, this is nothing but the usual left action of a monoid on a set. But interpreted in pRep k pGq, b k q with given (internal) monoid E, the category of (internal) left actions of E is nothing but pE ¸Gq-mod. Now, the equivalence (5) can be used to transfer the monoidal structure b k of Rep k pGq to a monoidal structure b 1 of À χPIrrpGq k-vec, as well as to transfer the (internal) monoid E to an (internal) monoid E 1 . The upshot is: the category of (internal) left actions of E 1 is still equivalent to pE ¸Gq-mod, but it is now built upon the concise data structure that represents G-equivariant maps as lists of small matrices. Thus, we have finally found our effective model in which we can compute with G-equivariant E-modules.

One last remark concerning the transfer of the monoidal structure from Rep k pGq to À χPIrrpGq k-vec: the computationally hardest part is the transfer of the associator isomorphisms. In our Cap implementation it involves Dixon's method [START_REF] Dixon | Constructing representations of finite groups, Groups and computation[END_REF] for constructing all irreducible representations of a group G as well as the computation of explicit decomposition isomorphisms of all pairs of tensor products between irreducible representations. Luckily, for any given group G, such an effort has only to be done once. We simply can store the resulting associators and reuse them in future sessions (this applies in particular to the associators that we computed for G " A 5 in order to perform the computations for this paper).

  2.4). It is shown in [Flø00, Theorem 9.1.2] that D b pCoh pPV ¸Gqq » TateppE ¸Gq-grmodq where D b pCoh pPV ¸Gqq is the bounded derived category of G-equivariant coherent sheaves on PV . Combining this result with TateppE ¸Gq-grmodq » E ¸G-grmod, we get our desired G-equivariant BGG correspondence D b pCoh pPV ¸Gqq » E ¸G-grmod.

3

  Up till now an installation of Magma[START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] is also needed for a proper usage of our packages, since Magma provides fast linear algebra over number fields. It is planned to resolve this dependency by switching to open software alternatives like Nemo[FHHJ15].

  0-th row. " 0, then the non-zero entries to its right can only lie in the dark gray range indicated in the following picture:

	. . .	. . .	. . .
	0	0	0	¨¨ḧ
		?	?	¨¨0
		?	?	¨¨.
	. .	. . .	. . .	n	`1
	0	?	?	¨¨0
		0	0	¨¨.
	. .	. . .	. . .
	. . .	. . .		. . .	. . .
	0	?	. . .	?	0
	h n l	?	. . .	?	0
	0	?	. . .	?	0
	. . .	. . .		. . .	. . .
	0	?	. . .	?	0
	0	?	. . .	?	h 0 r
	0	?	. . .	?	0
	. . .	. . .		. . .	. . .

Proof. We only have to show the "if" direction, which can be deduced from [EFS03, Lemma 7.4]. The reasoning is as follows. Given a cohomology table having an r-th column with only one non-zero entry h

  Example 7.3], Eisenbud, Fløystad, and Schreyer use this criterion to prove that a particular object in E-grmod gives rise to the famous Horrocks-Mumford bundle.

	. . .	. . .	. . .		. . .	. . .	. . .
	¨¨¨0	0	0	¨¨¨0	0	0	¨¨?
		h n l	?	¨¨¨?	0	0	¨¨0
		0	?	¨¨¨?	0	0	¨¨.
	. .	. . .	. . .		. . .	. . .	. . .
	¨¨¨0	0	?	¨¨¨?	0	0	¨¨0
		0	?	¨¨¨?	h 0 r	?	¨¨0
		0	0	¨¨¨0	0	0	¨¨.
	. .	. . .	. . .		. . .	. . .	. . .
	Thus, F corresponds to a vector bundle.
	In [EFS03,
					. .	. . .
			¨¨¨0	0	0
			¨¨¨?	?	0
				. . .	. . .	. . .
			¨¨¨?	?	0
			¨¨¨?	?	h
			¨¨¨0	0	0
				. . .	. . .	. . .

n `1

Applying these two restrictions to the two columns of our given cohomology table, we can conclude that the only non-zero entries lie in the following range:

Cap's core system is part of the current GAP release (as of October 2017).

They are part of the CAP_project repository: https://github.com/sebastianpos/CAP_project.

This work was supported by Project 'Derived categories of equivariant coherent sheaves' of SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German Research Foundation (DFG).

Appendix B. Sketch: computing with equivariant modules

We briefly explain how computations within the category pE ¸Gq-grmod become feasible on the computer (see [START_REF] Posur | Constructive category theory and applications to equivariant sheaves[END_REF] for a detailed explanation). To simplify our explanation, we will drop the Z-grading, but it is straightforward to add it to our setup in the end. The main ideas for computing in pE ¸Gq-mod are

(1) to reduce all computations to linear algebra, (2) to keep the resulting matrices small by exploiting G-equivariance. The first idea is easy to implement since every finitely generated G-equivariant E-module is in particular a finite dimensional vector space. But the resulting matrices describing the Gequivariant and E-linear operators within injective/projective resolutions quickly become very big. A reduction of this growth can be achieved by exploiting the G-equivariance using category theory in a way that we are now going to explain.

