Hunt's Formula for S U q (N) and U q (N)
aa r X i v : . [ m a t h . OA ] S e p HUNT’S FORMULA FOR SU q ( N ) AND U q ( N ) UWE FRANZ, ANNA KULA, J. MARTIN LINDSAY, MICHAEL SKEIDE
Abstract.
We provide a Hunt type formula for the infinitesimal generators of Lévy processon the quantum groups SU q ( N ) and U q ( N ) . In particular, we obtain a decomposition of suchgenerators into a gaussian part and a ‘jump type’ part determined by a linear functional thatresembles the functional induced by the Lévy measure. The jump part on SU q ( N ) decomposesfurther into parts that live on the quantum subgroups SU q ( n ) , n ≤ N . Like in the classical Huntformula for locally compact Lie groups, the ingredients become unique once a certain projectionis chosen. There are analogous result for U q ( N ) . Contents
1. Introduction 22. Preliminaries 52.1. Generating functionals of Lévy processes 52.2. Schürmann triples and the projection P SU q ( N ) and U q ( N ) U q ( N ) Introduction
Let G denote a locally compact Lie group. All information about a Lévy process with values in G can be captured (up to stochastic equivalence) by its infinitesimal generator, a densely definedlinear functional ψ on the C ∗ -algebra C ( G ) of bounded continuous functions on G vanishing atinfinity. The domain may be thought of as those functions that possess a second order Taylorexpansion around the neutral element of G , e . Hunt’s formula [15], a generalization of theLévy-Khintchine formula for R (see for instance Applebaum [3] or Sato [21]), asserts that ψ ( f ) = ψ G ( f ) + Z G\{ e } [ P ( f )]( g ) d L ( g ) . The gaussian part ψ G is a linear combination of first and second order derivatives at theneutral element, P is an arbitrary hermitian projection that takes away the linear terms, and L is the Lévy measure (which may have a singularity up to order two at e ). Defining the Lévyfunctional L ( f ) := R G\{ e } f ( g ) d L ( g ) on the functions which together with their first derivativesvanish at e and putting ψ L = L ◦ P , this reads(1.1) ψ = ψ G + ψ L . Since the integral may be viewed as a mixture of point evaluations, and since, for fixed g = e , agenerating functional of the form f f ( g ) − f ( e ) generate a jump process , L ◦ P is sometimesalso referred to as the jump part . (In the case G = R , we get a compound Poisson process ). Remark . Note that for the pure point evaluation, no P is necessary; P is necessary only todeal with the singularity at e of the Lévy measure. Note, too, that there is (usually) no canonicalchoice for P ; this is, why in literature the classical Lévy-Khintchine formula for G = R maylook quite different depending on the reference. The projection P will keep us quite busy; seeSubsections 2.2–2.5 and Sections 3 and 6. If G is compact, the well known Tannaka-Krein duality (see, for instance, Hewitt and Ross[14, Section VII.30]) asserts that the coefficient algebra R ( G ) (consisting of coefficients offinite-dimensional representations of G ) is a norm dense ∗ -subalgebra of the C ∗ -algebra C ( G ) ofcontinuous functions. Actually, R ( G ) is a commutative Hopf ∗ -algebra , and the structure ofthe topological group G may be recovered from the Hopf ∗ -algebra R ( G ) .More generally, a compact quantum group G = ( C ( G ) , ∆) in the sense of Woronowicz [35],which is roughly speaking a unital C ∗ -algebra C ( G ) with an additional structure reflecting thegroup properties on the level of functions on a group, always contains a dense ∗ -subalgebra R ( G ) that may be turned into a Hopf ∗ -algebra ([35, Theorem 1.2]). This opens up the way to applySchürmann’s theory of quantum Lévy processes on ∗ -bialgebras [23] to both situations.Like their classical counterparts, Lévy processes on ∗ -bialgebras are classified (up to quantumstochastic equivalence) by their generating functionals . A generating functional is a linearfunctional on R ( G ) fulfilling certain algebraic conditions; see Subsection 2.1. Finding generatingfunctionals amounts to the solution of a cohomological problem; see Subsection 2.2. Decom-posing a generating functional ψ into a sum, means, roughly speaking, that the cohomologicalproblem has to be solved for the constituents, individually; see Subsection 2.3. Of course, tomerit being called a Lévy-Khintchine decomposition , a decomposition as in (1.1) has to
UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) satisfy more: We have to say what a gaussian part is (a quadratic part, in the sense that itis on all monomials of degree and higher) and what a completely non-gaussian part is (nogaussian part can be subtracted); see Subsection 2.5.A “true” Hunt formula goes further. It includes an explicit description of the gaussiangenerating functionals. And it includes a certain approximation property that justifies to callthe completely non-gaussian part a jump part. The approximation property we require to callan approximated generating functional a jump part , is explained in Subsection 2.3; a justifica-tion/motivation is given in Remark 2.17.It is noteworthy that not all quantum groups allow to decompose every generating functionalinto a (maximal) gaussian and a completely non-gaussian part; see Franz, Gerhold, and Thom[13, Proposition 4.3]; therefore, already the answer to the question if the decomposition problemhas a solution or not, depends on the example under consideration.Schürmann and Skeide [27, 24] established Hunt’s formula in this sense on Woronowicz’s SU q (2) [33]. Skeide [28] applied Schürmann’s ideas to obtain a quick proof of Hunt’s formula forcompact Lie groups. In these notes, we deal with the case SU q ( N ) . We do not only find Hunt’sformula for SU q ( N ) . In Theorem 5.5 we find that a generating functional ψ decomposes (in asense, uniquely) as ψ = ψ G + L ◦ P + . . . + L N ◦ P , where again ψ G is a gaussian part, and where L n ( ≤ n ≤ N ) are (extensions to SU q ( N ) of)Lévy functionals on SU q ( n ) ⊂ SU q ( N ) . En passant , in Section 6, we derive similar results forthe quantum group U q ( N ) .The techniques are inspired quite a bit by [27, 24] for the decomposition and by [28] for thegaussian part. But the case of general N is more involved. It turns out that some resultson SU q (2) fail for N ≥ . For instance, for N ≥ in the gaussian case the cohomologicalproblem may not always be solved; see Corollary 3.6. Also, for N = 2 the Lévy functionals L without gaussian part can be parametrized by all vectors in a certain representation Hilbertspace, whereas for N ≥ this is no longer true; see Proposition 5.1.The paper is organized as follows. Section 2 presents preliminary results; some of themare new. (For instance, the treatment of the projection P , in particular, in connection withsubgroups, is new. Also, for the analytic key lemma in Subsection 2.7, though probably folklore ,we did not find a source; this lemma also drastically simplifies the case SU q (2) , and is responsiblefor that we need not reference to [27, 24] for more than motivation.) Section 3 deals with thegaussian case and our choice of P . Section 4 presents the actual decomposition for SU q ( N ) ,while Section 5 pushes forward to SU q ( N ) the parametrization results from Skeide [27, Section4.5] or [28, Section 4.3]. Section 6 deals with U q ( N ) . In the final section we discuss some openproblems for future work. Conventions and choices.
A ( topological ) compact quantum group G = ( C ( G ) , ∆) (Woronowicz [35]) is a unital C ∗ -algebra C ( G ) with a unital ∗ -homomorphism ∆ : C ( G ) → C ( G ) ⊗ min C ( G ) that is coassociative UWE FRANZ, ANNA KULA, J. MARTIN LINDSAY, MICHAEL SKEIDE ( (∆ ⊗ id ) ◦ ∆ = ( id ⊗ ∆) ◦ ∆ ) and that satisfies the quantum cancellation rules span ( C ( G ) ⊗ )∆( C ( G )) = C ( G ) ⊗ min C ( G ) = span ∆( C ( G ))( ⊗ C ( G )) .An ( algebraic ) compact quantum group or CQG-algebra (Dijkhuizen and Koornwinder[10]) is a Hopf ∗ -algebra ( G , ∆ , ε ) (see Subsection 2.1) that is spanned by the coefficients of itsfinite-dimensional unitary corepresentations. (Equivalently, a CQG-algebra is a Hopf ∗ -algebrawith a Haar state . But the original definition suits our situation better.)Algebraic and topological compact quantum groups are two sides of the same coin, compactquantum groups. (See the books by Klimyk and Schmüdgen [17, Section 11.3] or by Timmer-mann [31, Section 5.4].) Especially, a topological compact quantum group ( C ( G ) , ∆) containsa (unique) dense ∗ -subalgebra R ( G ) (the linear hull of the coefficients of its finite-dimensionalcorepresentations) such that the restriction of ∆ to R ( G ) maps into the algebraic tensor product R ( G ) ⊗ R ( G ) ; existence of counit and antipode are theorems. Schürmann’s theory of quantum Lévy processes , from which we take the notion of generatingfunctionals, is about ( algebraic ) quantum semigroups (or ∗ -bialgebras) and ( algebraic ) quantum groups (or Hopf ∗ -algebras). Even the notion quantum subgroups of a (topological)compact quantum group G is referring to R ( G ) rather than to C ( G ) . Therefore:In these notes we view compact quantum groups (like SU q ( N ) and U q ( N ) ) exclusively as CQG-algebras. (Our exposition of SU q ( N ) and U q ( N ) in Subsection 2.6 follows Koelinks expositionof U q ( N ) in [16]) We shall write G to mean R ( G ) ; the C ∗ -algebra C ( G ) does not occur.In Schürmann’s theory, ∗ -representations of G are by (possibly unbounded) operators on pre-Hilbert spaces – and (thinking, for instance, about the quantum groups constructed from Liealgebras) this is good so. But representations of CQG-algebras are all by bounded operators.After we introduced SU q ( N ) and U q ( N ) in Subsection 2.6, we will consequently complete theoccurring pre-Hilbert spaces. But before that, the discussion (some of it new) is quite general;it would be a pity to write it down in a way that is not directly quotable from future papersjust because we completed too early. (See also Footnote 3 in Subsection 2.5.)Last but surely not least, we emphasize already now that the property of a linear functionalon G to be a generating functional, makes reference only to the ∗ -algebra structure of G andto the counit ε ; no comultiplication is needed and no antipode is needed. Throughout thesenotes, with one exception (in the definition of quantum subgroup in Subsection 2.4), we ignorethe antipode. The comultiplication, though not strictly necessary, comes in useful in a coupleof places. (See the last sentence in Subsection 2.2 and the proof of Proposition 5.1.) Therefore,we carry it along. Frequently, in the literature R ( G ) is also denoted Pol ( G ) . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Preliminaries
Generating functionals of Lévy processes.
Classical Lévy processes, in the most gen-eral formulations, take values in a group or even only in a semigroup. The latter allows to saywhat the increments of the process are, and to define the most important property a Lévy processhas to satisfy: Namely, to have independent increments . The passage from the classical worldto the noncommutative world is frequently made by dualization : Replace spaces (for instance,probability spaces or semigroups) by ∗ -algebras of complex functions on these spaces, look howall the structures of the spaces are reflected by additional structures of these function algebrasand take these as axioms, but in the end forget that the algebras are commutative. Probabilityspaces become ∗ -algebras with a state, semigroups become ∗ -bialgebras, and semigroup-valuedrandom variables on a probability space become, under dualization, ∗ -algebra homomorphismsfrom the ∗ -bialgebra into the quantum probability space. A quantum Lévy process is, therefore,a family of such homomorphisms fulfilling certain extra properties.Fortunately, quantum Lévy processes are determined by their so-called generating functionals and, fortunately, the only scope of these notes is to examine the structure of such generatingfunctionals on the quantum groups SU q ( N ) and U q ( N ) . We do not really need to know what aquantum Lévy process is, but only the properties that make a functional a generating functional.For details about quantum Lévy processes we refer the reader to Schürmann [23], Meyer [19,Chapter VII], Franz [11] or the recent survey [12]. An abstract reconstruction of a Lévy processfrom a generating functional can be found in Schürmann [23] (basically, but in a more generalcontext, [23, Proposition 1.9.5] and the discussion preceding it and summarized as a theoremin [23, Corollary 1.9.7]), or (reducing it to Bhat and Skeide [5]) in Skeide [30]. A proof thatthe reconstruction can be done on a Fock space, can be found in Schürmann [23, Theorem2.3.5] (using quantum stochastic calculus) and in Schürmann, Skeide, and Volkwardt [25] (usingtechniques as for Trotter products and Arveson systems [4]).Recall from the conventions that we opted to set up compact quantum groups the algebraicway (as CQG-algebras), as these meet better Schürmann’s setting. Recall, too, that we optedto keep these preliminaries as applicable as possible also for quantum (semi)groups that are notnecessarily (locally) compact. Therefore, in the first few subsections (until 2.5), representationsare by (not necessarily bounded) operators on pre-Hilbert spaces. Only after we introduce U q ( N ) and SU q ( N ) in Subsection 2.6, we always assume pre-Hilbert spaces completed.An (algebraic) quantum semigroup G is a ∗− bialgebra , that is, G is a complex involutiveunital algebra G with a unital ∗ -homomorphism ∆ : G → G ⊗ G (the comultiplication ) from G into the algebraic tensor product G ⊗ G and a unital ∗ -homomorphism ε : G → C (the counit )from G into the complex numbers fulfilling coassociativity (∆ ⊗ id G ) ◦ ∆ = ( id G ⊗ ∆) ◦ ∆ and the counit property ( ε ⊗ id G ) ◦ ∆ = id G = ( id G ⊗ ε ) ◦ ∆ . An (algebraic) quantum group would be a Hopf ∗ -algebra, that is, a ∗ -bialgebra with an ad-ditional structure, the so-called antipode . As we do not need the antipode (its definition would UWE FRANZ, ANNA KULA, J. MARTIN LINDSAY, MICHAEL SKEIDE require to introduce the multiplication map on
