Uwe Franz
email: uwe.franz@univ-fcomte.fr

Anna Kula
email: anna.kula@math.uni.wroc.pl

AND J Martin Lindsay
email: j.m.lindsay@lancaster.ac.uk

Michael Skeide
email: skeide@unimol.it

HUNT'S FORMULA FOR SU q (N) AND U q (N)

Keywords: 2020 Mathematics Subject Classification. 46L53, 60G51 Quantum Lévy process, convolution semigroup, compact quantum group, CQG algebra, generating functional, Schürmann triple

For any Lévy process on the quantum group SU q (N ), where 0 < q < 1 and N ∈ N, a Lévy-Khintchine-type decomposition of its generating functional is given, together with an analogue of Hunt's formula. The non-gaussian component is shown to further decompose into generating functionals that live on the quantum subgroups SU q (n), for n N . Corresponding results are also given for the quantum groups U q (N ).

Introduction

Up to stochastic equivalence, a Lévy process with values in a locally compact Lie group G is determined by its generating functional. This is a (densely defined) linear functional γ on C 0 (G), the C * -algebra of continuous complex-valued functions on G which vanish at infinity, whose domain may be thought of as consisting of those functions that have a second order Taylor expansion around the identity element of the group. Hunt's formula ( [START_REF] Hunt | Semigroups of measures on Lie groups[END_REF]) is a generalization and extension of the Lévy-Khintchine formula ( [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF], [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]). It is equivalent to the assertion that

γ = γ D + γ G + γ L where γ L = L • P and L(f ) = G\{e} f (s) Π(ds) (1.1)
for the identity element e of G, in which P is a hermitian projection that kills the linear terms, the drift γ D and P -invariant gaussian part γ G are linear combinations of first and second order derivatives evaluated at e respectively, and Π is the so-called Lévy measure. The Lévy functional L is defined on the space of functions that, together with their first derivatives, vanish at e. The integral may be viewed as a mixture of point evaluations, moreover functionals of the form f → f (s) -f (e), for fixed s = e, generate jump processes. The functional γ L is also referred to as the jump part; in the case where G = R and Π is finite, it generates a compound Poisson process. The decomposition depends on the non-canonical projection P chosen; its role is to deal with any singularity of the measure Π at e. If G is compact, Tannaka-Krein duality ([10, Section VII.30]) asserts that the representative algebra R(G), generated by matrix coefficients of finite-dimensional representations of G, is a norm-dense * -subalgebra of the unital C * -algebra C(G). In fact, R(G) is a commutative Hopf * -algebra from which the topological group G may be fully recovered ( [START_REF] Neshveyev | Compact quantum groups and their representation categories[END_REF]). A compact quantum group in the sense of Woronowicz ([29]) is a unital C * -algebra-with-coproduct which enjoys density relations corresponding to the group cancellation law and contains a dense Hopf *algebra, the CQG algebra of the quantum group, whose role corresponds to that played by R(G) for a compact group G ( [START_REF] Dijkhuizen | CQG algebras: A direct algebraic approach to compact quantum groups[END_REF]). Schürmann's theory of quantum Lévy processes on * -bialgebras ( [START_REF]White noise on bialgebras[END_REF]) thereby applies. As with their classical counterparts, but now up to quantum stochastic equivalence, Lévy processes on * -bialgebras are classified by their generating functional, now a hermitian linear functional on the CQG algebra which is conditionally positive and vanishes at the identity element. The problem of finding a decomposition of generating functionals corresponding to (1.1) is expressible in cohomological terms. Of course meaning has to be given to drift, gaussian and jump parts in the quantum generalisation. Our Hunt formula includes an explicit description of the drifts and gaussian generating functionals and the specification of an approximation property that justifies calling the remainder a jump part (Proposition 2.8).

For some compact quantum groups every generating functional has such a decomposition but for others that is not so ( [START_REF] Franz | On the Lévy-Khinchin decomposition of generating functionals[END_REF], [START_REF] Das | Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups[END_REF]). A Hunt formula for Woronowicz's SU q (2) ( [START_REF] Woronowicz | Compact matrix pseudogroups[END_REF], [START_REF]Twisted SU(2) group. An example of a non-commutative differential calculus[END_REF]) was obtained in [START_REF] Skeide | The Lévy-Khintchine formula for the quantum group SU q (2)[END_REF], [START_REF] Schürmann | Infinitesimal generators on the quantum group SU q (2), Infin[END_REF]. This led to a short proof of the classical Hunt formula for compact Lie groups ( [START_REF]Hunt's formula for SU q (2) -a unified view[END_REF]). Here we tackle the case of SU q (N), obtaining a unique decomposition γ = γ D + γ G + γ N G where γ N G = γ 2 • P + • • • + γ N • P, in which P is a hermitian projection analogous to that of (1.1), γ D is a drift, γ G is a P -invariant gaussian generating functional and, for 2 n N, γ n is an extension to SU q (N) of a completely non-gaussian generating functional on SU q (n) which enjoys an irreducibility property. We also display the essentially classical structure of γ D and γ G , and show γ N G to be the limit of functionals of the form ω ξ(t) • π • P for a representation π and net of vector functionals (ω ξ(t) ) (Theorem 4.15). The case of general N turns out to be more involved than the case N = 2, and some results concerning SU q (2) fail for N 3. For instance, for N 3 the cohomological problem is not always solvable in the gaussian case (Corollary 2.13). Also, for N = 2 the completely non-gaussian generating functionals may be parametrized by the vectors in its associated representation Hilbert space, whereas for N 3 the situation is more subtle (Section 5).

The paper is organized as follows. Terminology and notations concerning the CQG algebra of a compact quantum group are set out below. Section 2 contains the basic definitions and preliminary results. The CQG algebras of the compact quantum groups SU q (N) and U q (N) are here respectively denoted SU q (N) and U q (N); the former is algebraically generated by a matrix of elements [u jk ] N j,k=1 (see Relations (2.6), et seq.). In Section 3 we deal with our choice of projection P , with respect to which we show that the gaussian generating functionals on SU q (N) are classified by a real (N -1)-vector and positive-definite real (N -1) × (N -1) matrix representing the drift and P -invariant diffusion-type second order term (Theorem 3.6). Unlike in lower dimensions, for N 3 there are cocycles of gaussian representations which have no associated generating functionals (Theorem 3.3). Every gaussian generating functional is induced from a gaussian generating functional that lives on the classical undeformed subgroup T N -1 of SU q (N), in the sense of Definition 2.21 (see Remark 3.7). In Section 4 we show that every representation π of SU q (N) has a unique full (representation) decomposition π 1 ⊕• • •⊕π N , where π 1 is its so-called gaussian part and, for 2 n N, π n lives on SU q (n) and π n (1 -u nn ) is injective. Completely non-gaussian cocycles η are approximated by coboundaries and determined by their values η(u nn ) (2 n N). From this we deduce a full (generating functional) decomposition γ = γ 1 +• • •+γ N for generating functionals, uniquely determined by the projection P , and conclude with our Hunt formula (Theorem 4.15). In Section 5 we show that, unlike in the case N = 2, if N > 2 then the values of η(u N N ) for cocycles η of representations π for which π(1 -u N N ) is injective, may not exhaust the representation space. We then indicate a completion process which yields a quasi-innerness property, and thereby full parameterisation, for completely non-gaussian cocycles. In Section 6 we briefly treat the quantum groups U q (N).

Our work suggests the investigation of Hunt formulae for other q-deformed compact Lie groups ( [START_REF] Neshveyev | Compact quantum groups and their representation categories[END_REF]).

Compact quantum groups and CQG algebras. A CQG algebra ( [START_REF] Dijkhuizen | CQG algebras: A direct algebraic approach to compact quantum groups[END_REF]), or algebraic compact quantum group, is a Hopf * -algebra G that is linearly spanned by the coefficients of its finitedimensional unitary corepresentations or, equivalently, has a faithful Haar state. Thus a CQG algebra is a unital * -algebra G, with unital * -algebra morphisms ∆ : G → G ⊗ G and ε : G → C, linear map κ : G → G and unital linear functional h : G → C, called respectively the coproduct, counit, coinverse or antipode, and Haar state, enjoying the coassociativity, counital, coinverse, invariance and positivity relations

(∆ ⊗ id) • ∆ = (id ⊗∆) • ∆; (ε ⊗ id) • ∆ = id = (id ⊗ε) • ∆; µ • (id ⊗κ) • ∆ = ι • ε = µ • (κ ⊗ id) • ∆; (id ⊗h) • ∆ = ι • h = (h ⊗ id) • ∆;
and h(a * a) > 0 for a = 0. Here µ : G ⊗ G → G denotes the linearisation of the algebra product, and ι the unital linear map C → G. The coinverse κ is uniquely determined by the bialgebra structure and any * -bialgebra morphism between CQG algebras respects coinverses and so is a CQG algebra morphism ([3, Remarks 4.2.3 and 4.2.5]); the Haar state h is also unique ([4, Proposition 3.2]). Compact quantum groups may also be viewed from the equivalent C *algebraic perspective, as was originally done by Woronowicz ([29]). The canonical (universal and reduced) Woronowicz algebras of a compact quantum group G are commonly denoted C u (G) and C r (G), and its CQG algebra is here denoted by R(G) in a further nod to their classical counterparts. The quantum space G itself is only manifested through one of its realisations. For more on this, we recommend [START_REF] Neshveyev | Compact quantum groups and their representation categories[END_REF], [START_REF] Klimyk | Quantum groups and their representations[END_REF]Section 11.3], and [START_REF] Timmermann | An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond[END_REF]Section 5.4]. For the purposes of this work, it suffices to operate exclusively within CQG algebras. In fact, in our analysis we need explicit recourse to none of the coproduct, coinverse or Haar state.