If our ground field k is a splitting field for our finite group G, then, due to Maschke's theorem, the category of finite dimensional k-representations of G (denoted by Rep k pGq) is equivalent as an abelian category to finitely many copies of the category of finite dimensional k-vector spaces (denoted by k-vec), where the copies are indexed by the irreducible representations IrrpGq of G:

A morphism α between two representations V and W in Rep k pGq is modeled by a matrix of dimension dimpV q ˆdimpW q which, under the equivalence (5), is transformed into an IrrpGq-indexed list of matrices whose χ-th entry is an m χ ˆnχ matrix, where m χ and n χ are the multiplicities of χ in V and W , respectively. Since this list of matrices may contain much less data then our original matrix α, we prefer computing in À χPIrrpGq k-vec to computing in Rep k pGq. But how can we model the category pE ¸Gq-grmod in terms of À χPIrrpGq k-vec? The answer lies in the concept of internalization. Roughly speaking, internalization means reformulating a (maybe classical set-theoretic) concept in the language of categories equipped with some extra structure. For example, we may speak about (internal) monoids in any category C with finite products. Such an (internal) monoid is then defined as an object M P C equipped with morphisms µ : M ˆM Ñ M and η : 1 Ñ M (where 1 denotes the terminal object in C, which is given by the empty product) such that the diagrams


Ahlem Bouhlel

Mohamed Bejaoui

Najet Ismail Ben Mosbah

Hadj Abdallah

Catherine Ribault

Roselyne Viel

Hassen Hentati

Anne Corlu

Ben Abdennebi Hassen

Abdennebi

Bouhlel Ahlem

Bejaoui Mohamed

Ben Mosbah Ismail

Abdallah Najet

Ribault Catherine

Viel Roselyne

Hentati Hassen

Thymoquinone protects rat liver after partial hepatectomy under ischemia/reperfusion through oxidative stress and endoplasmic reticulum stress prevention

Keywords: Ischemia reperfusion injury, hepatectomy, thymoquinone, oxidative stress, endoplasmic reticulum stress, apoptosis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Partial hepatectomy (PH) is considered as the operation of choice for the treatment of many liver malignancies [START_REF] Parkin | Cancer burden in the year 2000. The global picture[END_REF][START_REF] Kanematsu | Limited hepatic resection effective for selected cirrhotic patients with primary liver cancer[END_REF] and for living donor liver transplantation [START_REF] Cherqui | Laparoscopic living donor hepatectomy for liver transplantation in children[END_REF]. However, bleeding remains a major prognostic factor in liver resection. Complete clamping of the hepatic inflow at the hepatoduodenal ligament (the Pringle maneuver) is a safe strategy to alleviate this problem [START_REF] Kim | Ischemia-reperfusion injury of the human liver during hepatic resection[END_REF]. This maneuver, although effective in preventing excessive blood loss, is complicated by ischemia/reperfusion (I/R) injury, which compromises functional recovery of the remnant liver [START_REF] Kim | Ischemia-reperfusion injury of the human liver during hepatic resection[END_REF][START_REF] Serracino-Inglott | Hepatic ischemia-reperfusion injury[END_REF]. Clinically, liver I/R injury results in elevated liver transaminases, bile duct stricture and even liver failure [START_REF] Kim | Ischemia-reperfusion injury of the human liver during hepatic resection[END_REF][START_REF] Serracino-Inglott | Hepatic ischemia-reperfusion injury[END_REF][START_REF] Theodoraki | Beyond Preconditioning: Postconditioning as an Alternative Technique in the Prevention of Liver Ischemia-Reperfusion Injury[END_REF]. Furthermore, other organs can develop dysfunction secondary to the liver damage such as lungs, heart and kidneys [START_REF] Nastos | Global consequences of liver ischemia/reperfusion injury[END_REF]. Besides I/R, the surgical resection itself activates various complex cell signaling cascades of which many sides are still unresolved [START_REF] Michalopoulos | Liver regeneration[END_REF][START_REF] Fausto | Liver regeneration[END_REF]. For example, reactive oxygen species (ROS) are involved in the modulation of several signaling pathways that may influence liver regeneration after PH [START_REF] Forman | Redox signaling. In. Oxygen/Nitrogen Radicals: Cell Injury and Disease[END_REF][START_REF] Finkel | Oxidants, oxidative stress and the biology of ageing[END_REF]. Also, mitochondria have been shown to be an important player of liver regeneration [START_REF] Alexandrino | Mitochondrial bioenergetics and posthepatectomy liver dysfunction[END_REF]. Actually, high energy demand on the hepatocytes is required during the regenerative process after PH and this energy is provided by mitochondria through oxidative phosphorylation. In return, mitochondria release a large amount of ROS [START_REF] Guerrieri | Correlation between rat liver regeneration and mitochondrial energy metabolism[END_REF][START_REF] Mann | Human liver regeneration: hepatic energy economy is less efficient when the organ is diseased[END_REF] which activate proteins that inhibit the cell cycle [START_REF] Barnouin | H2O2 induces a transient multi-phase cell cycle arrest in mouse fibroblasts through modulating cyclin D and p21Cip1 expression[END_REF][START_REF] Mendelson | Independent regulation of JNK/p38 mitogen-activated protein kinases by metabolic oxidative stress in the liver[END_REF]. In addition, oxidative stress causes damage to cell components including proteins, lipids, and DNA resulting in cellular disorders and further organelle malfunction [START_REF] Moustafa | Effect of fenofibrate on the experimentally induced hepatic ischemia/reperfusion injury in rats: biochemical, light, and electron microscopic studies[END_REF].