G ⊗ G or the Sweedler notation), we do notaddress it in this paper.The comultiplication, instead, together with the induced notion of convolution we discuss ina second, is, though not strictly necessary for our results, frequently useful. In any case, for thebasic understanding why generating functionals occur as infinitesimal generators of convolutionsemigroups, the comultiplication is indispensable.Using the comultiplication, we define the convolution of two linear functionals ϕ and ψ on G as ϕ ⋆ ψ := ( ϕ ⊗ ψ ) ◦ ∆ . By coassociativity, the product ⋆ turns the dual G ′ = L ( G , C ) into an (associative) algebra, andby the counit property the counit ε is a unit for G ′ .The fundamental theorem of coalgebras asserts that every element of a coalgebra is containedin a finite-dimensional subcoalgebra; see, for instance, Abe [1, Corollary 2.2.14(i)]. This can beused easily to show that there is a complex functional calculus for entire functions, including(pointwise) differentiation and integration with respect to parameters. Therefore, there is a convolution exponential e ψ⋆ := P ∞ n =0 ψ ⋆n n ! and the formula ϕ t = e tψ⋆ establishes a one-to-one correspondence between (pointwise) continuous convolution semigroups ( ϕ t ) t ≥ and theirinfinitesimal generators ψ = ddt (cid:12)(cid:12) t =0 ϕ t .Like a classical Lévy process is determined by a convolution semigroup of probability measures,a quantum Lévy process is determined by a convolution semigroup of states. So, we are led tothe question, when does a convolution semigroup ϕ t = e tψ⋆ consist of states , that is, of positive ( ϕ t ( b ∗ b ) ≥ ) normalized ( ϕ t ( ) = 1 ) linear functionals? By looking at ddt (cid:12)(cid:12) t =0 ϕ t , we easilycheck that if all ϕ t are states, then ψ satisfies the following conditions: • ψ is hermitian , that is, ψ ( b ∗ ) = ψ ( b ) . • ψ is -normalized , that is, ψ ( ) = 0 . • ψ is conditionally positive , that is, ψ ( b ∗ b ) ≥ whenever b ∈ ker ε .The best way to show that these conditions are also sufficient for that the convolution semigroupgenerated by ψ consists of states, is by reconstructing from ψ a Lévy process that has thissemigroup as convolution semigroup, see Schürmann [23, Theorem 2.3.5] or Schürmann, Skeide,and Volkwardt [25]. We, therefore, say: Definition . A generating functional (for a quantum Lévy process) on a ∗ -bialgebra G is a linear functional ψ : G → C that is hermitian, -normalized, and conditionally positive.We see, to check if a linear functional on G is a generating functional, we only refer to the ∗ -algebra structure of G (in terms of positivity and being hermitian) and to the counit ε (in termsof its kernel ker ε ); there is no reference to the comultiplication. The kernel of ε , on the otherhand, and related structures are so important that we introduce already now the correspondingnotation we are going to use throughout. We define for all n ≥ K n := span (ker ε ) n , UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) the span of all products of n elements in ker ε . (In particular, K = ker ε .) We put K := G . Wealso put K ∞ := T n ≥ K n . Obviously, all K n ( ≤ n ≤ ∞ ) are ∗ -ideals and K n ⊃ K n +1 ⊃ K ∞ .2.2. Schürmann triples and the projection P . Like with other infinitesimal generators ofpositive-type semigroups (generators of semigroups of positive definite kernels, semigroups ofcompletely positive maps), there is a sort of GNS-construction also for generating functionals ofquantum Lévy processes, their so-called
Schürmann triple . Since every generating functional hasa Schürmann triple, finding all Schürmann triples will provide us with all generating functionals;this will be our strategy.Following Schürmann [23, Section 2.3], for a conditionally positive functional ψ on G wedefine a positive sesquilinear form on K by ( a, b ) ψ ( a ∗ b ) , we divide out the null-space N ψ := { b ∈ K : ψ ( b ∗ b ) = 0 } , and we form the quotient space D ψ := K / N ψ which inherits apre-Hilbert space structure by defining the inner product h a + N ψ , b + N ψ i := ψ ( a ∗ b ) .Let η ψ : G → D ψ denote the quotient map b b + N ψ ( b ∈ K ) extended by η ψ ( ) := 0 to allof G . This makes sense, because of the following crucial fact: Observation . Every b ∈ G can be written uniquely as k + λ with k ∈ K . Indeed,necessarily λ = ε ( b ) . So, if we define the canonical projection onto K as id − ε : b b − ε ( b ) ,then, necessarily, k = ( id − ε )( b ) . (Similar considerations are important in the discussion aroundLemma 2.7.)Of course, η ψ ( G ) = η ψ ( K ) = D ψ . As usual, an application of Cauchy-Schwarz inequalityshows that for each a ∈ G we (well)define a map π ψ ( a ) : η ψ ( b ) η ψ ( ab ) ( b ∈ K ) on D ψ . Oneeasily verifies that b π ψ ( b ) defines a unital ∗ -representation π ψ : G → L a ( D ) (the set of alladjointable maps D → D ). Using again that b = ( id − ε )( b ) + ε ( b ) for all b ∈ G , we see that η ψ is a π ψ - ε -cocycle , that is,(2.1) η ψ ( ab ) = π ψ ( a ) η ψ ( b ) + η ψ ( a ) ε ( b ) for all a, b ∈ G . Taking into account that ε is fixed, we frequently say η is a cocycle with respectto π . By construction, η ψ fulfills h η ψ ( a ) , η ψ ( b ) i = ψ ( a ∗ b ) for a, b ∈ K . Since ψ ( ) = 0 , this isthe same as(2.2) h η ψ ( a ) , η ψ ( b ) i = ψ ( a ∗ b ) − ψ ( a ∗ ) ε ( b ) − ε ( a ∗ ) ψ ( b ) for all a, b ∈ G . So, the ε - ε -2-coboundary of ψ is the map ( a, b )
7→ −h η ψ ( a ∗ ) , η ψ ( b ) i . We say the( ε - ε -)2-coboundary of ψ is η ψ -induced . Therefore: Definition . A Schürmann triple is a triple ( π, η, ψ ) consisting of • a unital ∗ -representation π : G → L a ( D ) on some pre-Hilbert space D , • a π - ε -cocycle η : G → D , and • a linear functional ψ : G → C whose ε - ε - -coboundary is η -induced. Of course, the prescription does determine a map π ( a ) – provided such map exists; the question with “defin-ing” by such determining prescriptions, is whether the map actually does exists. The term “ we (well)define ”(and analogous variants) is shorthand for “we attempt to define a map by the following prescription, and itturns out that such map is well-defined”. UWE FRANZ, ANNA KULA, J. MARTIN LINDSAY, MICHAEL SKEIDE
We say, ( π, η, ψ ) is cyclic if η is cyclic , that is, if η ( G ) = D .Every generating functional is part of the Schürmann triple ( π ψ , η ψ , ψ ) , the output of the GNS-construction preceding Definition 2.3. We say, ( π ψ , η ψ , ψ ) is the Schürmann triple associatedwith the generating functional ψ . Note that the Schürmann triple of ψ is cyclic by construction.Moreover, if ψ is part of any other Schürmann triple, say, ( π, η, ψ ) , then η ψ ( b ) η ( b ) defines anisometry v : D ψ → D , which intertwines the representation in the sense that π ( a ) v = vπ ψ ( a ) forall a ∈ G . If also ( π, η, ψ ) is cyclic, then v is even unitary and π = vπ ψ v ∗ . (Recall that a unitarybetween pre-Hilbert spaces has an adjoint, namely, its inverse.) In this sense, cyclic Schürmanntriples are determined by ψ up to unitary equivalence.It is noteworthy that the construction of a Schürmann triple for a generating functional ψ went ψ ( D ψ and) η ψ π ψ . For finding all generating functionals, we rather proceed theopposite way: Procedure . (1) Find all ∗ -representations π ;(2) find all cocycles η with respect to π ;(3) find all linear functionals ψ with η -induced -coboundaries;(4) exclude all those that are not generating functionals.Since every generating functional has a Schürmann triple, in that way we surely will find allgenerating functionals. (And even if, for some quantum semigroup, we should not succeed in fullgenerality for some of the steps, the procedure still promises to be a rich source for generatingfunctionals; this way was quite successful for quantum Lévy processes on the Lie algebra sℓ (2) in Accardi, Franz, and Skeide [2].) Remark . In Definition 2.3, the condition that π is unital (so that π ( ) = id D ) is justfor convenience. Indeed, if π is not unital, then π ( ) is still a projection, and (2.1) shows π ( ) η ( G ) = η ( G ) , so we may restrict to π ( ) D . By the same computation, we see that wemay actually restrict to the invariant subspace η ( G ) , making the triple cyclic. However, whilenonunital π is usually just annoying, for the purpose to proceed as π η ψ it is veryconvenient, for formal reasons (see the discussion following Proposition 2.13 and the beginningof Section 5), not to have to worry about cyclicity of η . To follow Procedure 2.4, we have to face the following questions, which are all related to eachother:(1) Can a given pair ( π, η ) be completed to a Schürmann triple?(2) If ( π, η, ψ ) is a Schürmann triple, how far can ψ be away from a generating functional?(3) Given two generating functionals ψ and ψ such that both ( π, η, ψ ) and ( π, η, ψ ) areSchürmann triples, how different can ψ and ψ be?All three questions root in the single question, given a pair ( π, η ) , what is fixed by the infor-mation/wish that a functional ψ completes the pair to a Schürmann triple? The coboundaryproperty in (2.2) fixes the values of ψ on K (by the line preceding (2.2)) and it fixes ψ to be UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) -normalized (simply plug in a = = b into (2.1) and (2.2) to see that η ( ) = 0 and, hence, ψ ( ) = 0 ).The first question is about existence: Does the prescription ab
7→ h η ( a ∗ ) , η ( b ) i (well)define alinear map on K (that may, then, be extended further, taking also into account ψ ( ) = 0 , toall of G )? It has a nontrivial answer, which depends on the quantum semigroup in question. Wecome back to it, later.The second and third question are about uniqueness. In both cases there exists a linearfunctional on G fulfilling (2.2), but we wish to know more about the remaining degrees offreedom. Note that a linear functional ψ on G satisfying (2.2), is, in particular, conditionallypositive. So, the only question to be answered for knowing if ψ is a generating functional, iswhether it is hermitian (this is automatic on K ).We are faced with a simple problem of linear algebra: Given a linear functional on K in howmany ways can it be extended to a linear functional on G ? In particular, given a Schürmanntriple ( π, η, ψ ) , can we choose the extension to G of ψ ↾ K in such a way that it becomes agenerating functional (Question (2)) and in how many different ways is this possible (Question(3))?Let us fix some notation. Definition . Let V be a vector space and let K be a subspace of V . We say, a family E ofvectors (necessarily in V \ K ) is a basis extension from K to V if E extends one basis of K (and, therefore, all bases of K ) to a basis of V . Equivalently (see also Observation 2.2), E is abasis extension if every element of V can be written as the sum of a unique linear combinationof vectors in E and a unique element k ∈ K .(Despite basis extensions E being, like bases, families of vectors, frequently we shall be sloppyand consider E just as a set. The only significant difference occurs if the family E has doubleelements – in which case it would not be a basis extension.)Without proof we state the following lemma from linear algebra. Lemma . For each basis extension E from K to V there is exactly one projection P E from V onto K fulfilling ker P E = span E . This projection has the form P E = id − X κ ∈ E κ ε ′ κ , where the linear functionals ε ′ κ are (well)defined by putting ε ′ κ ( κ ′ ) = δ κ , κ ′ for κ ′ ∈ E and ε ′ κ ( k ) = 0 for k ∈ K . We follow Skeide [28] and improve [28] quite a bit. Obviously, id − ε , our projection onto K = ker ε , is the projection associated with the basis extension { } from K to G , and ε ′ isjust ε . By the lemma, P { } = id − ε is the unique projection onto K that has kernel C . A linear functional ψ is -normalized ifand only ψ ◦ ( id − ε ) = ψ . Let E be a basis extension from K to K , so that { } ∪ E is a basis extension from K to G .A moments thought (taking also into account that { } , K , and K are ∗ -invariant) shows thatwe may (and, usually, will) assume that E is hermitian , that is, E consists of self-adjointelements. Then, also the functionals ε ′ κ ( κ ∈ E ) are hermitian. The associated (hermitian)projection from G onto K is(2.3) P := id − ε − X κ ∈ E κ ε ′ κ , = P E ◦ ( id − ε ) . Since it is the linear hull of the elements in a basis extension that determines the projection,we see that among all (hermitian) projections from G onto K , the projections that have thepreceding form are exactly those that satisfy the additional condition that ker P ∋ . For sucha projection P , a linear functional satisfying ψ ◦ P = ψ , hence, ψ ◦ ( id − ε ) = ψ is -normalized.And since P is hermitian, ψ ◦ P = ψ is hermitian if and only if ψ is hermitian on K . Definition . A Schürmann triple ( π, η, ψ ) for ψ is trivial if π = 0 . (Then D = { } and η = 0 .)Obviously, the Schürmann triple of a generating functional ψ is trivial if and only if ψ ↾ K = 0 .Generating functionals with trivial Schürmann triple are also called drifts . The followingstatements are fairly obvious; they answer Questions (2) and (3): • The drifts are precisely the real linear combinations of the functionals ε ′ κ ( κ ∈ E ). • Two linear functionals ψ and ψ coincide on K (this includes, in particular, two func-tionals completing the same pair ( π, η ) to a Schürmann triple) if and only if they differby a linear combination of the functionals ε , ε ′ κ ( κ ∈ E ). • In particular, two generating functionals complete the same pair ( π, η ) to a Schürmanntriple if and only if they differ by a drift.So, the answer to Question (1) is positive, if (and only if) for the pair ( π, η ) Equation (2.2)(well)defines a linear functional on K . Given such a functional, we extend it to G by puttingit at and at all κ ∈ E (so that ψ ◦ P = ψ ), obtaining a generating functional ψ turning ( π, η, ψ ) into a Schürmann triple. Invoking our answer to Question (3), we obtain all generatingfunctionals doing the same job, by adding to ψ any drift.To get a handier formulation, we repeat a statement from the end of [28, Subsection 2.2]. Proposition . For every generating functional ψ on G that is not a drift there is a projection P of the form (2.3) fulfilling ψ ◦ P = ψ. Every other generating functional ψ ′ having the same pair ( π, η ) in its Schürmann triple, isobtained from ψ as ψ ′ = ψ ◦ P ′ where P ′ is the some projection fulfilling ψ ′ ◦ P ′ = ψ ′ . Corollary . If ( π, η, ψ ) is a nontrivial Schürmann triple for ψ , then ψ is a generatingfunctional if and only if the unique projection P onto K such that ψ ◦ P = ψ is hermitian. In order to perform Procedure 2.4, we will have to find a suitable choice for E . This involves,first, the problem to show that the elements κ ∈ E are enough to span together with and UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) K everything and, then, to show that they are linearly independent. While the functionals ε ′ κ cannot be defined in the prescribed way before we actually know that the κ are linearly inde-pendent, in applications (including the present one, but not only) we actually, first, (well)definethe ε ′ κ in a different way and, then, use them to prove that the κ are linearly independent.The way we will define in Section 3 the ε ′ κ for SU q ( N ) , also explaining the notation, comesfrom the observation in Skeide [28, Example 2.2] that every drift d (like, for instance, ε κ ) can beobtained as derivative of the convolution semigroup of characters e td⋆ . (Being a drift is exactlywhat makes this a convolution semigroup of characters.) This is one of two places in thesenotes where the comultiplication of G is, though not strictly necessary, useful. If we, as we plan,forget about the comultiplication, then just the insight that drifts can be obtained by takingderivatives of (suitably parametrized) families of characters remains.We summarize (basically, [28, Example 2.2]): Proposition . Let ( ε θ ) θ ≥ be a family of characters with ε = ε . (1) If e θ is pointwise differentiable at θ = 0 , then ε ′ is an ε - ε -cocycle. (2) If e θ is pointwise twice differentiable at θ = 0 , then the ε - ε - coboundary of ε ′′ is ε ′ -induced. Clearly, ε ′ is on K ∪ { } . Corollary . Suppose we have a family ( ε ′ i ) i ∈ I of ε - ε -cocycles, all obtained (for suitablefamilies ( ε iθ ) θ ≥ ) as in the proposition, and we have a family ( κ i ) i ∈ I , indexed by the same set I , such that ε ′ j ( κ i ) = δ i,j . Then the κ i ∈ K \ K are linearly independent and may be extendedto a basis extension from K to K .If, moreover, the κ i and K generate K then the κ i are a basis extension from K to K . Generalities about decomposition and approximation.