Convention. In Schürmann's theory representations are by possibly-unbounded adjointable operators on pre-Hilbert spaces because he works in the more general setting of * -bialgebraswith-character. By contrast, representations of a CQG algebra G are all by bounded operators, and so may be extended to the Hilbert space completions. Accordingly, by a representation of G we always mean a unital * -algebra morphism π : G → B(h), for some Hilbert space h = h π .

Note. MS wishes to emphasise that revisions for this final version of the paper were done by the other authors, and that the original version is available on the arXiv ([9]).

Preliminaries

Generating functionals of quantum Lévy processes and Schürmann triples. Let G be a CQG algebra. A Lévy process on G is a family of * -algebra morphisms from G to a noncommutative probability space enjoying certain properties which encode the stationarity and independence of increments (see [START_REF]White noise on bialgebras[END_REF], [START_REF] Franz | Lévy processes on quantum groups and dual groups[END_REF] and [15, Chapter VII], or the survey [START_REF]Independence and Lévy processes in quantum probability[END_REF]). Definition 2.1. A generating functional for a quantum Lévy process on G is a linear functional γ on G which is hermitian: γ = γ † : a → γ(a * ), normalised : γ(1) = 0, and conditionally positive: γ(c * c) 0 for all c ∈ ker ε.

Quantum Lévy processes are determined up to quantum stochastic equivalence by their generating functionals, and may be reconstructed from their generating functional using quantum stochastic calculus on a symmetric Fock space ([20, Theorem 2.3.5], [14, Theorem 7.1]), or using Trotter products and Arveson (product) systems ( [START_REF] Schürmann | Transformations of quantum Lévy processes on Hopf algebras[END_REF]).

Definition 2.2 ([20]

). A Schürmann triple on G is an ordered triple (π, η, γ) consisting of a representation π of G, a π-ε-cocycle, or π-ε-derivation, that is, a linear mapping η :

G → h π satisfying η(ab) = π(a)η(b) + η(a)ε(b) (a, b ∈ G), (2.1
) and a linear functional γ on G satisfying

γ † = γ, γ(1) = 0 and γ(c * c) = η(c) 2 (c ∈ ker ε), (2.2)
equivalently,

γ † = γ and η(a), η(b) = γ(a * b) -γ(a)ε(b) -ε(a)γ(b) for a, b ∈ G. A linear functional γ on G completes a π-ε-cocycle η if (π, η, γ
) is a Schürmann triple; we then say that η, or (π, η), is completable.

A Schürmann triple (π, η, γ) or cocycle η, is called cyclic if η(G) = h π .
The third component of a Schürmann triple is a generating functional. Conversely, for any generating functional γ, there is a cyclic Schürmann triple with γ as its third component. If (π, η, γ) is a cyclic Schürmann triple then, for any linear isometry V from h π into a Hilbert space, (V π(•)V * , V η(•), γ) is a Schürmann triple (cyclic if and only if V is unitary), and every Schürmann triple having γ as its third component is of this form. Thus all cyclic Schürmann triples having γ as their third component are unitarily equivalent -we refer to any one of these as γ's (associated ) Schürmann triple ([20, Section 2.3]).

For K := ker ε, set

K n := span{c 1 • • • c n : c 1 , • • • , c n ∈ K} for n 1, and K ∞ := n 1 K n .
Thus (K n ) is a sequence of * -ideals of G decreasing to K ∞ . Also set P 2 (G) := {P ∈ L(G) : P is a hermitian projection, ran P = K 2 and 1 ∈ ker P },

where hermitian means P = P † : a → (P a * ) * for a ∈ G.

Definition 2.3. Let γ be a generating functional on G. Then γ is a drift if γ| K 2 = 0, equivalently, in terms of its associated Schürmann triple (π, η, γ), if h π = {0}. For P ∈ P 2 (G), we denote the drift γ -γ • P by γ P D , and call γ P -invariant if γ • P = γ. Remarks 2.4. The drifts form a real subspace of the linear dual of G. Any P ∈ P 2 (G) determines a unique resolution for generating functionals γ into a drift component plus a P -invariant one: γ = γ P D + γ • P -in this sense P -invariance may usefully be thought of as a P -driftless property (i.e. having zero drift component with respect to P ). If a cocycle η is completable then, for any particular generating functional γ which completes η, the set of all generating functionals which do so equals {γ + γ ′ : γ ′ is a drift} and the unique P -invariant one is γ • P = γ -γ P D . The P -invariant generating functionals on G are the maps of the form ψ • P for a linear functional ψ on K 2 which is nonnegative: ψ(c * c) 0 for all c ∈ K (and thus also hermitian).

There is no canonical choice of projection from P 2 (G). By contrast, since C1 and K are complementary subspaces of G, there is a unique projection in L(G) with range K and 1 in its kernel-namely (id -ι • ε : a → a -ε(a)1), moreover it is hermitian and compatible with the projections in P 2 (G). Definition 2.5. Let U be a subspace of a complex vector space V . A linearly independent subset E of V \U is a basis extension from U to V if its linear span is a complementary subspace of U. In case V is involutive, a basis extension is hermitian if it consists of selfadjoint elements.

For any hermitian basis extension E from K 2 to K, the functionals (ε ′ d ) d∈E on G given by ε

′ d λ1 + k 2 + e∈E λ e e = λ d , for λ ∈ C, k 2 ∈ K 2 and {λ e : e ∈ E} ⊂ C, (2.3)
form a basis for the real space of drifts on G, and

P E := id -ι • ε - d∈E d ε ′ d (•) ∈ P 2 (G) (2.4)
equals the projection onto K 2 along span({1} ∪ E) = C1 ⊕ spanE. The resulting map E → P E is surjective and

P E 1 = P E 2 if and only if spanE 1 = spanE 2 .
Procedure 2.6. For obtaining all generating functionals on G, one needs to identify:

(1) the representations π of G;

(2) for each representation π, the π-ε-cocycles η;

(3) for each such cocycle η, the generating functionals γ which complete it.

In the cases of the quantum groups SU q (N) and U q (N) the representation theory is known ( [START_REF] Koelink | On * -representations of the Hopf * -algebra associated with the quantum group U q (n)[END_REF]). Step (2) is a cohomological problem, as π-ε-cocycles form the first Hochschild cohomology group H 1 (G, π h ε ) for h = h π , and this may usually be computed in a straightforward way. The main problem lies in Step [START_REF] Dascalescu | Hopf algebras. An introduction[END_REF]. The basic constraint on a given cocycle η, for it to be completable, is that η(c) must equal η(d) whenever c, d ∈ K satisfy c * c = d * d; the task then amounts to solving ψ(c * c) = η(c) 2 (c ∈ K) for a linear functional ψ on K 2 since then, for any P ∈ P 2 (G), the prescription a → ψ(P a) defines a (P -invariant) generating functional which completes η.

Approximately inner cocycles. As just described, the problem of classifying generating functionals on G lies in the fact that there might be none which completes a given cocycle. In this section we identify a situation where such a completion does exist. Definition 2.7. A π-ε-cocycle is a coboundary, or inner derivation, if it is of the form

η π,ξ : = (π -ι • ε)(•)ξ : a → π(a)ξ -ξ ε(a)
for some vector ξ in h π , and is approximately inner if it is a pointwise limit of coboundaries (η π,ξ(λ) ) for some net (ξ(λ)) in h π .

For a vector ξ of a Hilbert space h, ω ξ denotes the vector functional T → ξ, T ξ on B(h). The following result is heavily used in Section 4.

Proposition 2.8. Approximately inner cocycles are completable. Specifically, let P ∈ P 2 (G), let π be a representation of G, and let (ξ(λ)) be a net in h π such that (η λ := η π,ξ(λ) ) converges pointwise to a map η. Then η is a π-ε-cocycle and the net (γ λ := ω ξ(λ) • π • P ) converges pointwise to a P -invariant generating functional γ which completes η.

Proof. For each λ, the P -invariant linear functional γ λ is hermitian and (π, η λ , γ λ ) is easily seen to satisfy (2.1) and (2.2). Therefore, since η is evidently a π-ε-cocycle and K 2 is both the range of P and the linear span of the set {c * c : c ∈ K}, the proposition follows from the fact that

γ λ (c * c) = π(c)ξ(λ) 2 = η π,ξ(λ) (c) 2 → η(c) 2 for each c ∈ K.
In the classical setting of (1.1) we see that the generating functional γ L is expressible as the limit of the functionals ω 1 G\U • π • P , as the neighbourhoods U of e shrink to {e}, π being the multiplication representation of R(G) on L 2 (G, Π) and 1 here denoting indicator function.

Gaussian generating functionals, cocycles and representations.

Definition 2.9. A generating functional γ, cocycle η, or representation π is called gaussian if it vanishes respectively on K 3 , K 2 , or K.

For components of a Schürmann triple, these are equivalent ([20, Proposition 5.1.1]). A representation π is gaussian if and only if π = ι h π • ε, where ι h π denotes the unital linear map from C to B(h π ).