Endoplasmic reticulum (ER) stress is triggered in several pathological events such as hypoxia, glucose deprivation and oxidative stress. ER homeostasis disruption leads to the accumulation of unfolded and misfolded proteins in the ER lumen. As a consequence, unfolded protein response (UPR) is activated in order to resolve this protein-folding defect and so to restore ER homeostasis [START_REF] Wang | Saturated fatty acids promote endoplasmic reticulum stress and liver injury in rats with hepatic steatosis[END_REF]. However, if the UPR is insufficient to alleviate the stress, caspase

Accepted Article

. dependent and independent cell death occurs [START_REF]Translational control in the endoplasmic reticulum stress response[END_REF]. Indeed, ER stress activates caspase 12 and mitochondrial apoptosis pathway [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF][START_REF] Mosbah | Endoplasmic reticulum stress inhibition protects steatotic and non-steatotic livers in partial hepatectomy under ischemia-reperfusion[END_REF]. ER stress also upregulated glycogen synthase kinase-3β (GSK3β) and voltage-dependent anion channel (VDAC) proteins which initiate the efflux of cytochrome C from the outer mitochondrial membrane, which in turn, activates the pro-apoptotic proteins caspase 9 and its effectors caspase 3 [START_REF] Panzarini | Timing the multiple cell death pathways initiated by Rose Bengal acetate photodynamic therapy[END_REF]. Thymoquinone (TQ) (2-isopropyl-5-methylbenzo-1, 4-quinone), is the major and the main active constituent of the essential oil of Nigella sativa seeds [START_REF] Saleh | Levels of selenium, dl-α-tocopherol, dl-γ-tocopherol, alltransretinol, thymoquinone and thymol in different brands of Nigella sativa seeds[END_REF]. TQ has several biological activities, especially antioxidant and free radical scavenging capacity [START_REF] Mansour | Effects of volatile oil constituents of Nigella sativa on carbon tetrachloride-induced hepatotoxicity in mice: evidence for antioxidant effects of thymoquinone[END_REF][START_REF] Osama | Thymoquinone Is a Potent Superoxide Anion Scavenger[END_REF]. Recently, it has been shown that TQ protects rat liver against I/R injury [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF]. However, its effect on liver subjected to resection under I/R is still unknown. Therefore, this study aimed to evaluate the effect of TQ administration in rat before a 2/3 hepatectomy (PH).

Results

In order to explore liver injury, we evaluated whether TQ pretreatment could preserve liver architecture and hepatocyte proliferation capacity after PH (Figure 1). As expected, a disorganized hepatic lobule aspect with focal necrosis was observed in rat livers subjected to PH under I/R. TQ treatment markedly attenuated the hepatic damage when compared to PH group (2.3 ± 0.3 vs. 3.9 ± 0.4, P < 0.001). We noted a relatively preserved hepatic architecture with less vacuolization and nuclear picnosis, and few zones of necrosis were detected.

Consistent with these observations, we found that ALT activity in serum of PH+TQ group was significantly decreased when compared to PH group. Values reached 161 ± 42 and 74 ± 46 U/L for PH and TQ+PH groups, respectively (P < 0.01). We noted also that TQ treatment in sham operated rats (TQ group) did not result in any modification in ALT activity in comparison to sham group. In addition, hepatocyte proliferation, assessed 24 hours after PH by quantitation of the number of Ki-67-positive hepatocytes, was greater in TQ+PH group

Accepted Article

. than in PH group (7.7 ± 5.8 vs. 16.0 ± 3.6 %, P < 0.01). Thus, we could assume that the administration of TQ decreased hepatic damage and promoted liver regeneration.