The results in this subsectionaddress the decomposition of a generating functional into a sum of two and how to (well)definefunctionals by suitably approximating their cocycles.The decomposition of a generating functional ψ into a sum ψ = ψ + ψ of two generatingfunctionals ψ and ψ is related to direct sum operations among the GNS-representation andGNS-cocycle of the latter two. Proposition . Let ψ , ψ , and ψ be generating functionals such that ψ = ψ + ψ , anddenote by ( π i , η i , ψ i ) the Schürmann triples of ψ i (with pre-Hilbert spaces D i ). Then η := η ⊕ η : b η ( b ) ⊕ η ( b ) ∈ D := D ⊕ D is a cocycle with respect to π := π ⊕ π and ( π, η, ψ ) is aSchürmann triple. We omit the obvious/simple proof. Observe, however, that ( π, η, ψ ) is, in general, not the Schürmann triple of ψ . (Just take ψ = ψ . Then the Schürmann triple of ψ is ( π , √ η , ψ ) .This problem is the same as for the GNS-representation of the sum of two positive functionals.) This is one of the main reasons why it would be extremely inconvenient for us to restrict ourattention to cyclic cocycles, only.Seeking a sort of converse of Proposition 2.13, we observe that cocycles behave nicely withrespect to decomposition of the representation space.
Proposition . Suppose π and π are unital ∗ -representations of G on pre-Hilbert spaces D and D , respectively, and suppose η is a cocycle with respect to π := π ⊕ π . Then, in theunique decomposition η = η ⊕ η , the η i are (unique) cocycles with respect to π i . We, again, omit the simple proof that follows simply by observing that π ( ) := π ( ) ⊕ π ( ) = id D ⊕ D , so that the maps η i := π i ( ) η do the job. Corollary . Suppose we have linear functionals ψ , ψ , and ψ satisfying ψ = ψ + ψ .If, in the situation of Proposition 2.14, two of the triples ( π , η , ψ ) , ( π , η , ψ ) , and ( π, η, ψ ) are Schürmann triples, then so is the third. If, moreover, two of the functionals are generating,then so is the third. Again, there is not really anything to prove. (If two of the three conditions h η i ( a ∗ ) , η i ( b ) i = ψ i ( ab ) and h η ( a ∗ ) , η ( b ) i = ψ ( ab ) are satisfied for all a, b ∈ K , then so is the third.) So, thiscorollary looks rather innocent. It is, however, a surprisingly crucial tool. A Lévy-Khintchineformula is, in the first place, a decomposition result for functionals, asserting (when true) that allgenerating functionals can be written as a sum of functionals from two simpler classes. What wecan do easily, is decomposing the GNS-representation, hence (by Proposition 2.14), the cocyclesof a generating functional into direct summands corresponding to the simpler classes. We areleft with the (tricky!) question, whether the corresponding components of the cocycle give riseindividually to generating functionals (summing up to the original one). The corollary tells: Yes,if we can guarantee existence of a functional for one of the two components of the cocycle; andthis is what we will do. For how we are going to do that, the following simple approximationresult is of outstanding importance: It tells that if we can approximate the cocycle in thepair ( π, η ) by coboundaries for π , then there is also a generating functional ψ completing theSchürmann triple ( π, η, ψ ) .Recall that for each ∗ -representation π of G on D and each vector η , the map η := ( πη ) ◦ ( id − ε ) : b π ( b − ε ( b )) η is a π - ε -cocycle. We say, a cocycle of this form is a coboundaryfor π or just a coboundary . We fix D and π .Recall, too, that we assume fixed the (hermitian!) projection P onto K . (Subsection 2.2.) Lemma . Suppose ( η n ) n ∈ N is a sequence of vectors in D such that the sequence composed ofthe coboundaries ( πη n ) ◦ ( id − ε ) for π converges pointwise to a map, say, η . Then: (1) h η n , π ( • ) η n i ◦ P converges pointwise to a map, say, ψ . (2) ( π, η, ψ ) is a Schürmann triple. (3) ψ ◦ P = ψ , so that, by Corollary 2.10, ψ is a generating functional.Proof. P maps into K and a typical element of K has the form a ∗ b for a, b ∈ K . So, h η n , π ( • ) η n i ◦ P ( a ∗ b ) = h π ( a ) η n , π ( b ) η n i , which converges on each side, as π ( b ) η n is just the UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) value of the coboundary generated by η n . By the same computation, ψ fulfills (2.2) and, clearly, η fulfills (2.1), so ( π, η, ψ ) is a Schürmann triple. The third statement is obvious. (cid:3) Remark . It is the property that ψ is the limit of expressions h η n , π ( • ) η n i ◦ P which werequire in order to call ψ a jump part . Every generating functional ψ = ψ ◦ P is the (pointwise)limit of ϕ t − εt ◦ P = ϕ t t ◦ P for t → , where ϕ t = e ψt⋆ is the generated convolution semigroup of states; every ϕ t has aGNS-construction ( π t , η t ) with the cyclic (unit) vector η t in the representation space H t . Whatmakes the sequence of positive functionals h η n , π ( • ) η n i different from the sequence nϕ n , is thefact that in the former the representation π is fixed, while only the vector η n is running.This is exactly the situation we have in the integral R G\{ e } f ( g ) d L ( g ) (see the introduction), ifwe approximate it by Z G\ U n ( e ) f ( g ) d L ( g ) , where U δ ( g ) is (for some metric on the classical group G ) the open δ -neighbourhood of the point g ∈ G . Here, H = L ( G , L ) , the representation π of f is by multiplication with f , and η n is theindicator function of G\ U n ( e ) . Observation . In view of Subsection 2.5 (and the terminology introduced there), we men-tion that if π has a gaussian part, then this part disappears under π ◦ P . Therefore, as a limitof functionals h η n , π ( • ) η n i ◦ P , a jump part is completely non-gaussian.2.4. Schürmann triples on quantum subsemigroups.
In the course of proving our resultsfor SU q ( N ) , we will decompose representations into components that “live” on the quantumsubgroups SU q ( n ) ( n ≤ N ) . Also, promoting our results about SU q ( N ) into results about U q ( N ) , is based on the fact that U q ( N ) sits in between SU q ( N ) and SU q ( N + 1) . Definition . A quantum subsemigroup of a quantum semigroup G is a pair ( H , s ) consisting of a quantum semigroup H and a surjective ∗ -bialgebra homomorphism s : G → H .We say that a map T from G to some space X lives on ( H , s ) if T factors through s , thatis, if there exist a map ˜ T from H to X such that T = ˜ T ◦ s. If it is clear from the context what s is, then we will just speak of the quantum subsemigroup H of G . Remark . If the quantum group H is a quantum subsemigroup of the quantum group G via s , then (see Dascalescu, Nastasescu, and Raianu [9, Proposition 4.2.5] ) s also respects theantipodes. That is, H is a quantum subgroup of G . Since s is surjective, the map ˜ T illustrating that T lives on H is unique.In the remainder of this subsection we fix a quantum semigroup G , one of its quantum sub-semigroups ( H , s ) , and a (pre-)Hilbert space D . Since s respects the counits, ε lives on H viathe counit ˜ ε of H . Also, K n ( n = 0 , , . . . , n, . . . , ∞ ) are mapped by s onto the corresponding ˜ K n of H . Proposition . Suppose we have maps π , η , and ψ , all defined on G , and maps ˜ π , ˜ η , and ˜ ψ such that π = ˜ π ◦ s, η = ˜ η ◦ s, ψ = ˜ ψ ◦ s. Then: (1) π is a ∗ -representation of G (obviously, living on H ) if and only if ˜ π is a ∗ -representationof H . (2) η is a π - ε -cocycle (obviously, living on H ) if and only if ˜ η is a ˜ π - ˜ ε -cocycle. (3) ψ is a generating functional on G (obviously, living on H ) if and only if ˜ ψ is a generatingfunctional on H . (4) ( π, η, ψ ) is a Schürmann triple if and only if (˜ π, ˜ η, ˜ ψ ) is a Schürmann triple.Proof. The if -direction is clear, while the only if -direction follows from s ( K n ) = ˜ K n . (cid:3) Of course, the projections onto K are compatible in the sense that ( id − ˜ ε ) ◦ s = s ◦ ( id − ε ) . Corollary . Suppose a π - ε -cocycle η on G can be approximated by coboundaries ( πη n ) ◦ ( id − ε ) . If π = ˜ π ◦ s lives on H , then ˜ η := lim n →∞ (˜ πη n ) ◦ ( id − ˜ ε ) exists (pointwise), is a ˜ π - ˜ ε -cocycle, and fulfills η = ˜ η ◦ s , so that η lives on H , too.Moreover, denoting ψ as in Lemma 2.16 and ˜ ψ the analogue for ˜ η and a projection P H onto ˜ K , we get Schürmann triples ( π, η, ψ ) and (˜ π, ˜ η, ˜ ψ ) , and ψ and ˜ ψ are related by ψ = ˜ ψ ◦ s ◦ P . Proof.
Almost everything follows, appealing to surjectivity of s , by writing arguments of “twid-dled” maps in the form s ( a ) . The only thing that needs a word, is the last formula. Clearly, ˜ ψ ◦ s defines a generating functional on G (that lives on H ). By (˜ πη n ) ◦ ( id − ˜ ε ) ◦ s = ( πη n ) ◦ ( id − ε ) , we see that ˜ ψ ◦ s coincides with ψ on K . From that, the formula follows. (cid:3) We may ask, whether the projections P and P H may be chosen compatible, too, so that inthe last formula we really get ψ = ˜ ψ ◦ s , without the correction of the drift part via compositionwith P . The answer is yes, as long as the G and its subsemigroup H are fixed; but the possiblechoices of P depend on H . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Lemma . Let E be the basis extension from K to K determining P and let E H be thebasis extension from ˜ K to ˜ K determining P H . Then P H ◦ s = s ◦ P if and only if span s ( E ) ⊂ span E H .Moreover, if span s ( E ) ⊂ span E H , then necessarily span s ( E ) = span E H .Proof. First of all, since P maps into K , since s maps K into (actually, onto) ˜ K , and since P H acts as identity on ˜ K , the right-hand side always coincides with P H ◦ s ◦ P . Therefore, wehave to examine, when P H ◦ s coincides with P H ◦ s ◦ P . Since P acts as identity on K , thetwo maps always coincide on K . Since K and E span K , it remains to examine, when thetwo maps coincide on span E .By Lemma 2.7, the restriction of P to K is the unique idempotent onto K with kernel span E . Therefore, P H ◦ s ◦ P is on span E . Likewise, the restriction of P H to ˜ K is theunique idempotent onto ˜ K with kernel span E H . Therefore, P H ◦ s is on span E if and onlyif span s ( E ) ⊂ span E H .As for the last statement, since E and K span K and since s is surjective, for any ˜ κ ∈ E H there are k ∈ span E and k ∈ K such that s ( k ) + s ( k ) = ˜ κ . Plugging this in into P H , takingalso into account that ˜ κ ∈ span E H = ker P H , under the hypothesis span s ( E ) ⊂ span E H weget P H ( ˜ κ ) = P H ◦ s ( k ) + P H ◦ s ( k ) = 0 + s ( k ) . It follows that ˜ κ = s ( k ) ∈ span s ( E ) . Therefore, span E H ⊂ span s ( E ) , too. (cid:3) Corollary . If span s ( E ) ⊂ span E H , then the generating functional ψ from Corollary2.22 lives on H . Corollary . Suppose s ( E ) = E H ∪ { } . Then P H ◦ s = s ◦ P . Corollary 2.22 will develop its full power only in Section 6, when we reduce the case U q ( N ) tothe case SU q ( N ) . In the decomposition of generating functionals on SU q ( N ) into componentsthat live on the subgroups SU q ( n ) ( n ≤ N ) , Corollary 2.22 does not help. We do get a decom-position of the representations into representations that live on the subgroups (Subsection 4.1).We also can show that (for n ≥ ) the corresponding components of the cocycles are limits ofcoboundaries (Subsection 4.3). But, before we can show the latter statement, we first have toshow that the components of the cocycles live on the subgroups without knowing they are limitsof coboundaries (Subsection 4.2).This leaves us with the general problem to check when maps π and η on G do live on H . Ouralgebras are generated (as algebras) by sets of generators, usually, arranged in matrices, anddividing out some relations on these generators. Already when (well)defining representationsand their cocycles, we are using the well-known facts that a representation (or, more generally, ahomomorphism) is well-defined by assigning its values on the generators, and checking whetherthe assigned values satisfy the relations. The same is true for a cocycle. All our quantum subgroups arise by adding more relations. Therefore, the ( ∗ -)algebra describ-ing the quantum subgroup can be obtained by taking the quotient of the ( ∗ -)algebra describingthe containing quantum group and the ( ∗ -)ideal generated by the extra relations. The follow-ing, purely algebraic, lemma tells us what we have to do to check if a representation and itscocycle live on a quantum subsemigroup. (Applying it to the free algebra and its quotients, wealso obtain a proof of the preceding statements about representations and cocycles on algebrasgenerated by relations.) Lemma . Let A be a unital algebra over C and let ε be a homomorphism into C . Suppose R = { r , . . . , r M } is a subset of ker ε , let I := span A R A denote the ideal generated by R , anddenote by s the canonical homomorphism A → A /I . Then:A representation (or homomorphism) π of A defines a representation (or homomorphism) ˜ π : s ( a ) π ( a ) of A /I if and only if π ( R ) = { } .If π ( R ) = { } , then a π - ε -cocycle η defines a ˜ π - ˜ ε -cocycle ˜ η : s ( a ) η ( a ) if and only if η ( R ) = { } .Proof. We have to show that π ( a ) = 0 for all a ∈ I . Clearly, by π ( br k c ) = π ( b ) π ( r k ) π ( c ) , this isfulfilled if and (recall that representations are assumed unital!) only if π ( R ) = { } .In particular, ε satisfies the condition, so there is ˜ ε .Finally, we have to show that η ( a ) = 0 for all a ∈ I . Clearly, by η ( br k c ) = π ( b ) π ( r k ) η ( c ) + π ( b ) η ( r k ) ε ( c ) + η ( b ) ε ( r k ) ε ( c ) , this is fulfilled if and only if η ( R ) = { } . (cid:3) And for making this lemma more applicable in our context:
Corollary . Under the hypotheses of Lemma 2.26, suppose that S is a subset of A /I anddenote by J = span ( A /I ) S ( A /I ) the ideal in A /I generated by S . For each s ∈ S choose a s ∈ A such that a s + I = s and denote by K the ideal in A generated by R ∪ { a s : s ∈ S } . Then ( a + I ) + J a + K defines an isomorphism ( A /I ) /J → A /K .Proof. Since R ⊂ R ∪ { a s : s ∈ S } ⊂ K , the canonical homomorphism π : a a + K vanisheson R , hence, defines a homomorphism ˜ π : a + I a + K . Since { a s : s ∈ S } ⊂ R ∪ { a s : s ∈ S } ⊂ K and since S = { a s + I : s ∈ S } , the homomorphism ˜ π vanishes on S , hence, defines ahomomorphism ˜˜ π : ( a + I ) + J a + K . Conversely, the homomorphism a ( a + I ) + J definesa homomorphism A /K → ( A + I ) + J , obviously the inverse of ˜˜ π , because ( r + I ) + J = 0 forall r ∈ R and because ( a s + I ) + J = s + J = 0 for all s ∈ S . (cid:3) Gaussian generating functionals and Lévy-Khintchine decomposition.