Proposition 2.10. Let E be a hermitian basis extension from K 2 to K. Then, for any Hilbert space h, the h-valued gaussian cocycles on G are precisely the maps of the form d∈E ξ d ε ′ d (•) for a family of vectors (ξ d ) d∈E in h, where the functionals ε ′ d are as in (2.3). Proof. Since gaussian cocycles vanish on 1 and on K 2 , this follows from the fact that elements a of G are uniquely expressible as ε(a

)1 + k 2 (a) + d∈E ε ′ d (a)d for some k 2 (a) ∈ K 2 .
It would be desirable to have a similarly concise description of gaussian generating functionals. For now we note that in general not all gaussian cocycles η admit a gaussian generating functional.

Definition 2.11. A cocycle η on G is hermitian if it satisfies η(c) = η(c * ) for all c ∈ K. A gaussian cocycle of the form η = d∈E ξ d ε ′ d is hermitian if and only if the Gram matrix ξ d , ξ d ′ is
real (and therefore symmetric). Proposition 2.10 has the following consequence.

Corollary 2.12. G has non-hermitian gaussian cocycles if and only if dim K/K 2 2.

For a gaussian cocycle η to be completable it is sufficient that it be hermitian ([20, Proposition 5.1.11]) but not necessary. It becomes necessary too under the additional assumption given in the next corollary, which applies to both SU q (N) (by Lemma 3.2 and part (d) of Lemma 3.1), and U q (N).

Corollary 2.13. Suppose that c * c -cc * ∈ K 3 for all c ∈ K. Then a gaussian cocycle is completable if and only if it is hermitian.

Proof. It is necessity that is to be proved, so assume that γ is a generating functional completing a gaussian cocycle η. Then η(c) 2

-η(c * ) 2 = γ(c * c -cc * ) = 0 for all c ∈ K, as required.
Complete non-gaussianness and Lévy-Khintchine decomposition. We next collect basic facts about when a generating functional can have a Lévy-Khintchine decomposition.

Lemma 2.14. Let π 1 ⊕ π 2 be a decomposition of a representation π of G, let V i denote the inclusion map h π i → h π for i = 1, 2, and let η be a π-ε-cocycle. Then the following hold (a)

η i := V * i η(•) is a π i -ε-cocycle for i = 1, 2. (b)
If two of the three cocylces η, η 1 and η 2 are completable then so is the third.

It is quite possible that η is completable, but η 1 and η 2 are not. Definition 2.15. For a representation π of G, set

h π G := c∈K ker π(c) and h π R := (h π G ) ⊥ . Then π is completely non-gaussian if h π G = {0}, equivalently, if h π R = h π .
We also call a π-ε-cocycle η completely non-gaussian if π is, and a generating functional γ completely non-gaussian if the representation component of its Schürmann triple is.

The above definition and its notations are amply justified by the following straightforward proposition.

Proposition 2.16 ([20]). Let π be a representation of G. Then h π G and h π R are invariant subspaces and, denoting the resulting decomposition of π as

π G ⊕ π R , π G is gaussian and π R is completely non-gaussian. Moreover, h (π R ) G = {0} = h (π G ) R . If η = η G ⊕ η R is the corresponding decomposition of a π-ε-cocycle η then η G is gaussian, and if η is cyclic then η G and η R are cyclic too.
Generating functionals of the form ω ξ • π • P , and their limits as in Proposition 2.8, are completely non-gaussian.

Definition 2.17. A Lévy-Khintchine decomposition for a generating functional γ with Schürmann triple Evidently (AC) implies both (GC) and (NC), and either of these implies (LK); none of the reverse implications hold ( [START_REF] Franz | On the Lévy-Khinchin decomposition of generating functionals[END_REF]). The following is an immediate consequence of Proposition 2.8. Proposition 2.20. (NAI) implies (NC), and thus (LK).

(π, η, γ) is a decomposition γ = γ 1 + γ 2 for which (π G , η G , γ 1 ) and (π R , η R , γ 2 )
Schürmann triples on quantum subgroups. In the course of proving our results for SU q (N), we will decompose representations into components that live on its quantum subgroups SU q (n) in the sense given below. One way of extending our results to U q (N) is by exploiting the quantum subgroup relations T N U q (N) SU q (N + 1); this is done in Section 6.

Definition 2.21. A compact quantum group H is a quantum subgroup of a compact quantum group G, written H G, if there is a CQG algebra epimorphism (equivalently, a * -bialgebra epimorphism) s : G → H; we also say that (H, s) is a quantum subgroup of G.

Given such a subgroup relation, we say that a linear map T from G to a vector space V lives on (H, s) if ker T ⊃ ker s, equivalently, if T factors (evidently uniquely) through the epimorphism s: T = T • s for some map T : H → V.

For the remainder of this subsection we fix a quantum subgroup (H, s) of G and use tildes for induced maps having domain H. Since s respects counits, the functional ε on H satisfying ε • s = ε is its counit, and s(K n ) = K n for all n. Also, a representation of G lives on the trivial CQG algebra C if and only if it is gaussian. The properties listed next are easily verified.

Lemma 2.22. Suppose that π = π • s, η = η • s and γ = γ • s, for maps π, • • • , γ, then (1) π is a representation of G if and only if π is a representation of H. (2) If (1) holds then η is a π-ε-cocycle if and only if η is a π-ε-cocycle.
(3) γ is a generating functional on G if and only if γ is a generating functional on H. (4) (π, η, γ) is a Schürmann triple on G if and only if ( π, η, γ) is a Schürmann triple on H. Moreover, for any representation π of G living on (H, s) and vector ξ in h π , h π G = h π G , η π,ξ lives on H and η π,ξ = η π,ξ .

(2.5)

This has the following useful corollary.

Proposition 2.23. The property (NAI) is hereditary.

We now show that an approximately inner cocycle lives on a subgroup if its approximating inner cocycles do. Proposition 2.24. Let π be a representation of G living on (H, s), let (ξ(λ)) be a net in h π such that (η π,ξ(λ) ) converges pointwise to η, and let P ′ ∈ P 2 (H). Then the following hold.

(a) η π,ξ(λ) , (ω ξ(λ) • π • P ) and (ω ξ(λ) • π • P ′ ) have pointwise limits η, γ and γ ′ , such that η = η • s, and γ and γ ′ are generating functionals completing η and η respectively.

(b) γ = γ ′ • s • P . Proof. (a) It follows from Identity (2.5) that η π,ξ(λ) • s = η π,ξ(λ)
for each λ, and so (a) follows from the surjectivity of s and Proposition 2.8.

(b) This follows since s(K 2 ) = K 2 = ran P ′ so P ′ • s • P = s • P and thus, for each λ,

(ω ξ(λ) • π • P ′ ) • (s • P ) = ω ξ(λ) • π • s • P = ω ξ(λ) • π • P .
The projections P ∈ P 2 (G) and P ′ ∈ P 2 (H) may be chosen to be compatible. This follows from the following straightforward lemma.

Lemma 2.25. Let P = P E and P ′ = P E ′ for hermitian basis extensions E from K 2 to K and E ′ from K 2 to K, according to (2.4). Then P ′ • s = s • P if and only if s(E) ⊂ span E ′ , in which case span s(E) = span E ′ and so the generating functional γ from Proposition 2.24 lives on H.

The quantum groups SU q (N) and U q (N). Let 0 < q < 1. We next collect the facts about SU q (N) and U q (N) for N 2 that are required. For convenience, we extend our definitions to the case N = 1: SU q (1) = SU(1) := {e}, the trivial group, and U q (1) := U(1) = T, the torus. For an element σ of the permutation group S N , let i(σ) denote the number of inversions of σ:

#{(j, k) : j < k, σ(j) > σ(k)}.
As a unital algebra, the CQG algebra U q (N) of the compact quantum group U q (N), is generated by indeterminates u jk (j, k = 1, • • • , N) and D -1 , subject to the following relations ([13, Section 2]):

u ij u kj = qu kj u ij if i < k, (2.6a
)

u ij u il = qu il u ij if j < l, (2.6b
)

u ij u kl = u kl u ij if i < k, j > l, (2.6c
)

u ij u kl = u kl u ij -(q -1 -q)u il u kj if i < k, j < l, (2.6d)
and

D -1 D q = 1 = D q D -1 ,
for the q-determinant of the matrix U = u jk n j,k=1 ,

D q = D q (U) := σ∈S N (-q) i(σ) u 1,σ(1) • • • u N,σ(N ) .
The jk-th q-minor is defined as the q-determinant of the (N -1) × (N -1)-matrix obtained from U by removing the j-th row and the k-th column,

D jk q = D jk q (U) := σ∈S jk N-1 (-q) i(σ) u 1,σ(1) • • • u j-1,σ(j-1) u j+1,σ(j+1) • • • u N,σ(N ) ,
where

S jk N -1 denotes the set of bijections σ from {1, • • • , j -1, j + 1 • • • , N} to {1, • • • , k - 1, k + 1, • • • , N}.
The involution, counit and coproduct of U q (N) are then determined by the requirements

u * jk = (-q) k-j D jk q D -1 , (D -1 ) * = D q , ε(u jk ) = δ jk and ∆u jk = l u jl ⊗ u lk .
The matrix of elements U satisfies the unitarity relations (2.7) below.