It is well known that PH is associated with ROS production. We therefore examined the possible involvement of TQ in stimulating antioxidant systems. As shown in figure 2, PH under I/R resulted in a significant reduction in the antioxidant enzyme activities and increase in oxidative stress parameters. Interestingly, when compared to PH group, rats pre-treated with TQ had significantly increased activity of glutathione peroxidase (GPX) (271 ± 43 vs. 209 ± 21 µmol oxided GSH/min/mg prot, P < 0.05), superoxide dismutase (SOD) (5.8 ± 0.2 vs. 5.0 ± 0.4 U/µg prot, P < 0.05) and catalase (CAT) (140 ± 18 vs. 96 ± 18 µmol H2O2/min/mg prot, P < 0.01), sulfhydryl proteins (SHP) (10 ± 1 vs. 6 ± 3 µg/mg prot, P < 0.05 ) whereas they had reduced malondialdehyde (MDA) (0.37 ± 0.08 vs. 0.70 ± 0.15 nmol/mg prot, P < 0.05) and conjugated dienes (CD) (0.04 ± 0.02 vs. 0.07 ± 0.0.02 nmol/mg prot, P < 0.05) concentrations. TQ treatment without PH did not result in any modification of oxidative stress parameters.

Given the central role for mitochondria to supply energy in cell, we assessed ATP concentration in livers (Figure 3). Our data showed that livers following ischemia and PH had a significant decrease in ATP content compared to livers from sham rats (0.09 ± 0.02 vs. 0.71 ± 0.10 µmol/mg prot, P < 0.001). However, pre-treatment with TQ enabled to preserve ATP content in comparison to PH group (0.19 ± 0.07 µmol/mg prot, P < 0.05).

In order to evaluate whether the hepatoprotective effect of TQ could be related to ER stress prevention, protein relative expression of activating transcription factor-4 (ATF4), activating transcription factor-6 (ATF6) and X-box-binding protein-1 (XBP1) and protein concentration of glucose regulated protein 78 (GRP78) and C/EBP homologous protein (CHOP) were assessed (Figure 4). As expected, ischemia and PH markedly intensified the activation of all

Accepted Article

. ER stress proteins when compared to sham group. As regarded to PH group, TQ pretreatment induced a significant drop in the expression of ATF4 (P < 0.01), ATF6 (P < 0.01) and XBP1 (P < 0.01) proteins and in the concentration of GRP 78 (519 ± 229 vs. 1143 ± 350 µg/mL, P < 0.001) and CHOP (280 ± 113 vs. 490 ± 60 µg/mL, P < 0.001).

Given that ER stress and mitochondrial dysfunction promote cell apoptosis [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF][START_REF] Cao | Mechanism of Hepatocyte Apoptosis[END_REF][START_REF] Weng | Alleviation of Ischemia-Reperfusion Injury in Liver Steatosis by Augmenter of Liver Regeneration Is Attributed to Antioxidation and Preservation of Mitochondria[END_REF], we explored apoptosis proteins after PH under I/R (Figure 5). In concordance with the previous results, ischemia and PH increased apoptosis when compared to sham operated rats.

Importantly, the apoptosis effectors were down-regulated by pretreatment with TQ. We found 289 ± 80 vs. 832 ± 327 ng/mL (P < 0.001), 4.23 ± 1.29 vs. 9.57 ± 1.58 (P < 0.001), 4.76 ± 0.84 vs. 9.54 ± 1.98 (P < 0.001) and 7.36 ± 1.12 vs. 16.37 ± 2.35 pg/mg prot (P < 0.001) for cytochrome c, caspase-9, caspase-12 and caspase-3 activities, respectively.

Discussion

This study demonstrates for the first time that TQ protects rat liver from injury caused by hepatectomy under I/R and promoted hepatic regeneration through the induction of antioxidant defense capacity, the prevention of oxidative stress and the attenuation of ER stress, mitochondrial damage and apoptosis.

It was previously reported that TQ has many pharmacological actions, including antioxidant and anti-inflammatory effects against several hepatotoxic molecules (28-31). Moreover, it has been shown that TQ suppresses liver fibrosis [START_REF] Oguz | Protective effects of thymoquinone against cholestatic oxidative stress and hepatic damage after biliary obstruction in rats[END_REF][START_REF] Bai | Thymoquinone attenuates liver fibrosis via PI3K and TLR4 signaling pathways in activated hepatic stellate cells[END_REF] and attenuates liver injury induced by ischemia [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF][START_REF] El-Ghany | Thymoquinone triggers anti-apoptotic signaling targeting death ligand and apoptotic regulators in a model of hepatic ischemia reperfusion injury[END_REF]. Nevertheless, TQ effects have not been examined in an experimental model combining PH and I/R. Partial hepatectomy under IR increased liver injury as evidenced by ALT release in serum.