In the clas-sical theory of Lévy processes with values in abelian Lie groups, the coefficient algebra can bethought of as polynomials in the coordinate functions. The ideals K n correspond to polynomi-als with expansion starting with monomials of degree at least n . Correspondingly, functionalsvanishing on K n may be viewed as having no contribution on n th and higher powers. We have UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) met the drifts that vanish on K ; they may, therefore, be called as linear . Consequently, thefunctionals that vanish on K may be referred to as quadratic . Quadratic generating function-als, in the classical theory, correspond to second order differential operators. They generate Brownian motions , and are called gaussian . A crucial part of the classical Lévy-Khintchineformula consists in splitting an arbitrary generating functional into a (maximal) gaussian partand a residue part (with no gaussian component remaining). The residue part, classically is a(topological) convex combination of generating functionals of pure jump processes. (On R weget a compound Poisson process .) It is, therefore, sometimes referred to as jump part . (RecallRemark 2.17.)In this subsection (following Schürmann [22] and Skeide [28]), we discuss gaussian parts andexplain what we expect from a Lévy-Khintchine decomposition in the general context of (alge-braic) quantum semigroups. We also report results from Franz, Gerhold, and Thom [13], wherethe basic problems are discussed and classified.We have already examined the generating functionals with trivial Schürmann triples, thedrifts. The one step less simple class would be Schürmann triples with representations that aremultiples id D ε of ε . From η ( ab ) = π ( a ) η ( b ) h η ( a ) , π ( b ) η ( c ) i = ψ ( a ∗ bc ) ( a, b, c ∈ K ) , we conclude: Proposition . Let ψ be a generating functional and let ( π, η, ψ ) be its Schürmann triple.Then the following conditions are equivalent: (1) ψ vanishes on K . (2) η vanishes on K . (3) π vanishes on K . Recall that η in the Schürmann triple is cyclic. Without that, π may fail to vanish on K , evenif the other two, still equivalent, conditions are satisfied. Definition . A generating functional, a cocycle, and a representation on a quantum semi-group are called gaussian if they vanish on K , K , and K , respectively.From π = π ◦ ( id − ε ) + π ( ) ε, we conclude: Proposition . A (unital!) ∗ -representation π is gaussian if and only π = id D ε . Recall that we fixed a hermitian basis extension E from K to K , and that we have thefunctionals ε ′ κ ( κ ∈ E ) as in Lemma 2.7. Proposition . A cocycle η is gaussian if and only if it has the form η = X κ ∈ E η κ ε ′ κ for vectors η κ ∈ D . The vectors are unique; in fact, η κ = η ( κ ) . Proof.
Express a ∈ G as ε ( a ) + k + P κ ∈ E κ ε ′ κ ( a ) . Such a decomposition is unique and,necessarily, fulfills k ∈ K . Taking into account that a gaussian η vanishes on and on K , theformula follows. The other statements are obvious. (cid:3) We described gaussian representations in an easy and concise way. We described gaussiancocycles in a similarly easy and concise way, provided we have found a hermitian basis extension E . It would be desirable to have a similar description of gaussian generating functionals.However, it turns out that the form of a general gaussian generating functional on a quantumsemigroup depends on the quantum semigroup. In particular already the answer to the question,which gaussian cocycles actually do admit a gaussian generating functional, does depend onthe quantum semigroup. Schürmann [23, Proposition 5.1.11] showed that a sufficient (but, ingeneral, not necessary) condition for that a gaussian cocycle η admits a generating functionalis that it be hermitian , that is, h η ( a ∗ ) , η ( b ) i = h η ( b ∗ ) , η ( a ) i . In other words, since the κ areself-adjoint, the matrix h η κ , η κ ′ i is real and symmetric. The following little consequence appliesto our case SU q ( N ) ; see Lemma 3.1 and its corollary. Corollary . Suppose that ab − ba ∈ K for all a, b ∈ K . Then for a gaussian cocycle η there is a Schürmann triple ( π, η, ψ ) if and only if η is hermitian.Proof. If ( π, η, ψ ) is a Schürmann triple for a gaussian cocycle η , then h η ( a ∗ ) , η ( b ) i = ψ ( ab ) = ψ ( ba ) = h η ( b ∗ ) , η ( a ) i for all a, b ∈ K . (The middle equality because, under the stated hypoth-esis, ψ ( ab − ba ) = 0 .) The other direction is Schürmann’s result. (cid:3) Remark . Schürmann’s proof of [23, Proposition 5.1.11] is direct. (It does use the comul-tiplication.) In the course of arriving at Theorem 3.7, we also recover Schürmann’s result for SU q ( N ) (giving an explicit and classifying form to its gaussian generating functionals), appealingto a procedure similar to Proposition 2.11. After these generalities about gaussian parts, let us come back to the main problem, decom-position into a gaussian and a completely non-gaussian part , the first scope of a Lévy-Khintchineformula. Almost as easy as it is to understand gaussian representations and gaussian cocycles,it is also easy to separate a representation and, accordingly (by means of Proposition 2.14), thecocycles into a (maximal) gaussian part and a remaining (completely non-gaussian) part. Wereport the output of the careful discussion preceding [13, Definition 2.4]:
Proposition . Let ( π, η, ψ ) be a cyclic Schürmann triple on a pre-Hilbert space D for a gen-erating functional ψ . Then there are pre-Hilbert subspaces D and D of D with representation π and π , respectively, such that: • D ⊂ D ⊕ D ( ⊂ D ) . • π = ( π ⊕ π ) ↾ D , so that also η = η ⊕ η . • η ( G ) = D and π is gaussian. • η ( G ) = D and π is completely non-gaussian , that is, the only invariant subspaceof D on which π restricts to a gaussian representation, is the trivial subspace { } . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) One may show that D and D fulfilling these conditions are unique. (In fact, one necessarily has D = p η ( G ) , where p ∈ B ( D ) is the projection onto the completion of D = T a ∈ K ker π ( a ) , and D = p η ( G ) , where p = id − p . One verifies that π i ( a ) : p i η ( b ) p i π ( a ) η ( b ) (well)defines thedesired presentations.) Also, π and π are maximally gaussian and maximally completelynon-gaussian , respectively. Definition . A Lévy-Khintchine decomposition for ψ is a pair of generating functionals ψ and ψ such that ψ = ψ + ψ , and, with the pairs ( π i , η i ) ( i = 1 , as in Proposition 2.34,each ( π i , η i , ψ i ) ( i = 1 , is a Schürmann triple.If ψ has Lévy-Khintchine decomposition, then we put ψ G := ψ and ψ L := ψ . Given P , amongall Lévy-Khintchine decompositions ψ = ψ G + ψ L there is a unique one satisfying ψ L ◦ P = ψ L .In view of Corollary 2.15, for getting a Lévy-Khintchine decomposition, it is enough to guar-antee existence of ψ i for one i . Franz, Gerhold, and Thom [13] have analyzed the correspondingconditions and showed that none of the following four properties are equivalent, though all ofthem imply the last one. Definition . We say that a quantum semigroup G has • property (AC) if for each pair ( π, η ) there exists a Schürmann triple ( π, η, ψ ) . • property (GC) if for each gaussian pair ( π, η ) there exists a Schürmann triple ( π, η, ψ ) . • property (NC) if for each completey non-gaussian pair ( π, η ) there exists a Schürmanntriple ( π, η, ψ ) . • property (LK) if every generating functional ψ on G admits a Lévy-Khintchine decom-position.One of our main results asserts that SU q ( N ) and U q ( N ) have Property (NC), hence, (LC). SU q ( N ) does not have Property (GC) for N ≥ (showed also, by different means, in Das,Franz, Kula, and Skalski [8, Proposition 2.3]) and it does have Property (GC), hence, also (AC)for N ≤ . U q (1) is equal to U (1) and has (AC), while for N ≥ , U q ( N ) does only have Property(NC), but not (GC), hence, nor (AC).2.6. The quantum groups SU q ( N ) and U q ( N ) . As a compact quantum group, SU q (2) wasintroduced by Woronowicz in [33], and in [34] he obtained the rest of the family SU q ( N ) , N ≥ ,as an application of a generalization of the Tannaka-Krein duality theorem. Rosso [20] extendedthese results further to q -deformations of other semi-simple compact Lie groups.Recall that in these notes we forget about the fact that G has an antipode. So we are lookingat SU q ( N ) and U q ( N ) rather as quantum semigroups. For simplicity, we also always assume < q < . (In general, one considers < | q | ≤ , where q = 1 corresponds to the classicalcases. In the, in some sense, degenerate case q = 0 , the antipode is missing. But, see also It is noteworthy that the proof of the proposition and the supplementary uniqueness statement do notreally gain much simplicity, if the representation operators are bounded (the case that interests us). Therefore,we prefer to formulate here the, not so well-known, general version from [13], which works for all (algebraic)quantum semigroups.
Number (2) in Section 7.) However, SU q ( N ) and U q ( N ) ( q = 0 ) are quantum groups and thevarious inclusions as quantum subsemigroups they satisfy, are actually inclusions as quantumsubgroups. So, we simply trust the literature and will say, from now on, quantum group andquantum subgroup despite never explicitly addressing their antipodes and properties referringto them.We also take, from now on, into account that (by the unitarity conditions in (2.4), below) all ∗ -representations of G are by bounded operators and, therefore, are assumed to act on a Hilbertspace, rather than only on pre-Hilbert space. A cocycle η with respect to a representation π ona Hilbert space H is, therefore, cyclic if η ( G ) = H .Our focus is on the two series of compact quantum groups SU q ( N ) and U q ( N ) ( N ≥ . Forformal reasons, we extend all our definitions to N = 1 . (Here SU q (1) = SU (1) = { e } , thetrivial group, and U q (1) = U (1) , the torus.) All SU q ( N ) and U q ( N ) are examples of compactquantum matrix groups ( CQMG s [10] or the topological version compact matrix pseudogroups [32]) of order N . This means that G is defined as the unital ∗ -algebra generated by N indeterminates arranged in a matrix U = [ u jk ] Nj,k =1 subject to the unitarity conditions (2.4) N X s =1 u js u ∗ ks = δ jk = N X s =1 u ∗ sj u sk and, depending on which quantum matrix group, further relations. The unitarity conditionsguarantee that all ∗ -representations map each u jk to a contraction. Comultiplication and counitare defined by extending ∆ : u jk N X s =1 u js ⊗ u sk (2.5a) ε : u jk δ j,k (2.5b)as unital ∗ -homomorphisms. (Of course, for each G one has to verify that the assignments ∆( u jk ) and ε ( u jk ) , respectively, satisfy all the defining relations of G .)Let S N := (cid:8) σ : σ a bijection on { , . . . , N } (cid:9) denote the symmetric group of order N , andfor any σ ∈ S N denote by i ( σ ) := { ( j, k ) : j < k, σ ( j ) > σ ( k ) } the number of inversions of σ . For every τ ∈ S N we define D qτ ( U ) := X σ ∈ S N ( − q ) i ( σ ) u σ (1) ,τ (1) u σ (2) ,τ (2) . . . u σ ( N ) ,τ ( N ) . Usually, SU q ( N ) is defined by adding to the unitarity conditions in (2.4), the twisted deter-minant conditions (2.6) D qτ ( U ) = ( − q ) i ( τ ) for all τ ∈ S N .Instead of the usual definition (unital ∗ -algebra generated by u jk subject to the relations in(2.4) and (2.6)), we prefer a different path and follow the exposition in Koelink [16, Section 2].We do not start with the unital ∗ -algebra generated by the u jk but with the unital algebra, onwhich, then, an involution is defined. This has the enormous advantage that, here and later on, UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) in (well)defining maps (representations and cocycles), we have to control relations only for thegenerators u jk but not for their adjoints; the saved amount of time is considerable.Let us recall that the quantum determinant of a matrix U is defined as D q ( U ) := X σ ∈ S N ( − q ) i ( σ ) u ,σ (1) u ,σ (2) . . . u N,σ ( N ) . The quantum minor D jkq ( U ) is defined as the quantum determinant of the ( N − × ( N − -matrix obtained from the matrix U by removing the j -th row and the k -th column. Thatis,(2.7) D jkq ( U ) := X σ ∈ S jkN − ( − q ) i ( σ ) u ,σ (1) . . . u j − ,σ ( j − u j +1 ,σ ( j +1) . . . u N,σ ( N ) , where S jkN − denotes the set of bijections σ : { , . . . , j − , j + 1 . . . , N } → { , . . . , k − , k +1 , . . . , N } . We, usually, abbreviate D := D q ( U ) and D jk := D jkq ( U ) .We define U q ( N ) to be the unital (complex, but not ∗ ) algebra generated by the N + 1 indeterminates u j,k ( j, k = 1 , . . . , N ) and D − subject to the relations u ij u kj = qu kj u ij for i < k, (2.8a) u ij u il = qu il u ij for j < l, (2.8b) u ij u kl = u kl u ij for i < k, j > l, (2.8c) u ij u kl = u kl u ij − ( q − q ) u il u kj for i < k, j < l, (2.8d)and(2.8e) D − D = = DD − . Recall that D is, by definition, in the subalgebra generated by the u jk alone, and that,by (2.8a) to (2.8d), D in central. (This also means that U q ( N ) is isomorphic to the centralextension of the algebra generated indeterminates u jk subject to (2.8), including the extrarelation D − D = = DD − .)One may check that Equations (2.5) plus ∆( D − ) := D − ⊗ D − and ε ( D − ) := 1 , turn U q ( N ) into a bialgebra. (There is also an antipode.) One may also check that D = D q id ( U ) .Defining(2.9) u ∗ jk := ( − q ) k − j D jk D − , one may show two things. Firstly, the map u jk u ∗ jk , D − D extends to an involution, turning U q ( N ) into a ∗ -bialgebra (even a Hopf ∗ -algebra); this concludesthe definition of U q ( N ) . Secondly, the u jk and u ∗ jk fulfill the unitarity conditions in (2.4). (To behonest, one, first, verifies that the elements in (2.9) would satisfy the unitarity condition which,then, motivates