As unital * -algebra, SU q (N) is generated by indeterminates u jk (j, k = 1, • • • , N), subject to the unitarity relations ( [START_REF]Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N ) groups[END_REF]):

N s=1 u js u * ks = δ jk 1 = N s=1 u * sj u sk (j, k ∈ {1, 2, • • • , N}), (2.7)
and the twisted determinant conditions

σ∈S N (-q) i(σ) u σ(1),τ (1) u σ(2),τ (2) • • • u σ(N ),τ (N ) = (-q) i(τ ) 1 (τ ∈ S N ).
The counit and coproduct are given by the same formulae as for U q (N).

Remark 2.26. We also use an alternative characterisation of SU q (N), namely as the quotient of U q (N) by the extra relation D q = 1; the involution then simplifies to u * jk := (-q) k-j D jk q , showing that, as an algebra, SU q (N) is generated by the u jk 's. This means that, when checking well-definedness of representations and cocycles, one only has to manage the relations of the generators u jk (namely (2.6) and D q ([u jk ]) = 1) and not those involving their adjoints.

The following commutation relations among the generators u jk of U q (N) and their adjoints, and therefore also those of SU q (N), are easily verified:

for i, j, k, l ∈ {1, • • • , N}, u ij u * kl = u * kl u ij if i = k and j = l, (2.8a
)

u ij u * kj = qu * kj u ij -(1 -q 2 ) m<j u im u * km if i = k, (2.8b
)

u ij u * il = q -1 u * il u ij + (q -1 -q) n>i u * nl u nj if j = l, (2.8c
)

u ij u * ij = u * ij u ij + (1 -q 2 ) n>i u * nj u nj -(1 -q 2 ) m<j u im u * im .
(2.8d)

We use the further consequences: for 1 j, k < N,

u N j u * N k = q -1 u * N k u N j if j = k, (2.9a
)

u jN u * kN = q -1 u * kN u jN if j = k, (2.9b) u * N N u N N = q 2 u N N u * N N + (1 -q 2 ) 1, (2.9c)
Identity (2.9a) follows from (2.8c). Identity (2.8b) with the unitarity condition (2.7) together imply that, for j = k,

u jN u * kN = qu * kN u jN -(1 -q 2 ) m<N u jm u * km = qu * kN u jN + (1 -q 2
)u jN u * kN , from which (2.9b) follows, and Identity (2.9c) follows from (2.8d):

u N N u * N N = u * N N u N N -(1 -q 2 ) m<N u N m u * N m = u * N N u N N -(1 -q 2 )(1 -u N N u * N N
). We next describe the relevant quantum subgroup relations. By definition, SU q (N) is a quantum subgroup of U q (N) via the CQG epimorphism determined by its action on generators as follows r N : u jk -→ u jk and D -1 -→ 1. Also U q (N) is a quantum subgroup of SU q (N + 1) via the epimorphism determined by

t N :     u 11 • • • u 1N u 1,N +1 . . . . . . . . . . . . u N 1 • • • u N N u N,N +1 u N +1,1 • • • u N +1,N u N +1,N +1     -→     u 11 • • • u 1N 0 . . . . . . . . . . . . u N 1 • • • u N N 0 0 • • • 0 D -1    
where, as in the definition of r N , the u jk on the left-hand side are the generators of SU q (N + 1) while those on the right-hand side are the generators of U q (N)); like r N , t N respects coproduct, counit and involution, and thus also coinverse. Composition gives the chain

SU q (1) U q (1) SU q (2) U q (2) • • • SU q (N) U q (N) • • •
Of particular interest for us is the epimorphism s N := r N -1 • t N -1 : SU q (N) → SU q (N -1), which is determined by

s N :     u 11 • • • u 1,N -1 u 1N . . . . . . . . . . . . u N -1,1 • • • u N -1,N -1 u N -1,N u N 1 • • • u N,N -1 u N N     -→     u 11 • • • u 1,N -1 0 . . . . . . . . . . . . u N -1,1 • • • u N -1,N -1 0 0 • • • 0 1     , (2.10)
and its iterates Proof. For m ∈ {n, N} let us abbreviate SU q (m) to A m and denote its algebra generators by u m jk (1 j, k m). We also write K for the ideal ker s n,N of A N . For σ ∈ S N and n < p N, u N p,σ(p) -δ p,σ(p) 1 ∈ S n,N ⊂ I so

s n,N := s n+1 • • • • • s N : SU q (N) → SU q (n) (n < N). ( 2
1 = D q [u N jk ] ∈ σ∈S N s.t. σ(p)=p for n<p N (-q) i(σ) u N 1,σ(1) • • • u N n,σ(n) + I = τ ∈Sn (-q) i(τ ) u N 1,τ (1) • • • u N n,τ (n) + I = D q [u N jk ] 1 j,k n + I.
It follows that the relation D q [u n jk ] = 1 in A n is preserved by the mapping from the set of generators of A n into the quotient algebra A N /I given by u n jk → u N jk + I (1 j, k n). Since this clearly also preserves the (remaining defining) relations (2.6), the mapping uniquely extends to an algebra morphism φ : A n → A N /I. Now the prescription a+I → a+K defines an algebra epimorphism ψ : A N /I → A N /K (since I ⊂ K) and, letting s n,N denote the canonically induced algebra isomorphism

A N /K → A n , (φ • s n,N • ψ)(u N jk + I) = (φ • s n,N )(u N jk ) = u N jk + I if 1 j, k n δ jk 1 + I if max{j, k} > n. Thus, since u N jk -δ jk 1 ∈ S n,N ⊂ I if max{j, k} > n, (φ • s n,N • ψ)(u N jk + I) = u N
jk + I for all j and k so φ • s n,N • ψ = id A N /I . It follows that the algebra epimorphism ψ is injective and thus an isomorphism. Since I ⊂ K, this implies that I = K.

We next establish relations between the values taken on generators, for a given cocycle on SU q (N).

Lemma 2.28. Let π be a representation of SU q (N) and let η be a π-ε-cocycle. For i < l N and j, k < N, η(

u il ) = -(I -qπ(u ll ))) -1 π(u il )η(u ll ), (2.12a
)

η(u li ) = -(I -qπ(u ll ))) -1 π(u li )η(u ll ), (2.12b
)

π(u N N -1)η(u jk ) = π(u jk -δ jk 1) -(q -1 -q)π(1 -q 2 u N N ) -1 π(u il u li ) η(u N N ). (2.12c)
In particular, by Remark 2.26, η is determined by its value η(u N N ) when π(1 -u N N ) is injective.

Proof. If a = u il or a = u li where i < l N, then a ∈ ker ε and, by Identities (2.6a) and (2.6b), a u ll = qu ll a. Hence, by the cocycle property, π(a)η(u ll ) + η(a) = qπ(u ll )η(a). Since π(u ll ) is a contraction, this is equivalent to the identity η(a) = -(I -qπ(u ll )) -1 π(a)η(u ll ).

By the cocycle property applied to Identity (2.6d), if j, k < N then

π(u jk )η(u N N ) + η(u jk ) = η(u jk u N N ) = η(u N N u jk ) -(q -1 -q)η(u jN u N k ) = π(u N N )η(u jk ) + η(u N N )ε(u jk ) -(q -1 -q)π(u jN )η(u N k ), so, π(u N N -1)η(u jk ) = π(u jk -δ jk 1)η(u N N ) + (q -1 -q)π(u jN )η(u N k ) = π(u jk -δ jk 1) -(q -1 -q)π(u jN ) I -qπ(u N N )) -1 π(u N k ) η(u N N ) = π(u jk -δ jk 1) -(q -1 -q) I -q 2 π(u N N )) -1 π(u jN u N k ) η(u N N ).
We end this section by characterising those representations and cocycles on SU q (N) that live on SU q (n), for n < N. Proposition 2.29. Let π be a representation of SU q (N), let η be a π-ε-cocycle and let n < N.

(a) The following are equivalent.

(i) π lives on SU q (n).

(ii) π(u kj ) = δ kj I if max{j, k} > n.

(iii) π(u jj ) = I for n < j ≤ N. (b) Suppose that π lives on SU q (n). Then the following are equivalent.

(i) η lives on SU q (n).

(ii) η(u kj ) = 0 if max{j, k} > n.

(iii) η(u jj ) = 0 for n < j ≤ N.

Proof. For both parts, the equivalence of (i) and (ii) follows from Proposition 2.27 because (ii) says π, respectively η, vanishes on the set S n,N (in the latter case, since cocycles kill the identity element), moreover (ii) obviously implies (iii).

(a) For all j = 1, • • • , N, the unitarity relations (2.7) imply the identities

π(u jj ) * π(u jj ) + k =j π(u kj ) * π(u kj ) = I = π(u jj )π(u jj ) * + k =j π(u jk )π(u jk ) *
so if π(u jj ) = I then π(u kj ) = 0 for k = j. Thus (iii) implies (ii). (b) By Identities (2.12a) and (2.12b), if η(u ll ) = 0 then η(u il ) = 0 = η(u li ) for i < l and so (iii) implies (ii).

Classification of gaussian generating functionals

In this Section we investigate the gaussian generating functionals on SU q (N) and their Schürmann triples. We follow Procedure 2.6 for gaussian representations, that is representations of the form ι h • ε : a → ε(a)I h . Since gaussian cocycles vanish on K 2 , we seek a hermitian basis extension E from K 2 to K (see Section 2).