Besides, histopathological evaluation of the livers has confirmed liver damage. In contrast, rats pretreated with TQ before PH showed a reduction in liver injury and an improvement of

Accepted Article

. the histopathological abnormalities. These findings consolidate previous reported works showing that TQ protects rat liver against IR injury [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF][START_REF] El-Ghany | Thymoquinone triggers anti-apoptotic signaling targeting death ligand and apoptotic regulators in a model of hepatic ischemia reperfusion injury[END_REF]. In line with the decrease of liver injury, we found that TQ promoted liver regeneration. This is consistent with previous reports showing that TQ had healing effects in bone [START_REF] Kirui | Effects of sustained delivery of thymoqiunone on bone healing of male rats[END_REF] and wound.

Thus far, the hepatoprotective effect of TQ is largely ascribed to its antioxidative properties.

Imbalance between oxidants and antioxidants is commonly termed as oxidative stress. In response to this state, cells react by their antioxidant defense machinery including both nonenzymatic and enzymatic antioxidants [START_REF] Kleikers | NADPH oxidases as a source of oxidative stress and molecular target in ischemia/reperfusion injury[END_REF]. In our study, we found that PH under IR led to a decreased of SOD, GPx and CAT activities, of SHP concentration while it was accompanied by an increased MDA and CD contents. These results are in line with those of previous studies suggesting that free radical damage occurs in the early phase of liver regeneration [START_REF] Steer | Liver regeneration[END_REF] resulting in lipid peroxidation [START_REF] Aguilar-Delfin | Selective enhancement of lipid peroxidation in plasma membrane in two experimental models of liver regeneration: partial hepatectomy and acute CCl4 administration[END_REF] and mitochondrial glutathione depression [START_REF] Vendemiale | Mitochondrial oxidative phosphorylation and intracellular glutathione compartmentation during rat liver regeneration[END_REF]. Furthermore, we noticed that TQ attenuated PH injury by preserving the anti-oxidant enzyme activities and enhancing the level of SHP. As a result, lipid peroxidation evidenced by both MDA and CD concentrations were lessened. In fact, the antioxidant effect of TQ has been assigned to its ability to scavenge free radical [START_REF] Khalife | Nonenzymatic reduction of thymoquinone in physiological conditions[END_REF][START_REF] Khattab | Thymoquinone supplementation attenuates hypertension and renal damage in nitric oxide deficient hypertensive rats[END_REF] and to up-regulate antioxidant gene expression [START_REF] Ismail | Nigella sativa thymoquinone-rich fraction greatly improves plasma antioxidant capacity and expression of antioxidant genes in hypercholesterolemic rats[END_REF][START_REF] Sayed-Ahmed | Thymoquinone attenuates diethylnitrosamine induction of hepatic carcinogenesis through antioxidant signaling[END_REF]. Moreover, the action of TQ could be promoted by its unrestricted crossing of membrane barriers to access subcellular compartments and thus to protect them [START_REF] Osama | Thymoquinone Is a Potent Superoxide Anion Scavenger[END_REF].

Further alterations touching cell organelles, including mitochondria, could be generated as a result of excessive ROS production. Previous data have hypothesized that oxidative stress induced by PH under IR impairs mitochondrial function and alters cell energy metabolism [START_REF] Steer | Liver regeneration[END_REF][START_REF] Vendemiale | Mitochondrial oxidative phosphorylation and intracellular glutathione compartmentation during rat liver regeneration[END_REF]. Moreover, hepatic regeneration could be affected by the change in mitochondrial energy production [START_REF] Horimoto | Uncoupling protein-2 deficiency promotes oxidant stress and delays liver regeneration in mice[END_REF][START_REF] Beyer | The cytoprotective Nrf2 transcription factor controls insulin receptor signaling in the regenerating liver[END_REF]. As previously reported, our study confirms the fact that PH under I/R decreases ATP production. In contrast, pre-treatment of rats with TQ led to the

Accepted Article

. preservation of ATP stores in liver, which is in concordance with other works [START_REF] Sayed-Ahmed | THYMOQUINONE SUPPLEMENTATION PREVENTS THE DEVELOPMENT OF GENTAMICIN-INDUCED ACUTE RENAL TOXICITY IN RATS[END_REF][START_REF] Nagi | Thymoquinone supplementation attenuates cyclophosphamide-induced cardiotoxicity in rats[END_REF]. Thus, we can presume that the prevention of ATP drop would be due, at least in part, to the ability of TQ to preserve mitochondrial integrity.