to define the involution in that way.) Now, SU q ( N ) is defined to be the quotient of U q ( N ) by the extra relation D = . Clearly,the homomorphisms ∆ and ε respect this extra relation, so, by Lemma 2.26, they survive thequotient. A similar argument shows that the involution survives the quotient, too. By Corollary2.27, SU q ( N ) is isomorphic to the algebra generated by indeterminates u jk subject to Relations(2.8) and D = .We briefly explore several homomorphisms, which illustrate how the several SU q ( n ) and U q ( m ) sit in each other as quantum subgroups. Of course, they reoccur when we examine whether mapslive on a quantum subgroup. (See Subsection 2.4.)By definition, SU q ( N ) is a quantum subgroup of U q ( N ) via the quotient map ˘ t N : U q ( N ) ∋ u jk u jk ∈ SU q ( N ) , D − . But also U q ( N ) is a quantum subgroup of SU q ( N + 1) . Indeed, quite obviously the map(2.10) t N : u . . . u N u ,N +1 ... ... ... u N . . . u NN u N,N +1 u N +1 , . . . u N +1 ,N u N +1 ,N +1 u . . . u N ... ... ... u N . . . u NN . . . D − determines a surjective homomorphism from SU q ( N + 1) onto U q ( N ) (where, in the same wayas in the definition of ˘ t N the u jk on the left-hand side are the generators of SU q ( N + 1) while the u jk on the right-hand side are the generators of U q ( N ) ), and a few computations show that t N respects comultiplication, counit, and involution. By iterating the t N and the ˘ t N appropriately,we get a 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 homomorphism s N := ˘ t N − ◦ t N − given by(2.11) s N : u . . . u ,N − u N ... ... ... u N − , . . . u N − ,N − u N − ,N u N . . . u N,N − u NN u . . . u ,N − ... ... ... u N − , . . . u N − ,N − . . . , which establishes the inclusion SU q ( N − ⊂ SU q ( N ) . This case is so important for us, thatwe rest for a moment to convince ourselves that this really defines a homomorphism. For thismoment, we distinguish between the generators u jk of SU q ( N ) on the left-hand side, and theirimages v jk := s N ( u jk ) on the right-hand side. (So, for j, k ≤ N − , the v jk are the generators u jk of SU q ( N − .) Clearly, s N respects the relations in (2.8). (Indeed, those relation thatregard only indices not bigger than N − are fulfilled, because the generators of SU q ( N − fulfill them. Those relation in (2.8a) and (2.8b) that have at least one index equal to N , alsocontain at least one factor of the type v kN or v Nk , hence, are identically . The same is true for(2.8c) and (2.8d) if k = l . Only the case k = l = N remains, which is also okay.) Clearly, the v jk satisfy the determinant condition D = . (Indeed, one easily checks s N ( D ) = s N ( D NN ) . And s N ( D NN ) = , because the generators of SU q ( N − fulfill the determinant condition.) Therefore s N is a well-defined algebra homomorphism. Clearly, s N is also a ∗ -algebra homomorphism.(Indeed, the matrix ˜ V := [ v ∗ kj ] jk fulfills ˜ V V = N = V ˜ V . Therefore, ˜ V = V ∗ .) Obviously, s N UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) respects the counits. (Indeed, the two counits are multiplicative, and s N intertwines the rightvalues on the generators. The same argument, though checking is slightly more involved, showsthat s N also respects the the comultiplication; but we do not need that.) Corollary . The algebra SU q ( N − is canonically isomorphic to the quotient of thealgebra SU q ( N ) by the extra relations u kN = δ kN = u Nk .Proof. The homomorphism s N , clearly, respects the extra relations. Therefore it defines a homo-morphism ˜ s N from the quotient of SU q ( N ) onto SU q ( N − . Reading the definition backwardson the ( N − × ( N − -submatrix, all of the relations in (2.8) (for SU q ( N − !) are fulfilled.Thanks to the extra relations, also the quantum determinant is sent to . (cid:3) Clearly, this ˜ s N is also a ∗ -isomorphism and counits (and comultiplications) are the same.This makes applicable Lemma 2.26 when we wish to check if representations of SU q ( N ) andtheir cocycles live on the quantum subgroup SU q ( N − . Corollary . Let π ba a ∗ -representation π of SU q ( N ) and η a π - ε -cocycle. (1) π lives on SU q ( N − if and only if π ( u kN ) = id H δ kN = π ( u Nk ) . (2) If π lives on SU q ( N − , then η lives on SU q ( N − if and only if η ( u kN ) = 0 = η ( u Nk ) . When, in Section 4, we also take into account operator theoretic statements, this corollaryimproves considerably. For all representations (see the beginning of Subsection 4.1) and at leastfor (all) cocycles with respect to certain representations (see the proof of Corollary 4.7), it issufficient to check only the respective condition regarding u NN .Throughout, we also will need the iterated homomorphisms(2.12) s n,N := s n +1 ◦ . . . ◦ s N = ˘ t n ◦ t n ◦ . . . ◦ ˘ t N − ◦ t N − ( n < N ) , which establish SU q ( n ) as a quantum subgroup of SU q ( N ) . In Section 6, we will also need ˘ s N := t N − ◦ ˘ t N : U q ( N ) → U q ( N − and its iterates(2.13) ˘ s n,N := ˘ s n +1 ◦ . . . ◦ ˘ s N = t n ◦ ˘ t n +1 ◦ . . . ◦ t N − ◦ ˘ t N ( n < N ) , which establish U q ( n ) as a quantum subgroup of U q ( N ) . Note that ˘ t n ◦ ˘ s n,N = s n,N ◦ ˘ t N , t n ◦ s n +1 ,N +1 = ˘ s n,N ◦ t N . (2.14)We close this subsection by collecting some more relations. By definition, the generators u jk of U q ( N ) (and, therefore, also the generators of SU q ( N ) ) satisfy the basic commutation relationsin (2.8). We will need frequently the following special cases. u jN u NN = qu NN u jN , (2.15a) u Nk u NN = qu NN u Nk , (2.15b) u jN u Nk = u Nk u jN , (2.15c) u jk u NN = u NN u jk − ( q − q ) u jN u Nk , (2.15d) for j, k < N . Recall that (as explained in brackets after Equation (2.9)) the convolution isdefined such that the unitarity conditions in (2.4) are fulfilled. Additionally, one may verify thefollowing commutation relations among the generators and their adjoints. u ij u ∗ kl = u ∗ kl u ij for i = k, j = l, (2.16a) u ij u ∗ kj = qu ∗ kj u ij − (1 − q ) X p A key lemma. The following little lemma is key for the approximation results in Section4. The idea for the approximation is taken from Schürmann and Skeide [27, 24]. But the lemmawould have simplified considerably also the proofs in [27, 24], which, in fact, will be reprovedhere. (For optical reasons, we write instead of id H .) Lemma . Let a be a contraction on a Hilbert space H . Then the following are equivalent: (1) ( − a ) H = H . (2) lim p ↑ − a − pa = , strongly. (3) lim p ↑ − a − pa = , weakly. (4) − a is injective. UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Moreover, under any of the conditions, lim p ↑ − p − pa = 0 , strongly.Proof. Obviously, (2) ⇒ (3). The approximation in (2) or (3) shows that every x is in the closureof the range of − a in the respective topology. So, clearly, (2) ⇒ (1) and (3) ⇒ ( − a ) H w = H .Since weak and norm closure of linear subspaces of Hilbert spaces coincide, we also get (3) ⇒ (1).For (1) ⇒ (2), let us start with the observations that, for all ≤ p < , − − a − pa = a − p − pa and (cid:13)(cid:13)(cid:13) − p − pa (cid:13)(cid:13)(cid:13) ≤ − p − p k a k ≤ − p − p = 1 . So, (cid:13)(cid:13) − a − pa (cid:13)(cid:13) ≤ and, therefore, it suffices to check strong convergence on the total subset ( − a ) H of H . So, let us choose y ∈ H and check strong convergence on x = ( − a ) y . We find (cid:13)(cid:13)(cid:13) x − − a − pa x (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) a − p − pa ( − a ) y (cid:13)(cid:13)(cid:13) ≤ − p )2 k y k → . This also shows the supplement − p − pa → , strongly.So, we have closed (1) ⇒ (2) ⇒ (3) ⇒ (1). To show equivalence with (4), it is enough to observethat, firstly, (1) and (4) are “dual under adjoint” to each other ( a injective if and only if a ∗ H isdense in H ), and that, secondly, (3) is invariant under taking adjoints. (cid:3) Let us note that the equivalence of (1) and (4) (the latter expressed in the form ( − a ∗ ) H = H ) is well-known. It can easily be shown directly; see, for instance, the elementary argumentthat occurs in the proof of Shalit [26, Theorem 8.2.7] or the occurrence as a corollary of dilationtheory in the survey by Levy and Shalit in [18]. But, we will neeed the implication (4) ⇒ (2),and for that we do not know a source.Actually the statement proved in the proof of [26, Theorem 8.2.7] is slightly better than(1) ⇔ (4). As we need it, we state it and also furnish a (different) proof. Lemma . Under the same hypothesis: ax = x ⇔ a ∗ x = x (for all x ∈ H ).Proof. Of course, we are only interested in the case, when x = 0 so that x would be an eigen-vector. But for the proof there is no difference. We have k ax − x k = k ax k − h x, a ∗ x i − h x, ax i + k x k , k a ∗ x − x k = k a ∗ x k − h x, a ∗ x i − h x, ax i + k x k . If the first row, is , then the second row would be negative if k a ∗ x k < k ax k . We get k x k = k ax k ≤ k a ∗ x k ≤ k a ∗ k k x k = k x k . Therefore, k a ∗ x − x k = k ax − x k = 0 . The other direction follows by a ↔ a ∗ . (cid:3) Corollary . For any contraction a ∈ B ( H ) , the Hilbert space H decomposes uniquely intoinvariant subspaces H and H such that a acts on H as identity and such that − a is injectiveon H . Moreover, H = ker( − a ) and − a − pa converges strongly to the projection onto H . Classification of gaussian generating functionals In this Section we investigate the gaussian generating functionals on SU q ( N ) and their Schür-mann triples. We shall see that gaussian generating functionals on SU q ( N ) are classified by ( N − real numbers, which captures the freedom in the choice of a drift term, and a positivereal ( N − × ( N − -matrix , which captures the freedom in choosing a gaussian generatingfunctional ψ that satisfies ψ ◦ P = ψ . Contrary to the case N = 2 , for N ≥ there exist gaussianpairs ( id H ε, η ) that cannot be completed to a Schürmann triple.The first thing we have to do, also in order to actually indicate a projection P , is to find ahermitian basis extension E from K to K . (See Subsection 2.2.) Our algebra is generated bythe elements u jk − δ jk ∈ K , their adjoints, and . Since the elements of E must be in K butnot in K , it is clear that we have to search them among the hermitian linear combinations ofthe u jk − δ jk and their adjoints. We put d j := u jj − u ∗ jj i . Lemma . (a) u jk , u ∗ jk ∈ K (actually, u jk , u ∗ jk ∈ K ∞ ), for j = k , (b) ( u jj − ) + ( u ∗ jj − ) ∈ K , (c) d + d + . . . + d N ∈ K , (d) d j d k − d k d j ∈ K (actually, d j d k − d k d j ∈ K ∞ ).Proof. (a) Uniting appropriately Relations (2.8a) and (2.8b), we obtain u jk u ll = qu ll u jk for j = k and l := max( j, k ) . Therefore (expanding the brackets on the right-hand side), u jk = q ( u ll − ) u jk − u jk ( u ll − )1 − q . Since u ll − , u jk ∈ K , we get u jk ∈ K . (By induction it follows that u jk ∈ K n for all integers n ≥ .)(b) By the unitarity relation in (2.4) we see that − u jj u ∗ jj = X p = j u jp u ∗ jp ∈ K ∞ . Hence ( u jj − 1) + ( u jj − ∗ = − ( − u jj u ∗ jj ) − ( u jj − u jj − ∗ ∈ K .(c) Putting v j := u jj − ∈ K , we obtain u . . . u NN = ( v + ) . . . ( v N + ) = + ( v + . . . + v N ) + terms in K . Therefore, v + . . . + v N + ( − u . . . u NN ) ∈ K , Since D = , we have − u . . . u NN = X σ ∈ S N ,σ =id ( − q ) i ( σ ) u ,σ (1) . . . u N,σ ( N ) . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Since for σ = id there is at least one j with j = σ ( j ) , we see by Part (a) that the right-handside is in K ∞ . So, v + . . . + v N ∈ K , hence, d + . . . + d N = ( v + . . . + v N ) − ( v + . . . + v N ) ∗ i ∈ K . (d) This follows from (2.8d), (2.16a), and Part (a). (cid:3) By (a), (b), and (c), we have: Corollary . Put E := { d , . . . , d N } . Then the set E ∪ K spans K . By (d) and Corollary 2.32, we have: Corollary . The gaussian cocycle η can be completed to a Schürmann triple ( id H ε, η, ψ ) ifand only if η is hermitian. Remark . Recall that, in the proof of Corollary 2.32, we did show that the cocycles ofgaussian generating functionals are hermitian under an extra condition (fulfilled by SU q ( N ) , bythe lemma). The backwards direction, still depends on Schürmann’s [23, Proposition 5.1.11] . Inthe sequel, after completing the discussion of E and P , we will construct (explicitly and in aclassifying way) for each hermitian gaussian cocycles of SU q ( N ) a generating functional, thus,making Corollary 2.32 (for SU q ( N ) ) independent of [23, Proposition 5.1.11] . To show that E is a basis extension, it remains to show that the elements d j of E are linearlyindependent and are not in K . To that goal, let us consider the family of characters defined by ε θ ,...,θ N ( u kl ) := e iθ k δ k,l , where θ , . . . , θ N ∈ R , and where θ is determined by P Nk =1 θ k = 0 . (Of course, one easilyverifies directly that this (well)defines a ∗ -homomorphism into C . But, see also Remark 3.10.)One easily verifies that the functionals ε ′ j := ∂ε θ ,...,θN ∂θ j (cid:12)(cid:12) θ = ... = θ N =0 ( j = 2 , . . . , N ) (pointwisederivative) vanish on K and satisfy ε ′ j ( d k ) = δ jk . Corollary . The elements d , . . . , d N in E are linearly independent and not in K . There-fore, by Corollary 2.12, E is a hermitian basis extension from K to K . The ε ′ k here coincide, indeed, with the functionals ε ′ k occurring in Subsection 2.5 from the basisextension. We, now, also can fix our P as in (2.3). By Proposition 2.31, we get all gaussiancocycles in the form η = N X j =2 η j ε ′ j . But what are the hermitian ones, and how do they give rise to a generating functional? Well, the first question is easy: η is hermitian if and only if the matrix with entries r jk := h η j , η k i is real (hence, symmetric). For N ≥ , it is easy to write down gaussian cocycles thatviolate this. (See also [8, Proposition 2.3].) Corollary . For N ≥ , the quantum group SU q ( N ) does not have Property (GC). This contrasts the fact that SU q (2) has Property (AC); essentially, [27, Lemma 2.6+Theorem2.8] or [24, Lemma 3.2+Theorem 3.3].As for the second question: Almost as easy as for the one-dimensional case in Proposition2.11, one checks by direct verification that, if the matrix ( r jk ) is real then the functional ψ := 12 N X j,k =2 r j,k ε ′′ j,k , with ε ′′ jk := ∂ ε θ ,...,θN ∂θ j ∂θ k (cid:12)(cid:12) θ = ... = θ N =0 ( j, k = 2 , . . . , N ) , has an η -induced -coboundary (being, there-fore, conditionally positive and -normalized). Note that ε ′′ jk ( d l ) = 0 as soon as j = l or k = l .