Lemma 3.1. Set v j := (u jj -1) ∈ K and d j := (2i) -1 (u jj -u * jj ) = (2i) -1 (v j -v * j ) ∈ K. Then the following hold. (a) u jk ∈ K 2 for j = k. (b) v j + v * j ∈ K 2 . (c) d 1 + • • • + d N ∈ K 2 . (d) d j d k -d k d j ∈ K 3 .
Proof. (a) Let j = k. Combining Relations (2.6a) and (2.6b), one has u jk u ll = qu ll u jk for j = k and l := max(j, k). Therefore, since u ll -1, u jk ∈ K, u jk = (1 -q) -1 q(u ll -1)u jk -u jk (u ll -1) ∈ K 2 .

(b) By the unitarity relation (2.7) we see that

1 -u jj u * jj = m =j u jm u * jm ∈ K 2 , so v j + v * j = (u jj -1) + (u jj -1) * = -(1 -u jj u * jj ) -(u jj -1)(u jj -1) * ∈ K 2 . (c) Observe that u 11 • • • u N N = (v 1 + 1) • • • (v N + 1) = 1 + (v 1 + • • • + v N ) + terms in K 2 . Therefore, v 1 + • • • + v N + (1 -u 11 • • • u N N ) ∈ K 2 . Since D q = 1, we have 1 -u 11 • • • u N N = σ∈S N ,σ =id (-q) i(σ) u 1,σ(1) • • • u N,σ(N ) . (3.1)
Now, for σ = id there is at least one j such that j = σ(j), so, from part (a), the right-hand side of (3.1) is in

K 2 . Thus v 1 + • • • + v N ∈ K 2 , hence, d 1 + • • • + d N = (2i) -1 (v 1 + • • • + v N ) -(v 1 + • • • + v N ) * ∈ K 2 .
(d) This follows from part (a), in view of the relations (2.6d) and (2.8a).

Now consider the family of characters determined by

ε θ 2 ,••• ,θ N (u kl ) := e iθ k δ k,l (k, l ∈ {1, • • • , N}), for θ 2 , • • • , θ N ∈ R
ε ′ j := ∂ ∂θ j θ 2 =•••=θ N =0 ε θ 2 ,••• ,θ N (j = 2, • • • , N) (3.2)
are drifts because they kill 1 (since each

ε θ 1 ,••• ,θ d is a character), vanish on K 2 (by Leibniz' rule, since ε 0,••• ,0 = ε) and are hermitian (since d * k = d k and ε ′ j (d k ) = δ jk ). Lemma 3.2. Set E := {d 2 , • • • , d N }.
Then the following hold.

(a) E is a hermitian basis extension from K 2 to K.

(b) {ε ′ j : j = 2, • • • , N} is a basis for the real space of drifts on SU q (N). Proof. The set E is hermitian and it follows from parts (a), (b) and (c) of Lemma 3.1 that

E ∪ K 2 spans K. For j, k = 2, • • • , N, ε ′ j (d k ) = δ
jk so E is linearly independent, and ε ′ j kills K 2 so E and K 2 are disjoint. Thus (a) holds, and so does (b) since drifts vanish on {1} ∪ K 2 .

In view of part (d) of Lemma 3.1 and Corollaries 2.13 and 2.12, we deduce the following. Theorem 3.3. SU q (N) does not have property (GC) unless N 2. This is also proved in [START_REF] Das | Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups[END_REF]. SU q (N) has Property (AC) if N = 2 ([23], [START_REF] Schürmann | Infinitesimal generators on the quantum group SU q (2), Infin[END_REF]). From now on, we fix the hermitian basis extension

E N := {d 2 , • • • , d N } from K 2 to
K, and thereby also the projection in P 2 (SU q (N)) as in (2.4), which we denote P N . The resulting family of projections is compatible with the subgroup relations SU q (N) SU q (n). Proof. The epimorphism s N (see (2.10)) sends d N to 0 and, for 2 n N -1, sends the d n of SU q (N) to the d n of SU q (N -1), so s N (E N ) = E N -1 ∪ {0}. Therefore, by Lemma 2.25, P N -1 • s N = s N • P N . By Identity (2.11) this iterates to yield the proposition.

Note that the ε ′ j obtained in (3.2) coincide with the functionals ε ′ d (d = d j ) defined in (2.3) from the basis extension E N . Thus Proposition 2.10 yields the following characterization.

Proposition 3.5. The gaussian cocycles on SU q (N) are precisely the maps of the form

η = N j=2 ξ j ε ′ j (•) (3.3)
for a family of vectors (ξ j ) N j=2 in a Hilbert space h. We next describe the gaussian generating functionals on SU q (N). Consider the pointwise defined functionals

ε ′′ jk := ∂ 2 ∂θ j ∂θ k θ 2 =•••=θ N =0 ε θ 2 ,••• ,θ N (j, k = 2, • • • , N).
Theorem 3.6. Letting M n (R) + denote the set of real nonnegative-definite n × n matrices, the prescription

(r, R) → N j=2 r j ε ′ j + 1 2 N j,k=2
r jk ε ′′ jk defines a bijection from R N -1 × M N -1 (R) + to the set of gaussian generating functionals γ on SU q (N) in which the second sum is the P N -invariant component γ • P N .

Proof. In view of Lemma 3.2, it suffices to verify that the prescription [r jk ] → 1 2 N j,k=2 r jk ε ′′ jk defines a bijection from M N -1 (R) + to the set of P N -driftless (i.e. P N -invariant) gaussian generating functionals γ.

First note that by Leibniz' rule,

ε ′′ jk (ab) = ε ′′ jk (a)ε(b) + ε ′ j (a)ε ′ k (b) + ε ′ k (a)ε ′ j (b) + ε(a)ε ′′ jk (b) (a, b ∈ SU q (N)).
It follows that ε ′′ jk vanishes on K 3 and, by direct computation,

ε ′′ jk (d l ) = 0 and ε ′′ jk (d l d m ) = δ jl δ km + δ jm δ kl for j, k, l, m = 2, • • • , N. In particular, ε ′′ jk • P N = ε ′′ jk and, for all c ∈ K and λ ∈ C N -1 , λ j ε ′′ jk (c * c)λ k = 2 λ k ε ′ k (c) 2
0 so, since nonnegative-definiteness is preserved under the Schur product, for any matrix

R = [r jk ] ∈ M N -1 (R) + the functional 1 2 r jk ε ′′
jk is conditionally positive and therefore a P N -invariant gaussian generating functional.

Conversely, if γ is a gaussian generating functional, its associated cocycle η is of the form (3.3) and so, by Corollary 2.13 and part (d) of Lemma 3.1, η is hermitian and hence the Gram matrix ξ j , ξ k is real and thus in M N -1 (R) + .

Remarks 3.7. The CQG algebra T N -1 of the torus T N -1 is generated, as unital * -algebra, by a family of commuting unitaries {u j :

j = 1, • • • , N} subject to the relation u 1 • • • u N = 1. The prescription u jk → δ jk u j determines a CQG epimorphism τ N : SU q (N) → T N -1 with respect to which the characters ε θ 2 ,••• ,θ N of SU q (N) live on T N -1 .
Therefore the gaussian generating functionals of SU q (N) live on T N -1 . It also follows that, for any compact quantum group G satisfying SU q (N) G T N -1 , the projection P ∈ P 2 (G) may be chosen to be compatible with those for SU q (N) and T N -1 , and the gaussian generating functionals of G correspond to those of T N -1 . Application of results on classical compact Lie groups in [START_REF]Hunt's formula for SU q (2) -a unified view[END_REF] to T N -1 gives an alternative proof of Theorem 3.6. The original preprint version of our paper has motivated generalisation of the theorem to all q-deformations of simply connected semisimple compact Lie groups ([7, Theorem 6.1]).

Decomposition

This is the central section of the paper. We decompose an arbitrary representation π of SU q (N) uniquely into a direct sum π 1 ⊕• • •⊕π N , in which π 1 = π G , as defined in Proposition 2.16 and, for 2 n N, π n lives on SU q (n) and π n (1 -u nn ) is injective. We then show that in the corresponding decomposition η 1 ⊕ • • • ⊕ η N of a π-ε-cocycle η, for 2 n N each cocycle η n is approximately inner and determined by the vector η(u nn ). This implies that SU q (N) has Property (NAI) and so also (LK). We deduce a Hunt formula for SU q (N) incorporating full decomposition for generating functionals.

The following elementary lemma plays a key role in the approximation of cocycles (part (a) is well-known, for example in ergodic theory). For bounded operators T , we write ran T for ran T . where P := P ker(I-C) . In particular, the following four conditions are equivalent.

(i) I -C is injective; (i) ′ I -C has dense range; (ii) (I -tC) -1 (I -C) SOT --→ I as t → 1 -; (ii) ′ (1 -t)(I -tC) -1 SOT --→ 0 as t → 1 -. Proof. (a) Let ξ ∈ ker(I -C) = ran(I -C * ) ⊥ = ran(C * -I) ⊥
. By symmetry it suffices to prove that ξ ∈ ker(I -C * ). This follows by Pythagoras:

ξ 2 + (C * -I)ξ 2 = C * ξ 2 ξ 2 . (b) For 0 < t < 1, (1) I -P (t) = tP ⊥ (t), (2) P (t)
1, and ( 3)

P (t)(I -C) = (1 -t)P ⊥ (t); thus (4) P (t)(I -C)
2(1 -t)/t. By (1), P (t) → I on ker P ⊥ (t) = ker(I -C) and, by ( 4) and (2), P (t) → 0 on ran(I -C). Hence P (t) SOT --→ P by (a), and so P ⊥ (t) SOT --→ P ⊥ by [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF].

Decomposition of representations and cocycles. We start by separating out the maximal subspace on which the operator π(1 -u N N ) acts injectively, for a given representation π.