The formation of ROS and the induction of ER stress are known to be closely linked processes [START_REF] Malhotra | Endoplasmic reticulum stress and oxidative stress: a vicious cycle or a double-edged sword?[END_REF]. As a consequence of ER dysfunction, a signal transduction cascade which progress from the ER lumen to the cytoplasm and nucleus known as the UPR is activated [START_REF] Wang | Saturated fatty acids promote endoplasmic reticulum stress and liver injury in rats with hepatic steatosis[END_REF]. The UPR is an adaptive attempt that aims at restoring the ER homeostasis. It is mediated by three signaling proteins located at the ER membrane termed protein kinase RNA-like ER kinase (PERK), inositol requiring 1 (IRE1) and ATF6. The activation of these UPR sensors is impeded by their binding with a luminal protein chaperone, the glucose regulated protein 78 (GRP78). Once released from GRP78, the luminal domain of IRE1 dimerizes and transautophosphorylates and is thus transformed into an activated form. Activated IRE1 produces a potent transcription factor XBP1 by cleavage of XBP1 mRNA. The activated XBP1 upregulates UPR genes by directly binding to their related promoters [START_REF] Yeganeh | Hepatitis B and C virus-induced hepatitis: Apoptosis, autophagy, and unfolded protein response[END_REF]. During ER stress, PERK is activated by a similar mechanism to IRE1. Activated PERK phosphorylates eukaryotic initiation factor 2 (eIF2α). Phosphorylated eIF2α attenuates global protein synthesis by preventing ribosomal initiation complexes formation. Meanwhile, eIF2α is necessary for translation of some mRNAs, including the mRNA that encodes the ATF4. ATF4 regulates several UPR target genes principally those engaged in antioxidative stress responses, autophagy and ER stress-induced apoptosis such as CHOP [START_REF] Harding | An integrated stress response regulates amino acid metabolism and resistance to oxidative stress[END_REF][START_REF] Harding | Regulated translation initiation controls stress-induced gene expression in mammalian cells[END_REF]. Upon GRP78 release, ATF6 translocates from the ER to the Golgi where it is cleaved to form an active transcription factor. Active ATF6 migrates then to the nucleus and up-regulates numerous ER chaperone genes including GRP78, GRP94, and CHOP [START_REF] Zeeshan | Endoplasmic reticulum stress and associated ROS[END_REF]. Our data revealed the activation of all three branches of UPR (PERK, ATF6, IRE1) and their downstream targets ATF4, ATF6α, XBP1, respectively. We also showed the presence of ER stress in PH under IR

Accepted Article

. condition by the induction of CHOP and the up-regulation of GRP78. These results are in line with those of a previous report which highlighted ER stress after PH [START_REF] Mosbah | Endoplasmic reticulum stress inhibition protects steatotic and non-steatotic livers in partial hepatectomy under ischemia-reperfusion[END_REF]. In our study, we demonstrated that the induction of ER stress parameters in response to PH under IR condition could be reduced by TQ pre-treatment. Our data are in keeping with our previous findings showing that TQ reduces ER stress induced by warm liver IR in rat [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF]. However, how TQ protects ER from PH under I/R damage remains unclear; we presume that it could be related to the attenuation of oxidative stress.

Mitochondrial damage and ER stress induction by PH under I/R condition prompted us to further evaluate apoptosis. In the present study, an increase of cytochrome C release was observed after PH. In addition, both caspase-9 and -3 were found activated which indicates that PH under I/R may stimulate apoptotic cell death [START_REF] Szegezdi | Caspase-12 and ER-stress-mediated apoptosis: the story so far[END_REF][START_REF] Badiola | Induction of ER stress in response to oxygenglucose deprivation of cortical cultures involves the activation of the PERK and IRE-1 pathways and of caspase-12[END_REF]. In parallel, our data showed a caspase-12 induction, which is known as an important protein implicated in ER-induced apoptosis [START_REF] Chung | Photodynamic therapy with 9hydroxypheophorbide α on AMC-HN-3 human head and neck cancer cells: induction of apoptosis via photoactivation of mitochondria and endoplasmic reticulum[END_REF]. In our experimental model, TQ induced down-regulation of cytochrome C release and a marked decrease in the expression of caspase-9, -12 and -3. These findings are in line with a previous research paper [START_REF] El-Ghany | Thymoquinone triggers anti-apoptotic signaling targeting death ligand and apoptotic regulators in a model of hepatic ischemia reperfusion injury[END_REF]. The anti-apoptotic effect of TQ could be attributed to its potent scavenger and antioxidant role that help to attenuate ER stress and further apoptosis.

In summary, this study is the first to evaluate the hepatoprotective effects of TQ in an experimental model of PH. Data reported here suggest that TQ plays an important role in the prevention of liver injury in conditions of hepatectomy under I/R. TQ protection is related to its capacity to prevent oxidative stress, mitochondrial damage, ER stress and apoptosis.

Accepted Article

.