(In particular, it is if j = k .) And ε ′′ jj ( d j ) = 0 , by direct computation. So, ψ also fulfills ψ ◦ P = ψ (being, therefore, hermitian).Any real positive matrix ( r jk ) may occur. Indeed, denote by ( q jk ) the unique positive squareroot, which, necessarily, also has real entries. (Just diagonalize by a real unitary.) Then, thegaussian cocycle η := P Nj =2 η j ε ′ j with η j ∈ C N − having ℓ -coordinate q ℓj ( ℓ = 2 , . . . , N ) has thematrix h η j , η k i = r jk .Obviously, ψ determines the r jk . Adding also a drift term, we, thus, obtain: Theorem . Gaussian generating functionals on SU q ( N ) are parametrized one-to-one by N − real numbers r j ( j = 2 , . . . , N ) and a positive real ( N − × ( N − -matrices ( r jk ) j,k =2 ,...,N as (3.1) ψ = N X j =2 r j ε ′ j + 12 N X j,k =2 r j,k ε ′′ j,k . We add that our projections P for SU q ( N ) and for SU q ( N − , are compatible with thesubgroup structure: Proposition . P ◦ s N = s N ◦ P .Proof. The homomorphism s N (see (2.11)) sends d N to and it sends the d n (2 ≤ n ≤ N − of SU q ( N ) to the d n of SU q ( N − . Therefore, s N ( E ) = E SU q ( N − ∪ { } , and the statementfollows from Corollary 2.25. (cid:3) And iterating: Corollary . P ◦ s n,N = s n,N ◦ P . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Remark . Note that the classical ( N − -torus T N − may be identified with the quantumgroup generated by N commuting unitaries u j subject to the relation u . . . u N = . Sending u jk to u j δ jk defines a ∗ -homomorphism τ N , identifying T N − as a quantum subgroup of SU q ( N ) .Moreover, the family ε θ ,...,θ N lives on T N − . This shows several things: (1) The gaussian generating functionals of SU q ( N ) live on T N − ; in this sense, SU q ( N ) and T N − have the same gaussian generating functionals. Effectively, we could have deducedTheorem 3.7, applying the results about classical compact Lie groups in [28] to T N − .(Our approach here is simpler and improves also on [28] .) Obviously, the projection P for SU q ( N ) lives on T N − , too, and the (unique!) map ˜ P that illustrates it, is theprojection P for T N − . (2) Every quantum (semi)group G sitting as SU q ( N ) ⊃ G ⊃ T N − , has the same gaussianparts. Moreover, the projection P for G may be chosen compatible with those for SU q ( N ) and T N − . Decomposition This is the central Section of these notes. We decompose an arbitrary representation π of SU q ( N ) into a unique direct sum π = π ⊕ π ⊕ . . . ⊕ π N , where π n lives on SU q ( n ) for ≤ n ≤ N , and where π is the maximal gaussian part (living on the trivial quantum subgroup SU q (1) ∼ = { e } , since π is the trivial representation π = id H ε ). Then, on the completelynon-gaussian part, we show that each cocycle η is determined by the vectors η ( u nn ) and canbe approximated by coboundaries. Therefore, SU q ( N ) possesses Property (NC), hence, (LK).Since the cocycles η n with respect to π n obtained from the decomposition of π are completelynon-gaussian whenever n ≥ , we also get a decomposition ψ = ψ + . . . + ψ N of ψ into generatingfunctionals ψ n that live on SU q ( n ) .4.1. Decomposition of representations. Obviously, a representation ρ of SU q ( N ) that liveson SU q ( N − sends u NN to id . But, this condition is also sufficient. Indeed, we have(4.1) − u ∗ NN u NN = N − X k =1 u ∗ kN u kN . So, if the left-hand side is sent by ρ to , then the sum of the positive operators to which ρ sendsthe right-hand side, must be as well. So, ρ ( u kN ) = 0 for all ≤ k ≤ N − . Starting from − u NN u ∗ NN , the same argument shows that ρ ( u Nk ) = 0 for all ≤ k ≤ N − . Clearly, theremaining matrix ( ρ ( u jk )) ≤ j,k ≤ N − has to be unitary. By Corollary 2.38, ρ lives on SU q ( N − . Lemma . Let π be a representation of SU q ( N ) . Then the subspace ker( id − π ( u NN )) is in-variant for π .Proof. Let f be in ker( id − π ( u NN )) . In other words, let f be such that π ( u NN ) f = f .Applying π to Equation (4.1) and, then, the positive functional h f, • f i , we get π ( u kN ) f = 0 for all ≤ k ≤ N − . Just as easy, from the other unitarity condition we get π ( u ∗ Nk ) f = 0 forall ≤ k ≤ N − . From Relation (2.16d), we get u ∗ Nk u Nk = u Nk u ∗ Nk + (1 − q ) X j Theorem . Let π be a representation of SU q ( N ) on a Hilbert space H . Then π decomposesuniquely into representations π n on invariant subspaces H n ( n = 1 , . . . , N ) such that • π is gaussian. • π n (2 ≤ n ≤ N ) lives on SU q ( n ) .Moreover, each π n ( − u nn ) is injective.Proof. For N = 2 , the statement follows directly from the lemma. And the corollary providesthe inductive step. (cid:3) Note that π ( − u ) is injective if and only if π = 0 .4.2. Decomposition of cocycles. Well, we know that if a ∗ -representation π decomposesinto representations π n , then a π - ε -cocycle η decomposes into π n - ε -cocycle η n . We now convinceourselves that not only (in the decomposition from the preceding subsection) π n lives on SU q ( n ) ,but that, for n = 2 , . . . , N , also η n lives on SU q ( n ) and is determined by η n ( u nn ) .Let us start with some general cocycle computations. Lemma . Let π be a ∗ -representation of SU q ( N ) and let η be a π - ε -cocycle. Then η ( u jN ) = − id − qπ ( u NN )) π ( u jN ) η ( u NN ) , (4.2a) η ( u Nk ) = − id − qπ ( u NN )) π ( u Nk ) η ( u NN ) , (4.2b) and (4.3) π ( u NN − ) η ( u jk ) = (cid:16) π ( u jk − δ jk ) − q − qπ ( − q u NN ) π ( u jN u Nk ) (cid:17) η ( u NN ) for any j, k < N .Proof. If a = u jN or a = u Nk for j, k < N , then a ∈ ker ε and au NN = qu NN a . Hence thecocycle property leads to π ( a ) η ( u NN ) + η ( a ) = qπ ( u NN ) η ( a ) . Since π ( u NN ) is a contraction, we know that id − qπ ( u NN ) is (boundedly) invertible. Thus, η ( a ) = − id − qπ ( u NN )) π ( a ) η ( u NN ) for any such element a .On the other hand, if a = u jk with j, k < N , then the cocycle property applied to (2.8d) reads π ( u jk ) η ( u NN ) + η ( u jk ) = η ( u jk u NN ) = η ( u NN u jk ) − ( q − q ) η ( u jN u Nk )= π ( u NN ) η ( u jk ) + η ( u NN ) ε ( u jk ) − ( q − q ) π ( u jN ) η ( u Nk ) , so, π ( u NN − ) η ( u jk ) = π ( u jk − δ jk ) η ( u NN ) + ( q − q ) π ( u jN ) η ( u Nk )= (cid:2) π ( u jk − δ jk ) − ( q − q ) π ( u jN ) id − qπ ( u NN )) π ( u Nk ) (cid:3) η ( u NN )= (cid:2) π ( u jk − δ jk ) − ( q − q ) id − q π ( u NN )) π ( u jN ) u Nk (cid:3) η ( u NN ) . (cid:3) Corollary . If π ( − u NN ) is injective, then any π - ε -cocycle η is determined by its value η ( u NN ) .Proof. If π ( − u NN ) is injective, then also η ( u jk ) ( j, k < N ) is determined by the lemma. (cid:3) Lemma . Suppose for n < N we have a ∗ -representation π of SU q ( N ) that lives on SU q ( n ) and such that π ( − u nn ) is injective. Then every π - ε -cocycle η satisfies η ( u mm ) = 0 for all m > n .Proof. Applying η to (2.8d) for i = j = n and k = l = m , we obtain π ( u nn ) η ( u mm ) + η ( u nn ) = π ( u mm ) η ( u nn ) + η ( u mm ) − ( q − q ) π ( u nm ) η ( u mn ) .π lives on SU q ( n ) , so, π ( u mm ) = id and π ( u nm ) = 0 . Hence, the equation simplifies to π ( u nn ) η ( u mm ) = η ( u mm ) . Since π ( − u nn ) is injective, we get η ( u mm ) = 0 . (cid:3) Corollary . Such η lives on SU q ( n ) , too.Proof. Since η ( U NN ) = 0 , by Equations (4.2) we get η ( u kN ) = 0 = η ( u Nk ) for all ≤ k ≤ N .By Corollary 2.38, η lives on SU q ( N − .The result follows, now, by induction. (Applying the same argument to the representation ˜ π N − and the cocycle ˜ η N − on SU q ( N − , we get that they live on SU q ( N − , and so forth.) (cid:3) Consequently: Theorem . In the notations of Theorem 4.3: Every π - ε -cocycle η decomposes (uniquely)into the direct sum over π n - ε -cocycle η n , where η is gaussian and where η n lives on SU q ( n ) for ≤ n ≤ N . Moreover, η is determined by its values η ( u nn ) (1 ≤ n ≤ N ) .Proof. Only the last statement needs a proof. It follows from η ( u nn ) = 0 ⊕ . . . ⊕ ⊕ η n ( u nn ) ⊕ . . . ⊕ η N ( u nn ) , because, by Corollary 4.5, each η n ′ ( u nn ) ( n ′ ≥ n ) is determined by η n ′ ( u n ′ n ′ ) . (cid:3) UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Remark . In the following subsection we will show that, for ≤ n ≤ N , the cocycles η n are limits of coboundaries. So, it is legitimate to ask, why, in order to show that η n lives on SU q ( n ) , we did not appeal to Corollary 2.22. The point is that our basic approximation result,Proposition 4.10, cannot be applied directly to the cocycle η n on SU q ( N ) , but only to a cocycleon SU q ( n ) . Only after having shown in the present subsection that η n lives on SU q ( n ) , we havegranted the cocycle ˜ η n on SU q ( n ) such that η n = ˜ η n ◦ s n,N to which Proposition 4.10 can beapplied. (See the proof of Theorem 4.11.) Approximation of cocycles. We now show that each of the cocycles η n ( n ≥ inTheorem 4.8 can be approximated by coboundaries. (For a gaussian cocycle this is, obviously,not so, because gaussian coboundaries are .) Proposition . Let π be a ∗ -representation of SU q ( N ) such that π ( − u NN ) is injective,and let η be a π - ε -cocycle. Then the coboundaries a π ◦ ( id − ε )( a ) − π ( − pu NN ) η ( u NN ) (0 < p < (4.4) converge (pointwise on SU q ( N ) ) to η for p → .Proof. This is essentially a consequence of Lemmata 2.39 and 4.4. By the cocycle property, itsuffices to control convergence on the generators u jk .For a = u NN , we get lim p → π ( − u NN ) π ( − pu NN ) η ( u NN ) = η ( u NN ) .For a = u kN or a = u kN ( k < N ) , by Lemmata 4.4 and 2.39 we get η ( a ) = − π ( − qu NN ) π ( a ) η ( u NN ) = lim p → − π ( − qu NN ) π ( a ) π ( − u NN ) π ( − pu NN ) η ( u NN )= lim p → π ( a ) − π ( − pu NN ) η ( u NN ) . Since a ∈ ker ε , we have π ( a ) = π ◦ ( id − ε )( a ) .After these two easy cases, here is the difficult one: For a = u jk ( j, k < N ) , by Lemmata 4.4and 2.39 we see that − π ( − pu NN ) × the right-hand side of (4.3) converges to η ( a ) . We are done ifwe show that the difference with the right-hand side of (4.4), that is, (cid:16) − π ( − pu NN ) (cid:0) π ( u jk − δ jk ) − q − qπ ( − q u NN ) π ( u jN u Nk ) (cid:1) − π ( u jk − δ jk ) − π ( − pu NN ) (cid:17) η ( u NN ) converges to . We observe that the terms with δ jk cancel out. Omitting η ( u NN ) , taking alsointo account that π ( − pu NN ) and π ( − q u NN ) commute, we remain with π ( u jk ) π ( − pu NN ) − π ( − pu NN ) π ( u jk ) + q − qπ ( − q u NN ) 1 π ( − pu NN ) π ( u jN u Nk ) . We will conclude the proof by showing that this converges to , strongly. Recall that q isfixed, and that multiplying the whole thing from the left with π ( − q u NN ) to make disappear π ( − q u NN ) does not change whether this expression converges for p → or not, nor does it, inthe case of convergence, change the answer to the question if the limit is or not. We get π ( − q u NN ) (cid:2) π ( u jk ) , π ( − pu NN ) (cid:3) + q − qπ ( − pu NN ) π ( u jN u Nk ) . (4.5)For simplicity, in the following computations (for fixed p < ) we omit the representation π . (In SU q ( N ) this does not make sense. But in the enveloping C ∗ -algebra it does, and afterreinserting π the result is the right one, because π is bounded.) Let us first compute thecommutator (cid:2) u jk , − pu NN (cid:3) . From (2.8d), we get by induction that [ u jk , u sNN ] = − ( q − q )(1 + . . . + q s − ) u s − NN u jN u Nk , where the sum . . . + q s − = P s − t =0 q t is understood to have s summands (also if s = 0 ); itcoincides with the well known q -number [ s ] q . We get (cid:2) u jk , − pu NN (cid:3) = − ( q − q ) ∞ X s =1 s − X t =0 q t p s u s − NN u jN u Nk . Inserting this in (4.5), expanding also − pu NN , and omitting the common factor ( q − q ) u jN u Nk ,weobtain ( q u NN − ) ∞ X s =1 s − X t =0 q t p s u s − NN + ∞ X s =0 ( pu NN ) s = ∞ X s =1 s − X t =0 q t +1) p s u sNN − ∞ X s =1 s − X t =0 q t p s u s − NN + + ∞ X s =1 ( pu NN ) s = + ∞ X s =1 s X t =0 q t ( pu NN ) s − p ∞ X s =1 s − X t =0 q t ( pu NN ) s − = ∞ X s =0 s X t =0 q t ( pu NN ) s − p ∞ X s =0 s X t =0 q t ( pu NN ) s = (1 − p ) ∞ X s =0 s X t =0 q t ( pu NN ) s . Reordering for powers of q , we get (1 − p ) ∞ X t =0 q t ∞ X s = t ( pu NN ) s = (1 − p ) ∞ X t =0 q t ( pu NN ) t ∞ X s =0 ( pu NN ) s = − q pu NN − p − pu NN . For p ↑ , the first factor converges in norm to − q u NN . Under π , by the supplement of Lemma2.39, the second factor converges to , strongly. (cid:3) Theorem . In the notations of Theorems 4.3 and 4.8: The cocycles η n (2 ≤ n ≤ N ) are(pointwise) limits η n = lim p ↑ π n ◦ ( id − ε ) − π n ( − pu nn ) η n ( u nn ) of coboundaries. UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Proof. Recall that by Theorem 4.8, η n lives on SU q ( n ) . The result follows by applying Proposi-tion 4.10 to the (unique) cocycle ˜ η n on SU q ( n ) such that η n = ˜ η n ◦ s n,N . (cid:3) Decomposition of generating functionals. We are now ready for the punch line. Proposition . Let π be a ∗ -representation of SU q ( N ) such that π ( − u NN ) is injective.Then, every π - ε cocycle η determines a unique generating functional ψ that completes ( π, η ) toa Schürmann triple and satisfies ψ ◦ P = ψ .Proof. This is a corollary of Proposition 4.10 and Lemma 2.16. (cid:3) Corollary . In