Lemma 4.2. Let π be a representation of SU q (N). Then π has a unique decomposition π N ⊕ π N for which π N lives on SU q (N -1), equivalently π N (1 -u N N ) = 0, and

π N (1 -u N N ) is injective. Moreover, h π N = ker π(1 -u N N ).
Proof. The equivalence is contained in Proposition 2.29. We first show that k := ker π(1 -u N N ) is an invariant subspace for π. Since the u jk generate SU q (N) as an algebra (Remark 2.26), to see this it suffices to fix ξ ∈ k and j, k ∈ {1, • • • , N}, and to verify that π jk ξ ∈ k (in the convenient abbreviation π jk := π(u jk )). For j = k = N this is obvious. For k < N, applying π to Identity (2.7) then the vector functional ω ξ , we see that π * N s ξ = 0 = π sN ξ for s < N so, by Identity (2.8d),

π * N k π N k ξ = π N k π * N k ξ + (1 -q 2 ) m<k π N m π * N m ξ = 0, thus π N k ξ = 0. Lastly, for j, k < N, π jk π N N ξ = π N N π jk ξ -(q -1 -q)π jN π N k ξ by Identity (2.6d), so π jk π N N ξ = π N N π jk ξ, in other words π jk ξ ∈ k, as required.
In the resulting decomposition π = π N ⊕ π N , π N (1 -u N N ) = 0 and π N (1 -u N N ) is injective. It remains to prove uniqueness. Thus let ρ ⊕ σ be another such decomposition of π; we must show that h ρ = k. This follows from Lemma 4.1:

h ρ = ker ρ(1 -u N N ) ⊂ k = ran π(1 -u N N ) ⊥ ⊂ ran σ(1 -u N N ) ⊥ = (h σ ) ⊥ = h ρ . Definition 4.3. A decomposition π 1 ⊕ • • • ⊕ π N of a representation of SU q (N) is full if (1) for 1 n < N, there is a representation π n of SU q (n) such that π n = π n • s n,N and, ( 2
) for n 2, π n (1 -u nn ) is injective.
For n = 1, (1) says that π n is gaussian, and for n 2, π n (1

-u nn ) = π n (1 -u n nn ) where u n nn denotes u nn in SU q (n); (2)
is equivalent to π(1 -u nn ) having dense range for n 2. This superscript convention, indicating which quantum subgroup is being referred to, continues below.

Theorem 4.4. Every representation of SU q (N) has a unique full decomposition.

Proof. We prove this by induction on N. For N = 1 there is nothing to prove. Suppose therefore that the proposition holds for N = K -1 for some K 2, and let π be a representation of SU q (K).

Existence.

By Lemma 4.2, π = π K ⊕ π K where π K (1 -u KK ) is injective and π K = π • s K for a representation π of SU q (K -1). By the induction hypothesis, π = ρ 1 ⊕ • • • ⊕ ρ K-1 where ρ 1 is gaussian and, for k = 2, • • • , K -1, ρ k (1 -u K-1 kk ) is injective and ρ k = ρ k • s k,K-1 , for some representation ρ k of SU q (k). Set π k := ρ k • s K for k = 1, • • • , K -1. Then π = π 1 ⊕ • • • ⊕ π K , where π 1 is gaussian, π K (1 -u KK ) is injective and, for k = 2, • • • , K -1, π k (1 -u kk ) equals ρ k (1 -u K-1
kk ) and so is injective, and

π k = ρ k • s k,K-1 • s K = ρ k • s k,K , so π k lives on SU q (k). Uniqueness. Suppose that π = ρ 1 ⊕ • • • ⊕ ρ K is another such decomposition. Then, by the uniqueness part of Lemma 4.2, ρ K = π K and ρ 1 ⊕ • • • ⊕ ρ K-1 = π 1 ⊕ • • • ⊕ π K-1 . Now, for k = 1, • • • , K -1, π k = π k • s K and ρ k = ρ k • s K for representations π 1 , • • • , ρ K-1 of SU q (K -1)
and, by the surjectivity of s

K , π 1 ⊕ • • • ⊕ π K-1 = ρ 1 ⊕ • • • ⊕ ρ K-1 . Since π 1 and ρ 1 are gaussian and, for k = 2, • • • , K -1, π k and ρ k live on SU q (k) and π k (1 -u K-1 kk ) and ρ k (1 -u K-1 kk ) are injective, it follows from the induction hypothesis that π k = ρ k for k = 1, • • • , K -1. Therefore π k = ρ k for k = 1, • • • , K, as required. Theorem 4.5. Let π 1 ⊕ • • • ⊕ π N be
the full decomposition of a representation π of SU q (N) and let η 1 ⊕ • • • ⊕ η N be the induced decomposition of a π-ε-cocycle η. Then η 1 is gaussian and, for n 2, η n lives on SU q (n).

Proof. For n = 1, the cocycle η n is gaussian since the representation π n is. For m > n 2, by part (a) of Proposition 2.29 applied to Identity (2.6d),

π n (u nn )η n (u mm ) + η n (u nn ) = π n (u mm )η n (u nn ) + η n (u mm ) -(q -1 -q)π n (u nm )η n (u mn ) = η n (u nn ) + η n (u mm ), so η n (u mm ) ∈ ker π n (1-u nn ) = {0} thus, by part (b) of Proposition 2.29, η n lives on SU q (n).
Approximation of cocycles and (NAI) for SU q (N). We now show that each of the cocycles η n (n 2) in Theorem 4.5 is approximately inner. Proof. In view of the cocycle relations and Remark 2.26, it suffices to prove that, for each of the algebra generators a = u jk , η(a) is the pointwise limit as t → 1 -of the following expression

-π(a -ε(a)1)π(1 -tu N N ) -1 η(u N N ). (4.1)
We prove this using Lemma 4.1 (the contraction operator lemma) and Lemma 2.28.

Case a = u N N . Lemma 4.1 implies that π(1 -tu N N ) -1 π(1 -u N N )η(u N N ) → η(u N N ). Case a = u kN or a = u N k (k < N).
Then a ∈ ker ε so π(a) = π(a -ε(a)1). Thus, using Relations (2.12a)-(2.12b), Lemma 4.1 implies that η(a) equals

-π(1 -qu N N ) -1 π(a)η(u N N ) = -lim t→1 -π(1 -qu N N ) -1 π(a)π(1 -tu N N ) -1 π(1 -u N N )η(u N N ) = -lim t→1 -π(a -ε(a))π(1 -tu N N ) -1 η(u N N ). Case a = u jk (j, k < N). We must show that -π(u jk -δ jk 1)π(1 -tu N N ) -1 η(u N N ) → η(u jk ). By the contraction operator lemma -π(1 -tu N N ) -1 π(u N N -1)η(u jk ) → η(u jk ). It therefore suffices to show that -π(1 -tu N N ) -1 π(u N N -1)η(u jk ) + π(u jk -δ jk 1)π(1 -tu N N ) -1 η(u N N ) → 0.
By Identity (2.12c) the first term equals and so, since the operators π(1 -q 2 u N N ) -1 and π(1 -tu N N ) -1 commute, after cancellation of the δ jk terms and multiplication through by the invertible operator π(1 -q 2 u N N ) we see that the task is equivalent to showing that the following converges to 0 on the vector η(u N N ):

π(1 -q 2 u N N ) π(u jk ), π(1 -tu N N ) -1 + (q -1 -q)π(1 -tu N N ) -1 π(u jN u N k ) (4.2)
-we show that it converges to 0 strongly. Let us abbreviate π(u il ) to π il for each i and l. It follows from Identity (2.6d) that

[π jk , π α N N ] = -(q -1 -q) α-1 ν=0 q 2ν π α-1 N N π jN π N k (α ∈ Z + ),
thus, taking the Neumann series for (I -tπ N N ) -1 , which is valid since tπ N N is a strict contraction,

π jk , (I -tπ N N ) -1 = -(q -1 -q) ∞ α=1 α-1 ν=0 q 2ν t α π α-1 N N π jN π N k .
Substituting this into (4.2) then gives the following operator composed with (q -1 -q)π jN π N k :

-(I -q 2 π N N ) ∞ α=1 α-1 ν=0 q 2ν t α π α-1 N N + (I -tu N N ) -1 = ∞ ν=0 ∞ α=ν+1 q 2(ν+1) (tπ α N N -tq 2ν (tπ N N ) α-1 + (I -tu N N ) -1 = ∞ ν=0 (q 2 tπ N N ) ν+1 ∞ β=0 (tπ N N ) β -t(q 2 tπ N N ) ν ∞ β=0 (tπ N N ) β + (I -tu N N ) -1 = (I -q 2 tu N N ) -1 q 2 tπu N N -tI + I -q 2 tu N N I -tu N N ) -1 = (I -q 2 tu N N ) -1 (1 -t)(I -tu N N ) -1
so the required convergence follows from Lemma 4.1.

Theorem 4.7. Let π 1 ⊕ • • • ⊕ π N be the full decomposition of a representation π of SU q (N) and let η 1 ⊕ • • • ⊕ η N be the induced decomposition of a π-ε-cocycle η. Then, for n 2, η n = pw-lim t→1 -η πn,ξ(n,t) where ξ(n, t) := -π n (1 -tu nn ) -1 η n (u nn ).

Thus, in terms of the decomposition

h π = h π G ⊕ h π R , η = pw-lim t→1 -η G ⊕ η π R ,ξ(t)
where

ξ(t) := -π 2 (1 -tu 22 ) -1 η 2 (u 22 ) ⊕ • • • ⊕ π N (1 -tu N N ) -1 η N (u N N ). (4.3)
Proof. Let n 2. By Theorem 4.5, η n = η n •s n,N for a cocycle η n on SU q (n) and, by Lemma 2.22, it suffices to prove that η πn,ξ(n,t) converges pointwise to η n . Now

π n (1 -u nn ) is injective (by Theorem 4.4), π n (1 -tu n nn ) = π n (1 -tu nn ) for all t ∈ [0, 1] and η n (u n nn ) = η n (u nn ) so π n (1 -u n nn ) is injective and ξ(n, t) = -π n (1 -tu n nn ) η n (u n nn ).
The theorem therefore follows by applying Proposition 4.6 with N = n.

Noting that if π is completely non-gaussian, so h π G = {0}, then (4.3) simplifies to the pointwise convergence η π.ξ(t) → η as t → 1 -, we draw the following immediate corollary. Theorem 4.8. SU q (N) has property (NAI), and thus also (LK).

Decomposition of generating functionals and Hunt formula for SU q (N). Lemma 4.9. Let (π ′ , η ′ ) and (π ′′ , η ′′ ) be cyclic representation-cocycle pairs on SU q (N) such that (π ′ , η ′ ) lives on SU q (N -1) and π ′′ (1 -u N N ) is injective. Then the following hold.

(a) The cocycle η ′ vanishes on (1

-u N N )K. (b) The set η ′′ ((1 -u N N )K) = π ′′ (1 -u N N )η ′′ (K) is dense in h π ′′ . (c) The cocycle η ′ ⊕ η ′′ is cyclic.
Proof. (a) This follows since π ′ (1-u N N ) = 0 because π ′ lives on SU q (N -1) and 1-u N N ∈ ker s N . (b) By Lemma 4.1, π ′′ (1 -u N N ) has dense range so this follows from the cyclicity of η ′′ . (c) The cyclicity of η ′ ⊕ η ′′ follows from that of η ′ and η ′′ since, for c 1 , c 2 ∈ K, by part (b) there is a sequence

(d p ) in K such that η ′′ ((1 -u N N )d p ) → η ′′ (c 2 -c 1 ), and by part (a) η ′ ((1 -u N N )d p ) = 0 for all p so (η ′ ⊕ η ′′ )(c 1 + (1 -u N N )d p ) = η ′ (c 1 ) η ′′ (c 1 ) + η ′′ ((1 -u N N )d p ) → η ′ (c 1 ) η ′′ (c 2 ) as p → ∞.
Definition 4.10. Let N 2. We say that a completely non-gaussian generating functional γ on SU q (N) is gf-irreducible if the following holds: for any generating functional decomposition γ = γ ′ + γ ′′ , if γ ′ lives on SU q (N -1) then it is a drift. Proposition 4.11. Let γ be a generating functional on SU q (N) for N 2, and let (π, η, γ) be its Schürmann triple. Then γ is gf-irreducible if and only if π(1 -u N N ) is injective.

Proof. Suppose first that γ is gf-irreducible. By Theorems 4.4 and 4.5 and Propositions 4.6 and 2.8, π and η decompose as π N ⊕ π N and η N ⊕ η N , where π N (1 -u N N ) is injective, η N lives on SU q (N -1) and η N is approximately inner and so completable by a P N -invariant generating functional γ N . The normalised hermitian functional γ 2 for all c ∈ K and so is a generating functional which completes η N and thus also lives on SU q (N -1), and satisfies γ N + γ N = γ. Thus γ N is a drift and so η N = 0. But η N is cyclic (since η is) and so h π N = {0} thus π = π N and so π(1 -u N N ) is injective.

N := γ -γ N satisfies γ N (c * c) = η(c) 2 -η N (c) 2 = η N (c)
Suppose conversely that π(1 -u N N ) is injective, and let γ ′ + γ ′′ be a generating functional decomposition of γ such that γ ′ lives on SU q (N -1). Let (π ′ , η ′ , γ ′ ) and (π ′′ , η ′′ , γ ′′ ) be the Schürmann triples of γ ′ and γ ′′ . Then (π ′ , η ′ , γ ′ ) lives on SU q (N -1), so η ′ vanishes on (1-u N N )K by part (a) of Lemma 4.9, also (π ′ ⊕ π ′′ , η ′ ⊕ η ′′ , γ) is a Schürmann triple so there is an isometry

V ∈ B(h π ; h π ′ ⊕ h π ′′ ) such that η ′ (c)
η ′′ (c) = V η(c) for all c ∈ K. In view of part (b) of Lemma 4.9, these together imply that η ′ = 0, so γ ′ is a drift. Therefore γ is gf-irreducible. Definition 4.12. A generating functional decomposition γ

= γ 1 + • • • + γ N on SU q (N) is full if (1) for 1 n < N, γ n = γ n • s n,N for a generating functional γ n on SU q (n), and (2)
for n 2, γ n is gf-irreducible and P n -invariant.

For n = 1, (1) says that γ n is gaussian. Given (1), letting ( π n , η n , γ n ) be γ n 's Schürmann triple, so that (π

n := π n •s n,N , η n := η n •s n,N , γ n := γ n •s n,N ) is γ n 's Schürmann triple, the condition (2) is equivalent to (2) ′ : π n (1 -u nn ) is injective and γ n is P N -invariant, by Proposition 4.11 (since π n (1 -u nn ) = π n (1 -u n nn ))
, and the compatibility of the family of projections (Proposition 3.4). Lemma 4.13. If a generating functional γ on SU q (N) has a full decomposition γ

1 + • • • + γ N then, in terms of each γ n 's Schürmann triple (π n , η n , γ n ), (a) π 1 ⊕ • • • ⊕ π N is a full (representation) decomposition, and (
b) the cocycle η 1 ⊕ • • • ⊕ η N is cyclic. (3) γ N G = pw-lim t→1 -ω ξ(t) • π R • P N where π R is the non-gaussian remainder of π and ξ(t) := -π 2 (1 -tu 22 ) -1 η 2 (u 22 ) ⊕ • • • ⊕ π N (1 -tu N N ) -1 η N (u N N ).
The realisation of γ N G in ( 3) is analogous to that of γ L in the classical Hunt formula (1.1) given in the remark following Proposition 2.8.

For the liming case q = 1 corresponding to the compact Lie group SU(N) the proofs of Lemma 3.1 and Theorem 4.7, on which our Hunt formula depends, are no longer vadid. However, the theorem as stated still holds. Indeed the gaussian/nongaussian decomposition and parameterisations (1) are statements of Hunt's results in the language of generating functionals, moreover (2) is seen by decomposing the Lévy measure into its restrictions to the corresponding subgroups of SU(N).

From parametrization by h π to quasi-innerness

Given a gf-irreducible generating functional on SU q (N), with Schürmann triple (π, η, γ), by Proposition 4.11 and Lemma 2.28 we know that π(1 -u N N ) is injective and so η is determined by its value η(u N N ). One may therefore ask which vectors of the representation space h π arise in this way. In case N = 2 every vector does, so the cocyles are parameterised by h π ([23, Theorem 2.8]; [START_REF] Schürmann | Infinitesimal generators on the quantum group SU q (2), Infin[END_REF]Theorem 3.3]). We now show this to be false for N = 3; the argument extends to higher values of N. The section ends with an indication of a positive counterpart to this, namely a quasi-innerness property of completely non-gaussian cocycles/π-ε-derivations. Proposition 5.1. There is a representation π of SU q (3) and vector ξ in h π such that π(1 -u 33 ) is injective but η(u 33 ) = ξ for every π-ε-cocycle η.

Proof. Following Woronowicz, we write the generators u jk of SU q (2) as

u 11 u 12 u 21 u 22 = α -qγ * γ α * .
Let ρ be the irreducible representation of SU q (2) on ℓ 2 (Z + ) defined, in terms of the standard orthonormal basis (e n ) n 0 by ρ(α) : e n → 1 -q 2n e n-1 and ρ(γ) : e n → q n e n

(where e -1 := 0). For k = 1, 2, set ρ k := ρ • r k for the CQG epimorphisms r k : SU q (3) → SU q (2) given by a n e n -1 -q 2(n+1) e n+1 = a 0 e 0 + n 1

a n -a n-1 1 -q 2n e n .

Thus a 0 = 1 and, for n 1, |a n | 2 = n k=1 (1 -q 2k ). Therefore, since |a n | 2 η(u 11 ) 2 < ∞, n k=1 (1 -q 2k ) → 0 as n → ∞ so q 2k diverges and we have our contradiction.

This leaves us with the question, which vectors in h π may occur as values η(u N N ) for a cocycle η. Every element in the dense subspace ran π(1 -u N N ) occurs; and the collection of cocycles determined by them is precisely the set of coboundaries. Indeed, for ξ ′ = -π(1 -u N N )ξ, by the contraction operator lemma we have (see Theorem 4.7) the following pointwise convergence:

-π • (id -ι • ε)(•)π(1 -tu N N ) -1 ξ ′ → π • (id -ι • ε)(•)ξ = η π,ξ as t → 1 -,
and the identity η π,ξ (u N N ) = π(u N N -1)ξ = ξ ′ . Proposition 5.2. Let (ξ(λ)) be a net in h π . Then the net of coboundaries (η π,ξ(λ) ) converges pointwise on SU q (N) provided that it converges on u jj for 1 j N.

Proof. This follows from Remark 2.26 since, for j = k, setting l := max(j, k). Relations (2.6a) or (2.6b) imply that η π,ξ(λ) (u jk ) = π(u jk )ξ(λ) = π(1 -qu ll ) -1 π(1 -qu ll )π(u jk )ξ(λ) = -π(1 -qu ll ) -1 π(u jk )π(u ll -1)ξ(λ) = -π(1 -qu ll ) -1 π(u jk )η π,ξ(λ) (u ll ).

We conclude this section with a quasi-innerness property enjoyed by all completely nongaussian cocycles.

Theorem 5.3. Let π be a completely non-gaussian representation of SU q (N), and let (h π , J) denote the completion of h π with respect to the norm |||•||| : ξ → N j=1 π(1 -u jj )ξ 2 1/2 . Then a net (ξ(λ)) in h π is |||•|||-Cauchy if and only if the corresponding net of π-ε-coboundaries η π,ξ(λ) ) converges pointwise. Moreover, the following hold.

(1) There is a unique operator π : K → B(h π ; h π ) which 'extends' the representation π in the sense that it satisfies π(ac) = π(a)π(c) and π(c)J = π(c) (a ∈ SU q (N), c ∈ K).

(2) The prescription χ → η π,χ := a → π(a -ε(a)1)χ defines a linear isomorphism from h π to the space of π-ε-cocycles.

There is also a unique operator π : This has the property: for all χ ∈ h π , the generating functional ω χ • π • P N completes (π, η π,χ ).

6. The case of U q (N)

A Hunt formula for U q (N) may be obtained by employing very simlar arguments to those used above for SU q (N). The upshot is the same as Theorem 4.15 except that it is with respect to the tower of subgroups U q (0) • • • U q (N) with U q (0) denoting the trivial compact quantum group, rather than the tower SU q (1) • • • SU q (N) (also starting at the trivial group), thus N replaces N -1 in (1), the decomposition in (2) starts at n = 0 rather than n = 1, and the components of ξ(t) in (3) start at n = 1 rather than n = 2. We therefore instead discuss only the (NAI) and (GC) questions for U q (N), as these may easily be deduced from our results and reasoning for the SU q (N) quantum groups.

Since SU q (N + 1) U q (N) T N , it follows from Remarks 3.7 that U q (N) has the same gaussian generating functionals as SU q (N + 1) and a hermitian projection P ′ for U q (N) compatible with that of SU q (N + 1) is the one corresponding to the following choice of basis extension:

E ′ = t N (d n ) : 2 n N ∪ t N (d N +1 ) = (2i) -1 (D -1 -D -1 * ) .
Theorem 6.1. U q (N) does not have property (GC), unless N = 1.

Proof. The reasoning used in the proof of the SU q (N) counterpart (Theorem 3.3) applies. By part (d) of Lemma 3.1, the basis extension E ′ again consists of elements whose commutators lie in K 3 , and dim K/K 2 = N 2 unless N = 1 so Corollaries 2.13 and 2.12 again apply.

Since the (NAI) property is hereditary (Proposition 2.23) and SU q (N +1) has it (Theorem 4.8), U q (N) does too. Theorem 6.2. U q (N) has property (NAI), and thus also (LK).

. 11 ) 2 . 27 .

 11227 Proposition Let 1 ≤ n < N. The kernel of s n,N equals the ideal I generated by the set S n,N := u kj -δ kj 1 : 1 j, k N, max{j, k} > n .

and θ 1

 1 given implicitly by N k=1 θ k = 0. The pointwise defined linear functionals

Proposition 3 . 4 .

 34 P n • s n,N = s n,N • P N for n < N.

Lemma 4 . 1 (

 41 Contraction operator lemma). For any contraction operator C on a Hilbert space, (a) ker(I -C * ) = ker(I -C), so also ran(I -C) = ker(I -C) ⊥ = ran(I -C * ), and (b) P (t) := (1 -t)(I -tC) -1 SOT --→ P and P ⊥ (t) := (I -tC) -1 (I -C) SOT --→ P ⊥ as t → 1 -,

Proposition 4 . 6 .

 46 Let η be a cocycle of a representation π of SU q (N) such that π(1 -u N N ) is injective. Then η = pw-lim t→1 -η π,ζ(t) where ζ(t) := -π(1 -tu N N ) -1 η(u N N ).

r 1 :(so r 1 = 3 i=1ρ 1

 1131 [u jk ] s 3 ). Then ρ 1 and ρ 2 are representations of SU q (3) and so, settingπ := ρ 1 ⋆ ρ 2 , π(u jk ) j,k = (u ji ) ⊗ ρ 2 (u ik ) ) ⊗ I -qρ(γ) * ⊗ ρ(α) q 2 ρ(γ) * ⊗ ρ(γ) * ρ(γ) ⊗ I ρ(α) * ⊗ ρ(α) -qρ(α) * ⊗ ρ(γ) * 0 I ⊗ ρ(γ) I ⊗ ρ(α) *   . Now π(1 -u 33 ) = I ⊗ ρ(1 -α * ) is injective because ρ(1 -α * ) is.Suppose for a contradiction that there is a π-ε-cocycle η satisfying η(u 33 ) = e 0 ⊗ e 0 . Since π(u 31 ) = 0 and ρ(α)e 0 = 0, Relation (2.12c) for j = 1 = k implies that(I ⊗ ρ(1 -α * ))η(u 11 ) = (I -π(u 33 ))η(u 11 ) = (I -π(u 11 ))η(u 33 ) = ρ(1 -α)e 0 ⊗ e 0 = e 0 ⊗ e 0 .For n 0, set a n := e 0 ⊗ e n , η(u 11 ) . Then, since ρ(α * )e n = 1 -q 2(n+1) e n+1 , e 0 = n 0 a n (I -ρ(α * ))e n = n 0

K 2 →

 2 B(h π ) such that π(c * c) = π(c) * π(c) and J * π(e)J = π(e) (c ∈ K, e ∈ K 2 ).

  are Schürmann triples (equivalently, by Lemma 2.14, one of them is). 2.18. With respect to a fixed projection P ∈ P 2 (G), if γ has such a Lévy-Khintchine decomposition then it has a unique one in which γ 1 = γ P D + γ G , γ 2 = γ R , and the generating functionals γ G and γ R are P -invariant.

	Remark

Definition 2.19. A CQG algebra, or its associated quantum group, is said to have property • (AC) if each cocycle η is completable. • (GC) if each gaussian cocycle η is completable. • (NC) if each completely non-gaussian cocycle η is completable. • (NAI) if each completely non-gaussian cocycle η is approximately inner. • (LK) if every generating functional admits a Lévy-Khintchine decomposition.

Acknowledgments. We gratefully acknowledge the MFO for a two-week Research in Pairs stay at Oberwolfach, where this work began. UF was supported by the French 'Investissements d'Avenir' programme, project ISITE-BFC (contract ANR-15-IDEX-03) and by an ANR project (No. ANR-19-CE40-0002). AK was supported by the Polish National Science Center grant SONATA 2016/21/D/ST1/03010 and by the Polish National Agency for Academic Exchange in the framework of POLONIUM programme PPN/BIL/2018/1/00197/U/00021. UF and JML were supported by the UK-France Alliance Hubert Curien Programme of research collaboration.

Proof. Let γ = γ 1 + • • • + γ N be such a decomposition. For each n denote by ( π n , η n , γ n ) the induced Schürmann triple on SU q (n), noting that for n = 2, • • • , N, γ n is gf-irreducible and, by (2) ′ , the operator π n (1 -u nn ) is injective and γ n is P N -invariant, in particular (a) holds.

(b) For N = 1 there is nothing to prove. Suppose therefore that the proposition holds for N = K -1 where K 2, and that a generating functional γ on SU q (K) has a full decomposition

Theorem 4.14. Every generating functional γ on SU q (N) has a unique full decomposition.

Proof. Existence. Let γ be a generating functional on SU q (N) and let (π, η, γ) be its Schürmann triple. By Theorem 4.4, π has a full decomposition

the corresponding decomposition of η. By Theorems 4.5 and 4.7, η n lives on SU q (n) for each n and, for n = 2, • • • , N, η n is approximately inner and thus completable by a P N -invariant generating functional γ n , so γ n also lives on SU q (n). Moreover, letting ( π, η, γ) be the induced Schürmann triple on SU q (n), π n (1 -u n nn ) equals π n (1 -u nn ) and so is injective, thus γ n is gf-irreducible by Proposition 4.11. Now the functional γ 2 for all c ∈ K and so is a generating functional which completes η 1 ; moreover it is gaussian because π 1 is. It follows that

N are full (representation) decompositions and (π, η, γ) and (π ′ , η ′ , γ) are cyclic Schürmann triples. Therefore there is a unitary operator

for each n. Thus, by the uniqueness part of Theorem 4.4, for each n, π

Combining the theorems of this section with Theorem 3.6 and Remarks 2.18 and 2.4, we deduce our main result. Theorem 4.15 (Hunt formula for SU q (N)). Let γ be a generating functional on SU q (N). Then there is a unique decomposition γ = γ D + γ G + γ N G , in which γ D is a drift, and γ G and γ N G are P N -invariant generating functionals which are respectively gaussian and completely nongaussian. Moreover, the following hold.

(1) γ G and γ D are uniquely parameterised by a matrix in M N -1 (R) + and vector in R N -1 .

(