Materials and methods

Experimental animals and Ethics Statement

Male Wistar rats weighing 200-230 g were used in this study. They were housed in an aircontrolled room with 12 h light and dark cycles, and a constant temperature (22 ± 2°C). They had free access to food and water. All procedures were carried out in accordance with the European Union Regulations (Directive 2010/63/EU) for animal experiments and approved by the local Experimental Animals Ethics Committee of the Faculty of Pharmacy of Monastir.

Surgical Procedure

Rats were anaesthetized with an intra-peritoneal (ip) injection of pentobarbital (5 %), and then subjected to PH (70 % of liver parenchyma) under 60 min of ischemia as described previously [START_REF] Mosbah | Endoplasmic reticulum stress inhibition protects steatotic and non-steatotic livers in partial hepatectomy under ischemia-reperfusion[END_REF]. Briefly, after dividing the ligaments of the hepatic lobes and resection of their left hepatic lobe, rats underwent 60 min of partial liver warm ischemia by clamping their portal triad supplying the median lobe. At the end of ischemia, the right and caudate lobes were resected, and reperfusion of the median lobe was achieved by the release of the clamp. Sham operated rats was subjected to anesthesia and pedicle dissection without resection or ischemia.

After surgery, rats were allowed food and water ad libitum. Blood and liver samples were collected after 24 h reperfusion and samples were stored at -80°C.

Experimental groups

Rats were allocated at random into four experimental groups, with six rats in each group: Group 1 (Sham group): Rats were subjected to only laparotomy without occlusion of hepatic pedicle and PH. Group 2 (TQ group): rats were orally pretreated with TQ (30 mg/kg) for ten consecutive days [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF]. Rat livers were then subjected to laparotomy without occlusion of hepatic pedicle and PH.

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. Group 3 (PH group): rats underwent 70% hepatectomy under 60 min of ischemia followed by 24 h of reperfusion. Group 4 (TQ+PH group): rats were orally pretreated with TQ (30 mg/kg) for ten consecutive days [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF]. Rat livers were then subjected to surgery as in group 3.

Chemicals

TQ (Sigma Aldrich) was dissolved in ethanol and aliquoted and stored at -20°C. Working solution was prepared daily before oral administration by diluting stock solution with distilled water (1:4). Ten days prior the surgery, sham and PH animals underwent a daily intragastric administration of vehicle solution (ethanol and distilled water) as described previously [START_REF] Bouhlel | Thymoquinone prevents endoplasmic reticulum stress and mitochondria-induced apoptosis in a rat model of partial hepatic warm ischemia reperfusion[END_REF].

Histology and Immunohistochemistry

Liver biopsies were fixed in 10 % formalin solution, embedded in paraffin and cut at 4 µm thickness. To appraise the severity of hepatic injury, stained sections with hematoxylin and eosin were graded with a point-counting method on a scale from 1 (excellent) to 5 (poor) as described previously [START_REF] Mosbah | Pretreatment with mangafodipir improves liver graft tolerance to ischemia/reperfusion injury in rat[END_REF]: (1) normal rectangular structure, (2) rounded hepatocytes with an increase of the sinusoidal spaces, (3) vacuolization, (4) nuclear picnosis and (5) necrosis.

Damage score was estimated semi-quantitatively by an experienced pathologist without having knowledge about the treatment groups.

For liver regeneration, liver samples were immunostained with a rabbit monoclonal antibody against Ki 67 (clone SP6, Abcam, Cambridge, MA), developed with diaminobenzidine, and counterstained with hematoxylin [START_REF] Schiffer | Hepatic regeneration is decreased in a rat model of sinusoidal obstruction syndrome[END_REF].

Transaminase determination

The serum activity of alanine aminotransferase (ALT) was determined using a commercial kit from DiaSys (Diagnostic System, Germany) following supplier's instructions.

Determination of hepatic adenosine triphosphate content

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. Hepatic adenosine triphosphate (ATP) measurements were performed using a firefly bioluminescence assay kit (ATP Bioluminescent Assay Kit FLAA-1KT, Sigma Aldrich St Quentin Fallavier, France) as described elsewhere [START_REF] Bejaoui | Acetazolamide protects steatotic liver grafts against cold ischemia reperfusion injury[END_REF].

Determination of oxidative stress parameters

Liver tissues were homogenized in ice-cold phosphate buffered saline (100 mM KH 2 PO 4 , 100 mM K 2 HPO 4 , pH 7.4) to estimate the content of SHP and the activities of GPx, SOD and CAT. For MDA determination, tissues homogenization was carried out in ice-cold trisbuffered saline (100 mM Tris, pH 7). For CD measurement, liver tissues were homogenized in distilled ice-water.

Liver GPx activity was estimated by the method of Floche and Gunzler 1984 (59). SOD activity assay was performed using the method previously described by Marklund and Marklund [START_REF] Marklund | Involvement of the Superoxide Anion Radical in the Autoxidation of Pyrogallol and a Convenient Assay for Superoxide Dismutase[END_REF]. CAT activity was determined according to Clairbone method [START_REF] Claiborne | Catalase activity. Handbook of methods for oxygen radical research[END_REF]. The determination of SHP level was achieved in agreement to the method of Sedlak [START_REF] Sedlak | Estimation of total, protein-bound, and nonprotein sulfhydryl groups in tissue with Ellman's reagent[END_REF]. MDA was measured by the thiobarbituric acid (TBA) assay [START_REF] Mosbah | Preservation of steatotic livers in IGL-1 solution[END_REF]. CD was assessed according to Srinivasan [START_REF] Srinivasan | Inhibition of protein and lipid oxidation in beef heart surimi-like material by antioxidants and combinations of pH, NaCl, and buffer type in the washing media[END_REF].

Determination of endoplasmic reticulum stress parameters

We looked for ER stress through the determination of GRP78 and CHOP using ELISA kits (MyBioSource, San Diego, CA) according to the manufacturer's protocol and by the determination of protein relative expression of ATF6α, ATF4 and XBP1 by western blot technique.

Briefly, liver tissues were homogenized in ice-cold lysis buffer (150 mM NaCl, 50 mM Tris-HCl (pH 7.5), 1 mM DTT, 50 mM NaF, 1 mM PMSF, 1 mM EDTA, 1 mM EGTA, 0.1 mM orthovanadate, 0.05% Triton-X 100 and 2% protease inhibitor cocktails). Equal amounts of proteins were separated by SDS-PAGE and transferred to PVDF membranes as described elsewhere [START_REF] Ayed Tka | Melatonin modulates endoplasmic reticulum stress and Akt/GSK3-beta signaling pathway in a rat model of renal warm ischemia reperfusion[END_REF]. Then, membranes were incubated over night at 4°C with primary antibodies

Accepted Article

. for ATF6α (sc-22799), ATF4 (sc-101663), XBP1 (sc-7160) and Hsc70 (sc-7298) acquired from Santa Cruz Biotechnology. After washing, the membranes were incubated for 2h at room temperature with appropriate horseradish peroxydase conjugated secondary antibody. Signals were detected by enhanced chemiluminescence and quantified by the quantity one software program (Bio-Rad Laboratories, Hercules, CA, USA).

Determination of apoptosis parameters

Cytochrome C level was evaluated using ELISA kit from MyBioSource (San Diego, CA). A fluorometric assay kit was used (Biovision Palo Alto, CA) to determine liver caspase 12 level. Caspase 9 and caspase 3 levels were measured using a colorimetric assay kit from Biovision (Palo Alto, CA).

Statistical analysis

Data were expressed as mean ± standard deviation (SD) and were compared statistically using Graph Pad Prism software (version 6.01) by variance of analysis (ANOVA) followed by Newman-Keuls multiple comparison test. P-value of less than 0.05 was considered statistically significant.

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Figure legends : Figure 1 .

 :1 Figure legends:

Figure 2 .

 2 Figure 2. Thymoquinone pretreatment attenuated oxidative stress after PH under I/R. Liver samples were assessed for MDA concentration (a), glutathione peroxidase activity (b), superoxide dismutase activity (c), catalase activity (d), sulfhydryl proteins content (e) and conjugated dienes concentration (f) in livers harvested from Sham, TQ, PH and TQ+PH groups. Results are expressed as mean ± SD (n = 6 for each group). *P<0.05 versus Sham. +P<0.05 versus TQ. #P<0.05 versus PH.

Figure 3 .

 3 Figure 3. Thymoquinone increased ATP content in liver after PH. ATP concentration in liver tissue of Sham, PH and TQ+PH groups. Results are expressed as mean ± SD (n = 6 for each group). *P<0.05 versus Sham. #P<0.05 versus PH.

Figure 4 .

 4 Figure 4. Thymoquinone preconditioning attenuated ER stress. GRP78 (a) and CHOP (b) concentration determined by ELISA kits and the relative expression of ATF6 (c), ATF4 (d) and XBP1 (e) proteins level evaluated by western blot in Sham, PH and TQ+PH groups. Results are expressed as mean ± SD (n = 6 for each group). *P<0.05 versus Sham. #P<0.05 versus PH.

Figure 5 .

 5 Figure 5. Thymoquinone pretreatment decreased liver apoptosis after PH. Caspase 12 (a), Cytochrome C (b) Caspase 9 (c) and Caspase 3 (d) concentrations were determined. Livers were harvested from Sham, PH and TQ+PH groups. Results are expressed as mean ± SD (n = 6 for each group). *P<0.05 versus Sham. #P<0.05 versus PH.

Acknowledgements

This work was supported by grant from the Tunisian Ministry of Higher Education and Scientific Research (UR12ES11).

The authors declare that they have no conflicts of interests concerning this article.