the notations of Theorems 4.3 and 4.8: For every ≤ n ≤ N , the cocycle η n determines a unique generating functional ψ n that completes ( π n , η n ) to a Schürmann tripleand satisfies ψ n ◦ P = ψ n . Moreover, ψ n lives on SU q ( n ) .Proof. Let ˜ ψ n denote the generating functional granted (for some P n on SU q ( n ) ) by Proposition4.12 for the cocycle ˜ η n on SU q ( n ) . Then ψ n := ˜ ψ n ◦ s n,N ◦ P does the job. By Proposition 3.8and its corollary, ψ n = ˜ ψ n ◦ s n,N . (cid:3) Corollary . SU q ( N ) has property (NC) , hence, (LK) .Proof. A cocycle η being completely non-gaussian, means precisely that η is . Then ψ := ψ + . . . + ψ N from the preceding corollary, does the job. (cid:3) The ψ n are determined uniquely by ψ n ◦P = ψ n and the requirement that ˜ ψ n has a Schürmanntriple where ˜ π n ( − u nn ) is injective. We wish to capture this property without explicit referenceto the Schürmann triple. Definition . A generating functional ψ on SU q ( N ) ( N ≥ ) is irreductible if it is com-pletely non-gaussian and if for every generating functional ˜ ψ on SU q ( N − , the statement ψ − ˜ ψ ◦ s N is a generating functional implies the statement ˜ ψ ◦ s N is a drift.This definition does the job: Proposition . Let ψ be a generating functional on SU q ( N ) ( N ≥ ) and ( π, η, ψ ) itsSchürmann triple. Then, ψ is irreductible if and only if, in Theorem 4.3’s decomposition, π = π N .Proof. Obviously, the statement is true if ψ is not completely non-gaussian. So, we assume that ψ is completely non-gaussian.We know from Theorem 4.3 that π = π N ⊕ ˜ π ◦ s N for some representation ˜ π of SU q ( N − .Moreover, ˜ π = 0 , because, otherwise, ˜ π ◦ s N would contribute to π . So, if N = 2 , then thereis nothing left to prove, so we assume N ≥ .By Corollary 4.7, the cocycle η − η N with respect to ˜ π ◦ s N also lives on SU q ( N − , hence,has the form ˜ η ◦ s N for a (unique) cocycle ˜ η with respect to ˜ π . By Corollary 4.14, both pairs ( π N , η N ) and (˜ π, ˜ η ) may be completed to Schürmann triples ( π N , η N , ψ N ) and (˜ π, ˜ η, ˜ ψ ) . Clearly, we may arrange ψ N such that ψ = ψ N + ˜ ψ ◦ s N .Clearly, ˜ ψ ◦ s N is a drift if and only if ˜ η ◦ s N = 0 . Of course, if ˜ π ◦ s N = 0 , that is, if π = π N ,then ˜ η ◦ s N = 0 so that ˜ ψ ◦ s N is a drift. Conversely, since η is cyclic, if ˜ π ◦ s N = 0 , that is, if π = π N , then ˜ η ◦ s N = 0 so that ˜ ψ ◦ s N is a not drift. (cid:3) As an immediate corollary, we get the main result: Theorem . Let ψ be a generating functional on SU q ( N ) . Then ψ decomposes uniquely intoa sum ψ := ψ + . . . + ψ N such that the ψ n satisfy: • ψ is gaussian. • ψ n = ˜ ψ n ◦ s n,N ( n ≥ for some irreductible generating functional ˜ ψ n on SU q ( n ) satisfying ˜ ψ n ◦ P = ˜ ψ n . Recall that, by definition, all ψ n (2 ≤ n ) , hence their sum, are completely non-gaussian. Corollary . ψ = ψ G + ψ L with ψ G := ψ and ψ L := ψ + . . . + ψ N is the unique Lévy-Khintchine decompositions satisfying ψ L ◦ P = ψ L . Remark . (1) Recall from Section 3 that SU q ( N ) , unlike SU q (2) , does not have property (GC) as soonas N ≥ . (2) Like for SU q (2) , completely non-gaussian cocycles on SU q ( N ) are all limits of cobound-aries and determined by N − vectors. However, unlike for SU q (2) , not all vectors inthe representation spaces of π n may occur. (This is subject of Section 5.) While for SU q (2) the cocycle would be defined as the limit in Proposition 4.10 (including a proofthat the limit for N = 2 exists for whatever vector we chose), for SU q ( N ) , the fact thatwe start with a given cocycle was used essentially in the proof that the limit for a = u jk ( j, k < N ) exists and gives the right result. (In fact, for the vector in the counter examplein Proposition 5.1, the limit cannot exist for all a , because, otherwise, it would define acocycle.) UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Parametrization of generating functionals In Section 4, we have shown that any Schürmann triple on SU q ( N ) can be decomposed(uniquely, in the sense explained in Theorem 4.17) into a gaussian part and a sum of generatingfunctionals that live on SU q ( n ) (2 ≤ n ≤ N ) and which, as functionals on SU q ( n ) , are irre-ductible. In Section 3, we have classified the gaussian generating functionals. In this section wealso parametrize the irreductible generating functionals. Putting everything together by meansof Theorem 4.17, we obtain a parametrization of all generating functionals on SU q ( N ) , and,thus, up to quantum stochastic equivalence (up to unitary equivalence), of all (cyclic) Lévyprocesses on this quantum group.The irreductible generating functionals on SU q ( N ) are exactly those generating functionals ψ that admit a Schürmann triple ( π, η, ψ ) where π ( − u NN ) is injective. Since, by Proposition4.12, every cocycle in such a triple determines the values of a generating functional on K ,we may classify the irreductible generating functionals (up to unitary equivalence) by cycliccocycles with respect to ∗ -representations π with injective π ( − u NN ) . However, concentratingon the cocycle and only, then, determining its representation (and check whether it fulfills thecondition) is not really very practicable. Not for nothing, the order in Procedure 2.4 is to startwith the representation π and, then, to determine all its cocycles. Not for nothing, did weexplain in Remark 2.5 that we do not usually require that the cocycle be cyclic. (This can bedone better under symmetry conditions on ψ ; see Das, Franz, Kula, and Skalski [7].)On the other hand, by Corollary 4.5 (or, better, by Lemma 4.4) we know that a cocycle η with respect to π with injective π (1 − u NN ) , is determined by its value η ( u NN ) . It is, therefore,tempting to parametrize such cocycles (and, then, the functionals they determine) by vectors η NN ∈ H such that η ( u NN ) = η NN . For N = 2 (where the irreductible generating functionalsare exactly the completely non-gaussian ones), we know from [27, 24] that every vector in H may occur as η ( u ) for a cocycle. However, for N ≥ , this is not so. Proposition . For every N ≥ there exists a ∗ -representation π of SU q ( N ) on H withinjective π ( − u NN ) and a vector η NN ∈ H such that no π - ε -cocycle η fulfills η ( u NN ) = η NN .Proof. For the N × N -matrix ( u jk ) of SU q ( N ) , let us refer by the [ m, n ] -block ( ≤ m < n ≤ N )to the submatrix with indices j, k ∈ { m, . . . , n } . So far, we always embedded SU q ( n ) by,roughly speaking, “identifying” its defining n × n -matrix with the [1 , n ] -block of SU q ( N ) . (Seethe definition of s N in (2.11) for the precise meaning of this in the case n = N − .) It isnoteworthy, that we may embed SU q ( n ) in the same way to any other [ m + 1 , m + n ] -block of SU q ( N ) . Here, we are interested in particular in the lower right n × n -square, that is, in the [ N − n + 1 , N ] -block of SU q ( N ) . Since a ∗ -representation π of SU q (3) gives, when embeddedinto the [ N − , N ] -block of SU q ( N ) , rise to a ∗ -representation π of SU q ( N ) , and since π hasinjective π ( − u NN ) if and only if π has injective π ( − u ) , we see that it is enough to showthe statement for N = 3 , only.Now, for N = 3 , every ∗ -representation ρ of SU q (2) gives rise to a ∗ -representation ρ of SU q (3) when embedding SU q (2) into the [1 , -block of SU q (3) and a ∗ -representation ρ of SU q (3) when embedding SU q (2) into the [2 , -block of SU q (3) . Recall that the generators u jk of SU q (2) are commonly written as (cid:18) u u u u (cid:19) = (cid:18) α − qγ ∗ γ α ∗ (cid:19) . For ρ let us choose the irreducible ∗ -representation on the Hilbert space H with ONB ( e k ) k ∈ N defined by ρ ( α ) : e k e k − p − q k , ρ ( γ ) : e k e k q (where e − := 0 ). We put π := ρ ⋆ ρ , so that π ( U ) = ( ρ ⊗ ρ ) (cid:16) X i =1 u ji ⊗ u ik (cid:17) = ( ρ ⊗ ρ ) α − qγ ∗ γ α ∗ 00 0 ⊗ α − qγ ∗ γ α ∗ = ρ ( α ) ⊗ id − qρ ( γ ) ∗ ⊗ ρ ( α ) q ρ ( γ ) ∗ ⊗ ρ ( γ ) ∗ ρ ( γ ) ⊗ id ρ ( α ) ∗ ⊗ ρ ( α ) − qρ ( α ) ∗ ⊗ ρ ( γ ) ∗ id ⊗ ρ ( γ ) id ⊗ ρ ( α ) ∗ . (The reader who diligently followed us when we said we do not (really) need the comultiplication,will now have to check that this assignment really defines a ∗ -representation of SU q (3) ; thereader who accepts that we have a comultiplication, may use the fact that the convolution of ∗ -representations is a ∗ -representation.) Now, ρ ( − α ∗ ) is injective, so π ( − u ) = id ⊗ ρ ( − α ∗ ) is injective, too. By Relation (4.3) for j = 1 = k , taking also into account that π ( u ) = 0 , weobtain(5.1) ( id ⊗ ρ ( α ∗ − )) η ( u ) = ( ρ ( α − ) ⊗ id ) η ( u ) for every cocycle with respect to π .Suppose there was a cocycle η with η ( u ) = e ⊗ e , so that ( ρ ( α − ) ⊗ id ) η ( u ) = − e ⊗ e .Inserting this into (5.1) and applying to the whole thing the map e ∗ ⊗ id : x ⊗ y 7→ h e , x i y , weobtain ρ ( α ∗ − ) η ( u ) = − e . However, examining what this means for the coefficients of the vector η ( u ) , taking also intoaccount that the products (1 − q ) . . . (1 − q k ) converge to a non-zero limit (see [27, TheoremA.4]), we would obtain k η ( u ) k = ∞ . Therefore, there is no such cocycle η . (cid:3) This leaves us with the question, which vectors in H may, actually, occur as values η ( u NN ) fora cocycle. First of all, there are many of them. More precisely, every element f in the (dense!)subspace π ( − u NN ) H of H may occur; and the (unique!) cocycles determined by them, are(exactly!) the coboundaries. Indeed, for g = − π ( − u NN ) f , by Lemma 2.39, we have (seeProposition 4.10 and around Lemma 2.16 for notation) lim p ↑ π ◦ ( id − ε ) (cid:2) − π ( − pu NN ) g (cid:3) = π ◦ ( id − ε ) f = ( πf ) ◦ ( id − ε ) =: η f , (5.2)pointwise on SU q ( N ) ; and η f ( u NN ) = π ( u NN − ) f = g . UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) In general, Proposition 4.10 tells us that, given an arbitrary cocycle η , putting f p := − π ( − pu NN ) η ( u NN ) , the cocycle η is the pointwise limit of the coboundaries η f p . In other words, for each cocyclethere is a sequence ( f m ) m ∈ N of elements in H such that the coboundaries ( η f m ) m ∈ N convergepointwise to η . Let us characterize better, what are the sequences ( f m ) m ∈ N that make thathappen.As in the proof of Proposition 4.10, it is, clearly, enough to check convergence of ( η f m ) m ∈ N onthe generators u jk . However, we can do better. Proposition . The sequence of coboundaries ( η f m ) m ∈ N converges (pointwise on SU q ( N ) ) if(and, of course, only if ) it converges on all u jj (1 ≤ j ≤ N ) .Proof. We have to show that, under the stated condition, (cid:0) η f m ( u jk ) (cid:1) m ∈ N converges for all j = k .Put l := max( j, k ) . Then, by Relation (2.8a) or by Relation (2.8b), we get η f m ( u jk ) = π ( u jk ) f m = π ( − qu ll ) π ( − qu ll ) π ( u jk ) f m = − π ( − qu ll ) π ( u jk ) π ( u ll − ) f m = − π ( − qu ll ) π ( u jk ) η f m ( u ll ) , which converges, because (cid:0) η f m ( u ll ) (cid:1) m ∈ N converges. (cid:3) By introducing an adequate norm k • k π on H , we may characterize the suitable sequences f m as Cauchy sequences in that norm and the elements in the completion H π with respect to thatnorm uniquely parametrize the cocycles.Recall that for ≤ a ∈ B ( H ) , the function k • k a : f p h f, af i = k√ af k is a seminorm on H ; it is a norm if and only if a is injective. Choosing a := P Nj =1 π ( − u jj ) ∗ π ( − u jj ) to define k • k π := k • k a , settles our problem. Indeed, because a ≥ π ( − u NN ) ∗ π ( − u NN ) and π ( − u NN ) is injective, k • k π is a norm (and not only a seminorm). By construction, a Cauchy sequence ( f m ) m ∈ N in that norm leads to a pointwise convergent sequence of coboundaries ( η f m ) m ∈ N . Andsince for every cocycle η the coboundaries η f p approximate it, the sequence ( f − m ) m ∈ N is Cauchyin k • k π and does the same job. We collect: Proposition . Let π be a representation of SU q ( N ) on H such that π ( − u NN ) is injective.Denote by H π the completion of H in the the norm k • k π . Then:For each a ∈ K , the operator π ( a ) extends continuously to a (unique, bounded) operator a π : H π → H, f lim m →∞ af m for an arbitrary Cauchy sequence ( f m ) m ∈ N converging to f in H π . Moreover, the formula a π f = η ( a ) establishes a one-to-one correspondence between elements f in H π and cocycles η with respect to π .For each a ∈ K and each f ∈ H π , the element ( a π f ) ∗ in H ∗ extends continuously to a(unique, bounded) linear functional g lim m →∞ h a π f, g m i on H π for an arbitrary Cauchy sequence ( g m ) m ∈ N converging to g in H π . Denoting by a π ∗ f the unique element of H π induc-ing that linear functional, we define an operator a π ∗ : H π → H π , fulfilling also h g, a π ∗ f i π =lim m →∞ h g m , π ( a ) f m i . In particular, if the cocycle η is determined by f ∈ H π , then the linearfunctional a 7→ h f, a π ∗ f i π on K may be extended to a generating functional ψ with Schürmanntriple ( π, η, ψ ) . This result generalizes the situation of SU q (2) as described in [27, Section 4.5] or [28, Section4.3]. Corollary . Let π be a completely non-gaussian representation (that is, in the decomposi-tion according to Theorem 4.17, π = 0 ). Then k • k π , defined as before, is a norm and, defining H π as before, Proposition 5.3 remains true also for π .Proof. For each nonzero h = h + . . . + h N ∈ H , there is at least one n such that h n = 0 . Andsince π n ( − u nn ) is injective on H n , the seminorm k • k π is, indeed, a norm. Furthermore, since π n = ˜ π n ◦ s n,N lives on SU q ( n ) , we have k • k π n = k • k ˜ π n . Now, the result follows by applyingProposition 5.3 to each ˜ π n , separately. (cid:3) Putting this together with Theorem 4.17, with Theorem 3.7, and with the uniqueness discus-sion for Schürmann triples following Definition 2.3, we obtain the following improvement of theparametrization following Procedure 2.4: Theorem . We obtain every generating functional ψ on SU q ( N ) in the following way as ψ = ψ G + ψ L , where: (1) ψ G is gaussian. (2) ψ L is completely non-gaussian.In the decomposition of a given ψ as ψ G + ψ L , both ψ G and ψ L are maximal. They are uniqueunder the condition ψ L ◦ P = ψ L . (1) The possible gaussian parts are classified uniquely by positive real ( N − × ( N − -matrices (encoding uniquely the part ψ G ◦ P of ψ G ) and N − real numbers (encodingthe remaining drift term). (2) The possible completely non-gaussing parts (satisfying ψ L ◦ P = ψ L ) are classified(uniquely up to cyclic cocycle intertwining unitary equivalence) by completely non-gaus-sian ∗ -representations π on a Hilbert space and elements f ∈ H π (such that the cocycledetermined by f is cyclic) as ψ L ( a ) = h f, P ( a ) π ∗ f i π . If we wish, we may decompose π and H uniquely further as in Theorem 4.3 into ∗ -representations π n on H n (2 ≤ n ≤ N ) such that π ( − u nn ) is injective and f n ∈ H π n n with ψ L = ψ + . . . + ψ N , where ψ n = h f n , P ( a ) π ∗ n f n i (preserving analogue uniqueness statements). UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) The case of U q ( N ) Recall the definition of U q ( N ) from Subsection 2.6. And recall that we have the inclusions SU q ( N + 1) ⊃ U q ( N ) ⊃ SU q ( N ) mediated by the ∗ -homomorphisms t N : SU q ( N + 1) → U q ( N ) and ˘ t N − : U q ( N ) → SU q ( N ) ,respectively. In this short section we show the Lévy-Khintchine decomposition results for gen-erating functionals on U q ( N ) , by reduction to those for SU q ( N + 1) ⊃ U q ( N ) . This is veryrapid. The price to be paid for being rapid, is that, by this method, we do not get the fulldecomposition result for the completely non-gaussian part into irreductible parts the live on U q ( n ) , but only the weaker Proposition 6.4. Remark . One may repeat the whole procedure in Section 4 almost verbatim for U q ( N ) toshow that every generating functional ψ on U q ( N ) decomposes (in a suitable sense uniquely)into a sum ψ = ψ + . . . + ψ N , where ψ n lives on U q ( n ) for n = 1 , . . . , N and where ψ is thegaussian part. (Some care is in place at points where in Section 4 we did use the determinantcondition D = that characterizes SU q ( N ) .) We opted not to include details. Since SU q ( N + 1) ⊃ U q ( N ) ⊃ T N , by Remark 3.10, U q ( N ) has the same gaussian generatingfunctionals as SU q ( N + 1) and the images t N ( d ) = d , . . . , t N ( d N ) = d N , t N ( d N +1 ) = D − − D − ∗ i form a suitable family E for U q ( N ) , defining also a projection P compatible with that of SU q ( N + 1) .Now if π is a ∗ -representation of U q ( N − , then ˆ π := π ◦ t N − is a ∗ -representation of SU q ( N ) that lives on U q ( N − . Obviously, if a representation of SU q ( N ) that lives on U q ( N − decomposes into a direct sum, then each direct summand lives on U q ( N − , separately.Therefore, all results about representations of SU q ( N ) in Subsection 4.1, turn over to U q ( N − (including the classical U q (1) = U (1) for N = 2 ).If ˆ η is a cocycle with respect to that ˆ π , then, by Subsection 4.3, the completely non-gaussianpart ˆ η NG := ˆ η ⊕ . . . ⊕ ˆ η N , is a limit of coboundaries with respect to a representation ˆ π NG :=ˆ π ⊕ . . . ⊕ ˆ π N that lives on U q ( N − . (Indeed, if a direct sum of representations lives ona quantum subgroup, then so does each its components.) By Corollaries 2.22 and 2.25, also ˆ η NG lives on U q ( N − and admits a generating functional ˆ ψ NG (unique, if ˆ ψ NG ◦ P = ˆ ψ NG )completing the Schürmann triple (ˆ π NG , ˆ η NG , ˆ ψ NG ) that lives on U q ( N − , too. Therefore, if ˆ η lives on U q ( N − , then so does the gaussian part ˆ η G := ˆ η . And if (ˆ π, ˆ η, ˆ ψ ) is obtainedfrom a Schürmann triple ( π, η, ψ ) on U q ( N − by composition with t N − (and, therefore, is aSchürmann triple on SU q ( N ) ), then the gaussian part ˆ ψ G := ˆ ψ − ˆ ψ NG lives on U q ( N − , too.Except for some care about N ↔ N − , taking also into account the analogue of Corollary5.4, we obtain word by word the analogue of the first part of Theorem 5.5. Theorem . For N ≥ , we obtain every generating functional ψ on U q ( N ) in the followingway as ψ = ψ G + ψ L , where: (1) ψ G is gaussian. (2) ψ L is completely non-gaussian.In the decomposition of a given ψ as ψ G + ψ L , both ψ G and ψ L are maximal. They are uniqueunder the condition ψ L ◦ P = ψ L . (1) The possible gaussian parts are classified uniquely by positive real N × N -matrices (en-coding uniquely the part ψ G ◦ P of ψ G ) and N real numbers (encoding the remaining driftterm). (2) The possible completely non-gaussing parts (satisfying ψ L ◦ P = ψ L ) are classified(uniquely up to cyclic cocycle intertwining unitary equivalence) by completely non-gaus-sian ∗ -representations π on a Hilbert space and elements f ∈ H π (such that the cocycledetermined by f is cyclic) as ψ L ( a ) = h f, P ( a ) π ∗ f i π . Corollary . U q ( N ) does have property (NC) , hence, (LK) . It has property (GC) if and onlyif N = 1 . What about the decomposition of the non-gaussian part into components living on U q ( n ) ?(This is the part from Theorem 5.5 that is missing in Theorem 6.2.) Well, returning to thenotation in front of Theorem 6.2 but now immediately for U q ( N ) and no longer for U q ( N − ,we know that ˆ ψ NG decomposes into a sum over ˆ ψ n +1 (1 ≤ n ≤ N ) , where ˆ ψ n +1 lives on SU q ( n +1) and on U q ( N ) . But, does it live on U q ( n ) ⊂ SU q ( n + 1) ? We have ˆ ψ n +1 = ψ n ◦ t N and we have,making also use of (2.12), (2.13), and (2.14), ˆ ψ n +1 = ˜ ψ n +1 ◦ s n +1 ,N +1 = ˜ ψ n +1 ◦ ˘ t n +1 ◦ ˘ s n +1 ,N ◦ t N . Therefore, by surjectivity of t N , the functional ψ n on U q ( N ) fulfills ψ n = ( ˜ ψ n +1 ◦ ˘ t n +1 ) ◦ ˘ s n +1 ,N and lives on U q ( n + 1) via the functional ˜ ψ n +1 ◦ ˘ t n +1 on U q ( n + 1) which, in turn, lives on SU q ( n + 1) .Every such ˜ ψ n +1 may occur; therefore, better than this is not possible as long as we work byreduction to the results for SU q ( N + 1) : Proposition . The completely non-gaussian part ψ NG of a generating functional on U q ( N )( N ≥ decomposes uniquely as ψ + . . . + ψ N +1 , where each ψ n lives on SU q ( n ) , hence, on U q ( n − , and where as functionals on SU q ( n ) , the ψ n are irreductible. UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) Final remarks and open problems We close by pointing out that several interesting open problems related to our decompositionresult.(1) Theorem 5.5 parametrizes all possible generating functionals on SU q ( N ) , but it is possiblethat two sets of parameters lead to the same generating functional. When can thishappen? What is the fundamental domain of this equivalence relation? That is, howto choose and characterize one representative for any equivalence class? The answer tothis question will establish the one-to-one correspondence between Lévy processes andthe parameters.(2) We assume in the paper that q ∈ (0 , , but the same holds for | q | < , q = 0 . The(classical) limit case q = 1 is known. It would be of great interest to see what happensfor q = − . The treatment of SU − (2) , that is, the anti-classical limit, was done in [29].(3) It would be interesting to generalize our results to q-deformations of other simple compactLie groups. Descriptions of the CQG-algebras of the compact quantum groups O q ( N ) , Sp q ( N ) , and SO q ( N ) can be found in Chapter 9 of [17].(4) As shown in [6], generating functionals on a given compact quantum group which satisfyadditional symmetry properties (KMS-symmetry) can be used to define Dirichlet forms,Laplace operators and Dirac operators, and so they can carry geometric informationabout the quantum group. A description of all generating functionals satisfying addi-tional properties (KMS-symmetric, central, etc.) will allow to construct “nice” Dirichletforms, Laplace operators and Dirac operators that reflect well the structure of the un-derlying quantum group.(5) We have decomposed generating functionals on SU q ( N ) into the sum of a gaussian partand of a completely non-gaussian part (Lévy-Khintchine decomposition), and we havedecomposed the completely non-gaussian part further into into a sum of generatingfunctionals that live on the quantum subgroups SU q ( n ) (2 ≤ n ≤ N ) and, there, areirreductible. As soon as we actually have examples of Lévy processes on SU q ( N ) thathave gaussian or irreductible generating functionals, we may ask how to “compose” themto get a Lèvy processs for the sum of the generators.Of course, this question makes sense independently of the special nature of the gen-erating functionals: Given two (or more) Lévy processes on a quantum semigroup withgenerating functionals ψ i , how to construct out of them a Lévy process that has thesum over the ψ i as generating functional? Answer: Plugging in the direct sum of theSchürmann triples of the ψ i into the construction in Schürmann, Skeide, and Volkwardt[25]. (See [25, Example 3.13] for details.) The result is that a certain convolution Trotterproduct of the individual Lévy processes gives a Lévy process that has as generatingfunctional the sum of the individual generating functionals. References [1] E. Abe. Hopf algebras . Cambridge University Press, 1980.[2] L. Accardi, U. Franz, and M. Skeide. Renormalized squares of white noise and other non-gaußian noises asLévy processes on real Lie algebras. Commun. Math. Phys. , 228:123–150, 2002. (Rome, Volterra-Preprint2000/0423).[3] D. Applebaum, Lévy Processes and Stochastic Calculus , Cambridge University Press, Cambridge 2004,pp.384.[4] W. Arveson. Continuous analogues of Fock space . Number 409 in Mem. Amer. Math. Soc. American Math-ematical Society, 1989.[5] B.V.R. Bhat and M. Skeide. Tensor product systems of Hilbert modules and dilations of completely positivesemigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 3:519–575, 2000. (Rome, Volterra-Preprint1999/0370).[6] F. Cipriani, U. Franz, A. Kula; Symmetries of Lévy processes on compact quantum groups, their Markovsemigroups and potential theory. Journal of Functional Analysis 266 (2014), 2789-2844.[7] B. Das, U. Franz, A. Kula, A. Skalski; One-to-one correspondence between generating functionals andcocycles on quantum groups in presence of symmetry. Mathematische Zeitschrift 21, Issue 3 (2015), pp.949-965.[8] B. Das, U. Franz, A. Kula, A. Skalski; Lévy–Khintchine decompositions for generating functionals onalgebras associated to universal compact quantum groups. Infin. Dimens. Anal. Quantum Probab. Relat.Top. 281, No. 3, Article Id 1850017 (2018).[9] S. Dascalescu, C. Nastasescu, S. Raianu, Hopf algebras. An introduction. Pure and Applied Mathematics,Marcel Dekker. 235. New York, NY (2001).[10] Dijkhuizen M.S. and T.H. Koornwinder. CQG algebras: A direct algebraic approach to compact quantumgroups. Lett. Math. Phys. , 32:315–330, 1994.[11] U. Franz, Lévy processes on quantum groups and dual groups. In: Quantum independent increment pro-cesses. II , 161-257, Lecture Notes in Math., 1866, Springer, Berlin, 2006.[12] U. Franz, Independence and Lévy processes in quantum probability. In: U. Franz, A. Skalski, Noncommu-tative Mathematics for Quantum Systems, Cambridge University Press, 2016.[13] U. Franz, M. Gerhold, A. 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A first course in functional analysis . CRC Press, 2017. UNT’S FORMULA FOR SU q ( N ) AND U q ( N ) [27] M. Skeide, The Lévy-Khintchine formula for the quantum group SU q (2) . PhD thesis, Heidelberg, 1994.Available at http://web.unimol.it/skeide/ .[28] M. Skeide, Hunt’s formula for SU q (2) ; a unified view. Information Dynamics & Open Systems , 6:1–27, 1999.(Rome, Volterra-Preprint 1995/0232).[29] M. Skeide, Infinitesimal generators on SU q (2) in the classical and anti-classical limit. Information Dynamics& Open Systems , 6:375–414, 1999. (Cottbus, Reihe Mathematik 1997/M-19).[30] M. Skeide. Lévy processes and tensor product systems of Hilbert modules. In M. Schürmann and U. Franz,editors, Quantum Probability and Infinite Dimensional Analysis — From Foundations to Applications , num-ber XVIII in Quantum Probability and White Noise Analysis, pages 492–503. World Scientific, 2005.[31] T. Timmermann, An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitariesand beyond. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. xx+407pp.[32] S.L. Woronowicz. Compact matrix pseudogroups. Commun. Math. Phys. , 111:613–665, 1987.[33] S.L. Woronowicz. Twisted SU (2) group. An example of a non-commutative differential calculus. Publ. Res.Inst. Math. Sci. , 23:117–181, 1987.[34] S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent.Math. 93, No. 1, 35-76 (1988).[35] S.L. Woronowicz, Compact quantum groups . Symétries quantiques (Les Houches, 1995), 845-884, North-Holland, Amsterdam, 1998. Acknowledgments: We gratefully acknowledge the MFO in Oberwolfach for fantastic twoweeks during a Research in Paris in February 2014, where this work started.U.F. was supported by the French ‘Investissements d’Avenir’ program, 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 frame of POLONIUMprogram PPN/BIL/2018/1/00197/U/00021.Uwe Franz: Laboratoire de mathématiques de Besançon, University of Bourgogne Franche-Comté, France ,E-mail: [email protected] ,Homepage: http://lmb.univ-fcomte.fr/uwe-franz Anna Kula: Institute of Mathematics, University of Wroclaw, Poland ,E-mail: [email protected] ,Homepage: J. Martin Lindsay: Department of Mathematics and Statistics, Lancaster University, UK ,E-mail: [email protected] Michael Skeide: Dipartimento di Economia, Università degli Studi del Molise, Via de Sanctis,86100 Campobasso, Italy ,E-mail: [email protected] ,Homepage:,Homepage: