# Hunt's Formula for $SU_q(N)$ and $U_q(N)$

Uwe Franz, Anna Kula, J. Martin Lindsay, Michael Skeide

► To cite this version:

Uwe Franz, Anna Kula, J. Martin Lindsay, Michael Skeide. Hunt's Formula for $SU_q(N)$ and $U_q(N)$. Indiana University Mathematics Journal, 2023, 72 (4), pp.1717-1748. 10.1512/iumj.2023.72.9485. hal-03116703

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# HUNT'S FORMULA FOR $SU_q(N)$ AND $U_q(N)$

UWE FRANZ, ANNA KULA, J. MARTIN LINDSAY, AND MICHAEL SKEIDE

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

## 1. INTRODUCTION

Up to stochastic equivalence, a Lévy process with values in a locally compact Lie group $G$ is determined by its generating functional. This is a (densely defined) linear functional $\gamma$ on $C_0(G)$, the $C^*$-algebra of continuous complex-valued functions on $G$ which vanish at infinity, whose domain may be thought of as consisting of those functions that have a second order Taylor expansion around the identity element of the group. Hunt's formula ([11]) is a generalization and extension of the Lévy-Khintchine formula ([1], [18]). It is equivalent to the assertion that

$$ \gamma = \gamma_D + \gamma_G + \gamma_L \quad \text{where} \quad \gamma_L = L \circ P \quad \text{and} \quad L(f) = \int_{G \setminus \{e\}} f(s) \Pi(ds) \quad (1.1) $$

for the identity element $e$ of $G$, in which $P$ is a hermitian projection that kills the linear terms, the drift $\gamma_D$ and $P$-invariant gaussian part $\gamma_G$ are linear combinations of first and second order derivatives evaluated at $e$ respectively, and $\Pi$ is the so-called Lévy measure. The Lévy functional $L$ is defined on the space of functions that, together with their first derivatives, vanish at $e$. The integral may be viewed as a mixture of point evaluations, moreover functionals of the form $f \mapsto f(s) - f(e)$, for fixed $s \ne e$, generate jump processes. The functional $\gamma_L$ is also referred to as the jump part; in the case where $G = \mathbb{R}$ and $\Pi$ is finite, it generates a compound Poisson process. The decomposition depends on the non-canonical projection $P$ chosen; its role is to deal with any singularity of the measure $\Pi$ at $e$.

If $G$ is compact, Tannaka-Krein duality ([10, Section VII.30]) asserts that the representative algebra $R(G)$, generated by matrix coefficients of finite-dimensional representations of $G$, is a norm-dense $*$-subalgebra of the unital $C^*$-algebra $C(G)$. In fact, $R(G)$ is a commutative Hopf $*$-algebra from which the topological group $G$ may be fully recovered ([16]). A compact quantum group in the sense of Woronowicz ([29]) is a unital $C^*$-algebra-with-coproduct which enjoys density relations corresponding to the group cancellation law and contains a dense Hopf $*$-algebra, the CQG algebra of the quantum group, whose role corresponds to that played by $R(G)$ for a compact group $G$ ([4]). Schürmann's theory of quantum Lévy processes on $*$-bialgebras ([20]) thereby applies. As with their classical counterparts, but now up to quantum stochastic equivalence, Lévy processes on $*$-bialgebras are classified by their generating functional, now

<footnote n="2020">2020 Mathematics Subject Classification. 46L53, 60G51.</footnote>

<footnote>Key words and phrases. Quantum Lévy process, convolution semigroup, compact quantum group, CQG algebra, generating functional, Schürmann triple.</footnote>
a hermitian linear functional on the CQG algebra which is conditionally positive and vanishes at the identity element. The problem of finding a decomposition of generating functionals corresponding to (1.1) is expressible in cohomological terms. Of course meaning has to be given to drift, gaussian and jump parts in the quantum generalisation. Our Hunt formula includes an explicit description of the drifts and gaussian generating functionals and the specification of an approximation property that justifies calling the remainder a jump part (Proposition 2.8).

For some compact quantum groups every generating functional has such a decomposition but for others that is not so ([8], [2]). A Hunt formula for Woronowicz's $SU_q(2)$ ([26], [27]) was obtained in [23], [21]. This led to a short proof of the classical Hunt formula for compact Lie groups ([24]). Here we tackle the case of $SU_q(N)$, obtaining a unique decomposition $\gamma = \gamma_D + \gamma_G + \gamma_{NG}$ where $\gamma_{NG} = \gamma_2 \circ P + \cdots + \gamma_N \circ P$, in which $P$ is a hermitian projection analogous to that of (1.1), $\gamma_D$ is a drift, $\gamma_G$ is a $P$-invariant gaussian generating functional and, for $2 \le n \le N$, $\gamma_n$ is an extension to $SU_q(N)$ of a completely non-gaussian generating functional on $SU_q(n)$ which enjoys an irreducibility property. We also display the essentially classical structure of $\gamma_D$ and $\gamma_G$, and show $\gamma_{NG}$ to be the limit of functionals of the form $\omega_{\xi(t)} \circ \pi \circ P$ for a representation $\pi$ and net of vector functionals $(\omega_{\xi(t)})$ (Theorem 4.15). The case of general $N$ turns out to be more involved than the case $N=2$, and some results concerning $SU_q(2)$ fail for $N \ge 3$. For instance, for $N \ge 3$ the cohomological problem is not always solvable in the gaussian case (Corollary 2.13). Also, for $N=2$ the completely non-gaussian generating functionals may be parametrized by the vectors in its associated representation Hilbert space, whereas for $N \ge 3$ the situation is more subtle (Section 5).

The paper is organized as follows. Terminology and notations concerning the CQG algebra of a compact quantum group are set out below. Section 2 contains the basic definitions and preliminary results. The CQG algebras of the compact quantum groups $SU_q(N)$ and $U_q(N)$ are here respectively denoted $SU_q(N)$ and $U_q(N)$; the former is algebraically generated by a matrix of elements $[u_{jk}]_{j,k=1}^N$ (see Relations (2.6), et seq.). In Section 3 we deal with our choice of projection $P$, with respect to which we show that the gaussian generating functionals on $SU_q(N)$ are classified by a real $(N-1)$-vector and positive-definite real $(N-1) \times (N-1)$ matrix representing the drift and $P$-invariant diffusion-type second order term (Theorem 3.6). Unlike in lower dimensions, for $N \ge 3$ there are cocycles of gaussian representations which have no associated generating functionals (Theorem 3.3). Every gaussian generating functional is induced from a gaussian generating functional that lives on the classical undeformed subgroup $\mathbb{T}^{N-1}$ of $SU_q(N)$, in the sense of Definition 2.21 (see Remark 3.7). In Section 4 we show that every representation $\pi$ of $SU_q(N)$ has a unique full (representation) decomposition $\pi_1 \oplus \cdots \oplus \pi_N$, where $\pi_1$ is its so-called gaussian part and, for $2 \le n \le N$, $\pi_n$ lives on $SU_q(n)$ and $\pi_n(1-u_{nn})$ is injective. Completely non-gaussian cocycles $\eta$ are approximated by coboundaries and determined by their values $\eta(u_{nn})$ ($2 \le n \le N$). From this we deduce a full (generating functional) decomposition $\gamma = \gamma_1 + \cdots + \gamma_N$ for generating functionals, uniquely determined by the projection $P$, and conclude with our Hunt formula (Theorem 4.15). In Section 5 we show that, unlike in the case $N=2$, if $N > 2$ then the values of $\eta(u_{NN})$ for cocycles $\eta$ of representations $\pi$ for which $\pi(1-u_{NN})$ is injective, may not exhaust the representation space. We then indicate a completion process which yields a quasi-innerness property, and thereby full parameterisation, for completely non-gaussian cocycles. In Section 6 we briefly treat the quantum groups $U_q(N)$.

Our work suggests the investigation of Hunt formulae for other $q$-deformed compact Lie groups ([16]).
**Compact quantum groups and CQG algebras.** A CQG algebra ([4]), or algebraic compact quantum group, is a Hopf $*-algebra $G$ that is linearly spanned by the coefficients of its finite-dimensional unitary corepresentations or, equivalently, has a faithful Haar state. Thus a CQG algebra is a unital $*-algebra $G$, with unital $*-algebra morphisms $\Delta: G \to G \otimes G$ and $\varepsilon: G \to C$, linear map $\kappa: G \to G$ and unital linear functional $h: G \to C$, called respectively the coproduct, counit, coinverse or antipode, and Haar state, enjoying the coassociativity, counital, coinverse, invariance and positivity relations

$$ (\Delta \otimes \mathrm{id}) \circ \Delta = (\mathrm{id} \otimes \Delta) \circ \Delta; \quad (\varepsilon \otimes \mathrm{id}) \circ \Delta = \mathrm{id} = (\mathrm{id} \otimes \varepsilon) \circ \Delta; $$

$$ \mu \circ (\mathrm{id} \otimes \kappa) \circ \Delta = \iota \circ \varepsilon = \mu \circ (\kappa \otimes \mathrm{id}) \circ \Delta; \quad (\mathrm{id} \otimes h) \circ \Delta = \iota \circ h = (h \otimes \mathrm{id}) \circ \Delta; $$

and $h(a^*a) > 0$ for $a \neq 0$. Here $\mu: G \otimes G \to G$ denotes the linearisation of the algebra product, and $\iota$ the unital linear map $C \to G$. The coinverse $\kappa$ is uniquely determined by the bialgebra structure and any $*-bialgebra$ morphism between CQG algebras respects coinverses and so is a CQG algebra morphism ([3, Remarks 4.2.3 and 4.2.5]); the Haar state $h$ is also unique ([4, Proposition 3.2]). Compact quantum groups may also be viewed from the equivalent $C^*$-algebraic perspective, as was originally done by Woronowicz ([29]). The canonical (universal and reduced) Woronowicz algebras of a compact quantum group $G$ are commonly denoted $C_u(G)$ and $C_r(G)$, and its CQG algebra is here denoted by $R(G)$ in a further nod to their classical counterparts. The quantum space $G$ itself is only manifested through one of its realisations. For more on this, we recommend [16], [12, Section 11.3], and [25, Section 5.4]. For the purposes of this work, it suffices to operate exclusively within CQG algebras. In fact, in our analysis we need explicit recourse to none of the coproduct, coinverse or Haar state.

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

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

## 2. PRELIMINARIES

**Generating functionals of quantum Lévy processes and Schürmann triples.** Let $G$ be a CQG algebra. A *Lévy process on $G$* is a family of $*-algebra$ morphisms from $G$ to a noncommutative probability space enjoying certain properties which encode the stationarity and independence of increments (see [20], [5] and [15, Chapter VII], or the survey [6]).

**DEFINITION 2.1.** A *generating functional* for a quantum Lévy process on $G$ is a linear functional $\gamma$ on $G$ which is *hermitian*: $\gamma = \gamma^\dagger: a \mapsto \overline{\gamma(a^*)}$, *normalised*: $\gamma(1) = 0$, and *conditionally positive*: $\gamma(c^*c) \ge 0$ for all $c \in \ker \varepsilon$.

Quantum Lévy processes are determined up to quantum stochastic equivalence by their generating functionals, and may be reconstructed from their generating functional using quantum stochastic calculus on a symmetric Fock space ([20, Theorem 2.3.5], [14, Theorem 7.1]), or using Trotter products and Arveson (product) systems ([22]).
**DEFINITION 2.2** ([20]). A *Schürmann triple* on $\mathcal{G}$ is an ordered triple $(\pi, \eta, \gamma)$ consisting of a representation $\pi$ of $\mathcal{G}$, a $\pi$-$\varepsilon$-cocycle, or $\pi$-$\varepsilon$-derivation, that is, a linear mapping $\eta : \mathcal{G} \to \mathfrak{h}^{\pi}$ satisfying

$$ \eta(ab) = \pi(a)\eta(b) + \eta(a)\varepsilon(b) \quad (a, b \in \mathcal{G}), \qquad (2.1) $$

and a linear functional $\gamma$ on $\mathcal{G}$ satisfying

$$ \gamma^{\dagger} = \gamma, \quad \gamma(1) = 0 \quad \text{and} \quad \gamma(c^{*}c) = \|\eta(c)\|^{2} \quad (c \in \ker \varepsilon), \qquad (2.2) $$

equivalently, $\gamma^{\dagger} = \gamma$ and $\langle\eta(a), \eta(b)\rangle = \gamma(a^{*}b) - \overline{\gamma(a)}\varepsilon(b) - \overline{\varepsilon(a)}\gamma(b)$ for $a, b \in \mathcal{G}$.

A linear functional $\gamma$ on $\mathcal{G}$ completes a $\pi$-$\varepsilon$-cocycle $\eta$ if $(\pi, \eta, \gamma)$ is a Schürmann triple; we then say that $\eta$, or $(\pi, \eta)$, is *completable*.

A Schürmann triple $(\pi, \eta, \gamma)$ or cocycle $\eta$, is called *cyclic* if $\overline{\eta(\mathcal{G})} = \mathfrak{h}^{\pi}$.

The third component of a Schürmann triple is a generating functional. Conversely, for any generating functional $\gamma$, there is a cyclic Schürmann triple with $\gamma$ as its third component. If $(\pi, \eta, \gamma)$ is a cyclic Schürmann triple then, for any linear isometry $V$ from $\mathfrak{h}^{\pi}$ into a Hilbert space, $(V\pi(\cdot)V^{*}, V\eta(\cdot), \gamma)$ is a Schürmann triple (cyclic if and only if $V$ is unitary), and every Schürmann triple having $\gamma$ as its third component is of this form. Thus all cyclic Schürmann triples having $\gamma$ as their third component are unitarily equivalent — we refer to any one of these as $\gamma$'s (*associated*) *Schürmann triple* ([20, Section 2.3]).

For $K := \ker \varepsilon$, set

$$ K_n := \operatorname{span}\{c_1 \cdots c_n : c_1, \cdots, c_n \in K\} \text{ for } n \ge 1, \text{ and } K_{\infty} := \bigcap_{n \ge 1} K_n. $$

Thus $(K_n)$ is a sequence of $^*$-ideals of $\mathcal{G}$ decreasing to $K_{\infty}$. Also set

$$ P_2(\mathcal{G}) := \{P \in L(\mathcal{G}) : P \text{ is a hermitian projection, ran } P = K_2 \text{ and } 1 \in \ker P\}, $$

where *hermitian* means $P = P^{\dagger} : a \mapsto (Pa^{*})^{*}$ for $a \in \mathcal{G}$.

**DEFINITION 2.3.** Let $\gamma$ be a generating functional on $\mathcal{G}$. Then $\gamma$ is a *drift* if $\gamma|_{K_2} = 0$, equivalently, in terms of its associated Schürmann triple $(\pi, \eta, \gamma)$, if $\mathfrak{h}^{\pi} = \{0\}$.

For $P \in P_2(\mathcal{G})$, we denote the drift $\gamma - \gamma \circ P$ by $\gamma_D^P$, and call $\gamma$ *P-invariant* if $\gamma \circ P = \gamma$.

**REMARKS 2.4.** The drifts form a real subspace of the linear dual of $\mathcal{G}$. Any $P \in P_2(\mathcal{G})$ determines a unique resolution for generating functionals $\gamma$ into a drift component plus a $P$-invariant one: $\gamma = \gamma_D^P + \gamma \circ P$—in this sense $P$-invariance may usefully be thought of as a $P$-driftless property (i.e. having zero drift component with respect to $P$). If a cocycle $\eta$ is completable then, for any particular generating functional $\gamma$ which completes $\eta$, the set of all generating functionals which do so equals $\{\gamma + \gamma' : \gamma' \text{ is a drift}\}$ and the unique $P$-invariant one is $\gamma \circ P = \gamma - \gamma_D^P$.

The $P$-invariant generating functionals on $\mathcal{G}$ are the maps of the form $\psi \circ P$ for a linear functional $\psi$ on $K_2$ which is nonnegative: $\psi(c^{*}c) \ge 0$ for all $c \in K$ (and thus also hermitian).

There is no canonical choice of projection from $P_2(\mathcal{G})$. By contrast, since $\mathbb{C}1$ and $K$ are complementary subspaces of $\mathcal{G}$, there is a unique projection in $L(\mathcal{G})$ with range $K$ and $1$ in its kernel—namely (id $- \iota \circ \varepsilon : a \mapsto a - \varepsilon(a)1$), moreover it is hermitian and compatible with the projections in $P_2(\mathcal{G})$.

**DEFINITION 2.5.** Let $U$ be a subspace of a complex vector space $V$. A linearly independent subset $E$ of $V \setminus U$ is a *basis extension from* $U$ to $V$ if its linear span is a complementary subspace of $U$. In case $V$ is involutive, a basis extension is *hermitian* if it consists of selfadjoint elements.
For any hermitian basis extension E from K₂ to K, the functionals (ε'₉)d₉∈E on G given by

$$
\epsilon'_{d}(\lambda 1 + k_2 + \sum_{e \in E} \lambda_e e) = \lambda_d, \quad \text{for } \lambda \in \mathbb{C}, k_2 \in K_2 \text{ and } \{\lambda_e : e \in E\} \subset \mathbb{C}, \qquad (2.3)
$$

form a basis for the real space of drifts on $\mathcal{G}$, and

$$
P^E := \mathrm{id} - \iota \circ \varepsilon - \sum_{d \in E} d \varepsilon'_d(\cdot) \in P_2(\mathcal{G}) \quad (2.4)
$$

equals the projection onto $K_2$ along $\operatorname{span}(\{1\} \cup E) = \mathbb{C}1 \oplus \operatorname{span}E$. The resulting map $E \mapsto P^E$
is surjective and $P^{E_1} = P^{E_2}$ if and only if $\operatorname{span}E_1 = \operatorname{span}E_2$.

PROCEDURE 2.6. For obtaining all generating functionals on $\mathcal{G}$, one needs to identify:

(1) the representations $\pi$ of $\mathcal{G}$;

(2) for each representation $\pi$, the $\pi$-$\varepsilon$-cocycles $\eta$;

(3) for each such cocycle $\eta$, the generating functionals $\gamma$ which complete it.

In the cases of the quantum groups SU<sub>q</sub>(N) and U<sub>q</sub>(N) the representation theory is known
([13]). Step (2) is a cohomological problem, as π-ε-cocycles form the first Hochschild cohomology
group H<sup>1</sup>(G, πh<sub>e</sub>) for h = h<sup>π</sup>, and this may usually be computed in a straightforward way. The
main problem lies in Step (3). The basic constraint on a given cocycle η, for it to be completable,
is that ||η(c)|| must equal ||η(d)|| whenever c, d ∈ K satisfy c*c = d*d; the task then amounts to
solving ψ(c*c) = ||η(c)||<sup>2</sup> (c ∈ K) for a linear functional ψ on K<sub>2</sub> since then, for any P ∈ P<sub>2</sub>(G),
the prescription a ↦→ ψ(Pa) defines a (P-invariant) generating functional which completes η.

**Approximately inner cocycles.** As just described, the problem of classifying generating functionals on $\mathcal{G}$ lies in the fact that there might be none which completes a given cocycle. In this section we identify a situation where such a completion does exist.

**DEFINITION 2.7.** A $\pi$-$\varepsilon$-cocycle is a *coboundary*, or *inner derivation*, if it is of the form

$$
\eta_{\pi, \xi} := (\pi - \iota \circ \varepsilon)(\cdot)\xi : a \mapsto \pi(a)\xi - \xi \varepsilon(a)
$$

for some vector $\xi$ in $\mathfrak{h}^{\pi}$, and is *approximately inner* if it is a pointwise limit of coboundaries
($\eta_{\pi, \xi(\lambda)}$) for some net ($\xi(\lambda)$) in $\mathfrak{h}^{\pi}$.

For a vector $\xi$ of a Hilbert space $\mathfrak{h}$, $\omega_{\xi}$ denotes the vector functional $T \mapsto \langle \xi, T\xi \rangle$ on $B(\mathfrak{h})$.
The following result is heavily used in Section 4.

**PROPOSITION 2.8.** *Approximately inner cocycles are completable.* Specifically, let $P \in P_2(\mathcal{G})$, let $\pi$ be a representation of $\mathcal{G}$, and let $(\xi(\lambda))$ be a net in $\mathfrak{h}^\pi$ such that $(\eta_\lambda := \eta_{\pi, \xi(\lambda)})$ converges pointwise to a map $\eta$. Then $\eta$ is a $\pi$-$\varepsilon$-cocycle and the net $(\gamma_\lambda := \omega_{\xi(\lambda)} \circ \pi \circ P)$ converges pointwise to a $P$-invariant generating functional $\gamma$ which completes $\eta$.

Proof. For each $\lambda$, the $P$-invariant linear functional $\gamma_\lambda$ is hermitian and $(\pi, \eta_\lambda, \gamma_\lambda)$ is easily seen to satisfy (2.1) and (2.2). Therefore, since $\eta$ is evidently a $\pi$-$\varepsilon$-cocycle and $K_2$ is both the range of $P$ and the linear span of the set $\{c^*c : c \in K\}$, the proposition follows from the fact that $\gamma_\lambda(c^*c) = \|\pi(c)\xi(\lambda)\|^2 = \|\eta_{\pi,\xi(\lambda)}(c)\|^2 \to \|\eta(c)\|^2$ for each $c \in K$. $\square$

In the classical setting of (1.1) we see that the generating functional $\gamma_L$ is expressible as the
limit of the functionals $\omega_{1_{G\setminus U}} \circ \pi \circ P$, as the neighbourhoods $U$ of $e$ shrink to $\{e\}$, $\pi$ being the
multiplication representation of $R(G)$ on $L^2(G, \Pi)$ and 1 here denoting indicator function.
**Gaussian generating functionals, cocycles and representations.**

**DEFINITION 2.9.** A generating functional $\gamma$, cocycle $\eta$, or representation $\pi$ is called gaussian if it vanishes respectively on $K_3$, $K_2$, or $K$.

For components of a Schürmann triple, these are equivalent ([20, Proposition 5.1.1]). A representation $\pi$ is gaussian if and only if $\pi = \iota_{h^\pi} \circ \varepsilon$, where $\iota_{h^\pi}$ denotes the unital linear map from $\mathbb{C}$ to $B(h^\pi)$.

**PROPOSITION 2.10.** Let $E$ be a hermitian basis extension from $K_2$ to $K$. Then, for any Hilbert space $h$, the $h$-valued gaussian cocycles on $\mathcal{G}$ are precisely the maps of the form $\sum_{d \in E} \xi_d \varepsilon'_d(\cdot)$ for a family of vectors $(\xi_d)_{d \in E}$ in $h$, where the functionals $\varepsilon'_d$ are as in (2.3).

*Proof.* Since gaussian cocycles vanish on 1 and on $K_2$, this follows from the fact that elements of $\mathcal{G}$ are uniquely expressible as $\varepsilon(a)1 + k_2(a) + \sum_{d \in E} \varepsilon'_d(a)d$ for some $k_2(a) \in K_2$. $\square$

It would be desirable to have a similarly concise description of gaussian generating functionals. For now we note that in general not all gaussian cocycles $\eta$ admit a gaussian generating functional.

**DEFINITION 2.11.** A cocycle $\eta$ on $\mathcal{G}$ is hermitian if it satisfies $\|\eta(c)\| = \|\eta(c^*)\|$ for all $c \in K$.

A gaussian cocycle of the form $\eta = \sum_{d \in E} \xi_d \varepsilon'_d$ is hermitian if and only if the Gram matrix $[\langle \xi_d, \xi'_d \rangle]$ is real (and therefore symmetric). Proposition 2.10 has the following consequence.

**COROLLARY 2.12.** $\mathcal{G}$ has non-hermitian gaussian cocycles if and only if $\dim K/K_2 \ge 2$.

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

**COROLLARY 2.13.** Suppose that $c^*c - cc^* \in K_3$ for all $c \in K$. Then a gaussian cocycle is completable if and only if it is hermitian.

*Proof.* It is necessity that is to be proved, so assume that $\gamma$ is a generating functional completing a gaussian cocycle $\eta$. Then $\|\eta(c)\|^2 - \|\eta(c^*)\|^2 = \gamma(c^*c - cc^*) = 0$ for all $c \in K$, as required. $\square$

**Complete non-gaussianness and Lévy-Khintchine decomposition.** We next collect basic facts about when a generating functional can have a Lévy-Khintchine decomposition.

**LEMMA 2.14.** Let $\pi_1 \oplus \pi_2$ be a decomposition of a representation $\pi$ of $\mathcal{G}$, let $V_i$ denote the inclusion map $h^{\pi_i} \to h^\pi$ for $i = 1, 2$, and let $\eta$ be a $\pi$-$\varepsilon$-cocycle. Then the following hold

(a) $\eta_i := V_i^*\eta(\cdot)$ is a $\pi_i$-$\varepsilon$-cocycle for $i = 1, 2$.

(b) If two of the three cocylces $\eta, \eta_1$ and $\eta_2$ are completable then so is the third.

It is quite possible that $\eta$ is completable, but $\eta_1$ and $\eta_2$ are not.

**DEFINITION 2.15.** For a representation $\pi$ of $\mathcal{G}$, set

$$h^{\pi_G} := \bigcap_{c \in K} \ker \pi(c) \quad \text{and} \quad h^{\pi_R} := (h^{\pi_G})^{\perp}.$$

Then $\pi$ is completely non-gaussian if $h^{\pi_G} = \{0\}$, equivalently, if $h^{\pi_R} = h^\pi$.

We also call a $\pi$-$\varepsilon$-cocycle $\eta$ completely non-gaussian if $\pi$ is, and a generating functional $\gamma$ completely non-gaussian if the representation component of its Schürmann triple is.
The above definition and its notations are amply justified by the following straightforward proposition.

**PROPOSITION 2.16 ([20]).** Let $\pi$ be a representation of $\mathcal{G}$. Then $h^{\pi_G}$ and $h^{\pi_R}$ are invariant subspaces and, denoting the resulting decomposition of $\pi$ as $\pi_G \oplus \pi_R$, $\pi_G$ is gaussian and $\pi_R$ is completely non-gaussian. Moreover, $h^{(\pi_R)_G} = \{0\} = h^{(\pi_G)_R}$.

If $\eta = \eta_G \oplus \eta_R$ is the corresponding decomposition of a $\pi$-$\varepsilon$-cocycle $\eta$ then $\eta_G$ is gaussian, and if $\eta$ is cyclic then $\eta_G$ and $\eta_R$ are cyclic too.

Generating functionals of the form $\omega_\xi \circ \pi \circ P$, and their limits as in Proposition 2.8, are completely non-gaussian.

**DEFINITION 2.17.** A Lévy-Khintchine decomposition for a generating functional $\gamma$ with Schürmann triple $(\pi, \eta, \gamma)$ is a decomposition $\gamma = \gamma_1 + \gamma_2$ for which $(\pi_G, \eta_G, \gamma_1)$ and $(\pi_R, \eta_R, \gamma_2)$ are Schürmann triples (equivalently, by Lemma 2.14, one of them is).

**REMARK 2.18.** With respect to a fixed projection $P \in P_2(\mathcal{G})$, if $\gamma$ has such a Lévy-Khintchine decomposition then it has a unique one in which $\gamma_1 = \gamma_D^P + \gamma_G$, $\gamma_2 = \gamma_R$, and the generating functionals $\gamma_G$ and $\gamma_R$ are $P$-invariant.

**DEFINITION 2.19.** A CQG algebra, or its associated quantum group, is said to have *property*

- (AC) if each cocycle $\eta$ is completable.

- (GC) if each gaussian cocycle $\eta$ is completable.

- (NC) if each completely non-gaussian cocycle $\eta$ is completable.

- (NAI) if each completely non-gaussian cocycle $\eta$ is approximately inner.

- (LK) if every generating functional admits a Lévy-Khintchine decomposition.

Evidently (AC) implies both (GC) and (NC), and either of these implies (LK); none of the reverse implications hold ([8]). The following is an immediate consequence of Proposition 2.8.

**PROPOSITION 2.20. (NAI) implies (NC), and thus (LK).**

**Schürmann triples on quantum subgroups.** In the course of proving our results for $SU_q(N)$, we will decompose representations into components that live on its quantum subgroups $SU_q(n)$ in the sense given below. One way of extending our results to $U_q(N)$ is by exploiting the quantum subgroup relations $\mathbb{T}^N \le U_q(N) \le SU_q(N+1)$; this is done in Section 6.

**DEFINITION 2.21.** A compact quantum group $\mathbb{H}$ is a quantum subgroup of a compact quantum group $\mathcal{G}$, written $\mathbb{H} \le \mathcal{G}$, if there is a CQG algebra epimorphism (equivalently, a $*$-bialgebra epimorphism) $s: \mathcal{G} \to \mathbb{H}$; we also say that $(\mathbb{H}, s)$ is a quantum subgroup of $\mathcal{G}$.

Given such a subgroup relation, we say that a linear map $T$ from $\mathcal{G}$ to a vector space $V$ *lives on* $(\mathbb{H}, s)$ if $\ker T \supset \ker s$, equivalently, if $T$ factors (evidently uniquely) through the epimorphism $s$:

$$T = \tilde{T} \circ s \quad \text{for some map } \tilde{T}: \mathbb{H} \to V.$$

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

**LEMMA 2.22.** Suppose that $\pi = \tilde{\pi} \circ s$, $\eta = \tilde{\eta} \circ s$ and $\gamma = \tilde{\gamma} \circ s$, for maps $\pi, \dots, \tilde{\gamma}$, then
(1) $\pi$ is a representation of $\mathcal{G}$ if and only if $\tilde{\pi}$ is a representation of $\mathcal{H}$.

(2) If (1) holds then $\eta$ is a $\pi$-$\varepsilon$-cocycle if and only if $\tilde{\eta}$ is a $\tilde{\pi}$-$\tilde{\varepsilon}$-cocycle.

(3) $\gamma$ is a generating functional on $\mathcal{G}$ if and only if $\tilde{\gamma}$ is a generating functional on $\mathcal{H}$.

(4) $(\pi, \eta, \gamma)$ is a Schürmann triple on $\mathcal{G}$ if and only if $(\tilde{\pi}, \tilde{\eta}, \tilde{\gamma})$ is a Schürmann triple on $\mathcal{H}$.

Moreover, for any representation $\pi$ of $\mathcal{G}$ living on $(\mathcal{H}, s)$ and vector $\xi$ in $\mathbf{h}^{\pi}$,

$$ \mathbf{h}^{\tilde{\pi}\mathcal{G}} = \mathbf{h}^{\pi\mathcal{G}}, \quad \eta_{\pi,\xi} \text{ lives on } \mathcal{H} \text{ and } \tilde{\eta}_{\pi,\xi} = \eta_{\tilde{\pi},\xi}. \qquad (2.5) $$

This has the following useful corollary.

**PROPOSITION 2.23.** *The property (NAI) is hereditary.*

We now show that an approximately inner cocycle lives on a subgroup if its approximating inner cocycles do.

**PROPOSITION 2.24.** Let $\pi$ be a representation of $\mathcal{G}$ living on $(\mathcal{H}, s)$, let $(\xi(\lambda))$ be a net in $\mathbf{h}^{\pi}$ such that $(\eta_{\pi,\xi(\lambda)})$ converges pointwise to $\eta$, and let $P' \in P_2(\mathcal{H})$. Then the following hold.

(a) $(\eta_{\tilde{\pi},\xi(\lambda)})$, $(\omega_{\xi(\lambda)} \circ \pi \circ P)$ and $(\omega_{\xi(\lambda)} \circ \tilde{\pi} \circ P')$ have pointwise limits $\tilde{\eta}$, $\gamma$ and $\gamma'$, such that $\eta = \tilde{\eta} \circ s$, and $\gamma$ and $\gamma'$ are generating functionals completing $\eta$ and $\tilde{\eta}$ respectively.

(b) $\gamma = \gamma' \circ s \circ P$.

*Proof.* (a) It follows from Identity (2.5) that $\eta_{\tilde{\pi},\xi(\lambda)} \circ s = \eta_{\pi,\xi(\lambda)}$ for each $\lambda$, and so (a) follows from the surjectivity of $s$ and Proposition 2.8.

(b) This follows since $s(K_2) = \tilde{K}_2 = \text{ran } P'$ so $P' \circ s \circ P = s \circ P$ and thus, for each $\lambda$, $(\omega_{\xi(\lambda)} \circ \tilde{\pi} \circ P') \circ (s \circ P) = \omega_{\xi(\lambda)} \circ \tilde{\pi} \circ s \circ P = \omega_{\xi(\lambda)} \circ \pi \circ P$. $\square$

The projections $P \in P_2(\mathcal{G})$ and $P' \in P_2(\mathcal{H})$ may be chosen to be compatible. This follows from the following straightforward lemma.

**LEMMA 2.25.** Let $P = P^E$ and $P' = P^{E'}$ for hermitian basis extensions $E$ from $K_2$ to $K$ and $E'$ from $\tilde{K}_2$ to $\tilde{K}$, according to (2.4). Then $P' \circ s = s \circ P$ if and only if $s(E) \subset \text{span } E'$, in which case $\text{span } s(E) = \text{span } E'$ and so the generating functional $\gamma$ from Proposition 2.24 lives on $\mathcal{H}$.

**The quantum groups** $SU_q(N)$ and $U_q(N)$. Let $0 < q < 1$. We next collect the facts about $SU_q(N)$ and $U_q(N)$ for $N \ge 2$ that are required. For convenience, we extend our definitions to the case $N = 1$: $SU_q(1) = SU(1) := \{e\}$, the trivial group, and $U_q(1) := U(1) = \mathbb{T}$, the torus. For an element $\sigma$ of the permutation group $\mathcal{S}_N$, let $i(\sigma)$ denote the number of inversions of $\sigma$: $\#\{(j, k): j < k, \sigma(j) > \sigma(k)\}$.

As a unital algebra, the CQG algebra $U_q(N)$ of the compact quantum group $U_q(N)$, is generated by indeterminates $u_{jk}$ ($j, k = 1, \dots, N$) and $D^{-1}$, subject to the following relations ($|13, \text{Section 2}|$):

$$ u_{ij}u_{kj} = qu_{kj}u_{ij} \qquad \text{if } i < k, \qquad (2.6a) $$

$$ u_{ij}u_{il} = qu_{il}u_{ij} \qquad \text{if } j < l, \qquad (2.6b) $$

$$ u_{ij}u_{kl} = u_{kl}u_{ij} \qquad \text{if } i < k, j > l, \qquad (2.6c) $$

$$ u_{ij}u_{kl} = u_{kl}u_{ij} - (q^{-1} - q)u_{il}u_{kj} \qquad \text{if } i < k, j < l, \qquad (2.6d) $$

and

$$ D^{-1}D_q = 1 = D_qD^{-1}, $$
for the q-determinant of the matrix $U = [u_{jk}]_{j,k=1}^n$,

$$D_q = D_q(U) := \sum_{\sigma \in \mathcal{S}_N} (-q)^{i(\sigma)} u_{1,\sigma(1)} \cdots u_{N,\sigma(N)}.$$

The jk-th q-minor is defined as the q-determinant of the $(N-1) \times (N-1)$-matrix obtained from U by removing the j-th row and the k-th column,

$$D_q^{jk} = D_q^{jk}(U) := \sum_{\sigma \in \mathcal{S}_{N-1}^{jk}} (-q)^{i(\sigma)} u_{1,\sigma(1)} \cdots u_{j-1,\sigma(j-1)} u_{j+1,\sigma(j+1)} \cdots u_{N,\sigma(N)},$$

where $\mathcal{S}_{N-1}^{jk}$ denotes the set of bijections $\sigma$ from $\{1, \dots, j-1, j+1, \dots, N\}$ to $\{1, \dots, k-1, k+1, \dots, N\}$. The involution, counit and coproduct of $\mathcal{U}_q(N)$ are then determined by the requirements

$$u_{jk}^* = (-q)^{k-j} D_q^{jk} D^{-1}, \quad (D^{-1})^* = D_q, \quad \varepsilon(u_{jk}) = \delta_{jk} \quad \text{and} \quad \Delta u_{jk} = \sum_l u_{jl} \otimes u_{lk}.$$

The matrix of elements U satisfies the unitarity relations (2.7) below.

As unital $^*$-algebra, $SU_q(N)$ is generated by indeterminates $u_{jk}$ ($j, k = 1, \dots, N$), subject to the unitarity relations ([28]):

$$\sum_{s=1}^{N} u_{js} u_{ks}^{*} = \delta_{jk} \mathbf{1} = \sum_{s=1}^{N} u_{sj}^{*} u_{sk} \quad (j, k \in \{1, 2, \dots, N\}), \qquad (2.7)$$

and the twisted determinant conditions

$$\sum_{\sigma \in \mathcal{S}_N} (-q)^{i(\sigma)} u_{\sigma(1), \tau(1)} u_{\sigma(2), \tau(2)} \cdots u_{\sigma(N), \tau(N)} = (-q)^{i(\tau)} \mathbf{1} \quad (\tau \in \mathcal{S}_N).$$

The counit and coproduct are given by the same formulae as for $\mathcal{U}_q(N)$.

REMARK 2.26. We also use an alternative characterisation of $SU_q(N)$, namely as the quotient of $\mathcal{U}_q(N)$ by the extra relation $D_q = 1$; the involution then simplifies to

$$u_{jk}^* := (-q)^{k-j} D_q^{jk},$$

showing that, as an algebra, $SU_q(N)$ is generated by the $u_{jk}$'s. This means that, when checking well-definedness of representations and cocycles, one only has to manage the relations of the generators $u_{jk}$ (namely (2.6) and $D_q([u_{jk}]) = 1$) and not those involving their adjoints.

The following commutation relations among the generators $u_{jk}$ of $\mathcal{U}_q(N)$ and their adjoints, and therefore also those of $SU_q(N)$, are easily verified: for $i, j, k, l \in \{1, \dots, N\}$,

$$u_{ij} u_{kl}^* = u_{kl}^* u_{ij} \quad \text{if } i \neq k \text{ and } j \neq l, \qquad (2.8a)$$

$$u_{ij} u_{kj}^* = q u_{kj}^* u_{ij} - (1 - q^2) \sum_{m<j} u_{im} u_{km}^* \quad \text{if } i \neq k, \qquad (2.8b)$$

$$u_{ij} u_{il}^* = q^{-1} u_{il}^* u_{ij} + (q^{-1} - q) \sum_{n>i} u_{nl}^* u_{nj} \quad \text{if } j \neq l, \qquad (2.8c)$$

$$u_{ij} u_{ij}^* = u_{ij}^* u_{ij} + (1 - q^2) \sum_{n>i} u_{nj}^* u_{nj} - (1 - q^2) \sum_{m<j} u_{im} u_{im}^* \qquad (2.8d)$$

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

$$u_{Nj} u_{Nk}^* = q^{-1} u_{Nk}^* u_{Nj} \quad \text{if } j \neq k, \qquad (2.9a)$$

$$u_{jN} u_{kN}^* = q^{-1} u_{kN}^* u_{jN} \quad \text{if } j \neq k, \qquad (2.9b)$$

$$u_{NN}^* u_{NN} = q^2 u_{NN} u_{NN}^* + (1 - q^2) \mathbf{1}, \qquad (2.9c)$$
Identity (2.9a) follows from (2.8c). Identity (2.8b) with the unitarity condition (2.7) together
imply that, for $j \neq k$,

$$
u_{jN} u_{kN}^* = q u_{kN}^* u_{jN} - (1-q^2) \sum_{m<N} u_{jm} u_{km}^* = q u_{kN}^* u_{jN} + (1-q^2) u_{jN} u_{kN}^* ,
$$

from which (2.9b) follows, and Identity (2.9c) follows from (2.8d):

$$
u_{NN} u_{NN}^{*} = u_{NN}^{*} u_{NN} - (1 - q^{2}) \sum_{m<N} u_{Nm} u_{Nm}^{*} = u_{NN}^{*} u_{NN} - (1 - q^{2})(1 - u_{NN} u_{NN}^{*}).
$$

We next describe the relevant quantum subgroup relations. By definition, $SU_q(N)$ is a quantum subgroup of $U_q(N)$ via the CQG epimorphism determined by its action on generators as follows

$$
r_N: u_{jk} \mapsto u_{jk} \text{ and } D^{-1} \mapsto 1.
$$

Also $\mathcal{U}_q(N)$ is a quantum subgroup of $\mathcal{SU}_q(N+1)$ via the epimorphism determined by

$$
t_N: \left[
\begin{array}{cccc}
u_{11} & \cdots & u_{1N} & u_{1,N+1} \\
\vdots & \ddots & \vdots & \vdots \\
u_{N1} & \cdots & u_{NN} & u_{N,N+1} \\
u_{N+1,1} & \cdots & u_{N+1,N} & u_{N+1,N+1}
\end{array}
\right]
\mapsto
\left[
\begin{array}{cccc}
u_{11} & \cdots & u_{1N} & 0 \\
\vdots & \ddots & \vdots & \vdots \\
u_{N1} & \cdots & u_{NN} & 0 \\
0 & \cdots & 0 & D^{-1}
\end{array}
\right]
$$

where, as in the definition of $r_N$, the $u_{jk}$ on the left-hand side are the generators of $SU_q(N+1)$
while those on the right-hand side are the generators of $U_q(N)$; like $r_N$, $t_N$ respects coproduct,
counit and involution, and thus also coinverse. Composition gives the chain

$$
SU_q(1) \le U_q(1) \le SU_q(2) \le U_q(2) \le \dots \le SU_q(N) \le U_q(N) \le \dots
$$

Of particular interest for us is the epimorphism $s_N := r_{N-1} \circ t_{N-1} : \mathcal{S}U_q(N) \to \mathcal{S}U_q(N-1)$,
which is determined by

$$
s_N: \left[
\begin{array}{cccc}
u_{11} & \cdots & u_{1,N-1} & u_{1N} \\
\vdots & \ddots & \vdots & \vdots \\
u_{N-1,1} & \cdots & u_{N-1,N-1} & u_{N-1,N} \\
u_{N1} & \cdots & u_{N,N-1} & u_{NN}
\end{array}
\right] \mapsto \left[
\begin{array}{cccc}
u_{11} & \cdots & u_{1,N-1} & 0 \\
\vdots & \ddots & \vdots & \vdots \\
u_{N-1,1} & \cdots & u_{N-1,N-1} & 0 \\
0 & \cdots & 0 & 1
\end{array}
\right], \quad (2.10)
$$

and its iterates

$$
s_{n,N} := s_{n+1} \circ \dots \circ s_N : \mathcal{S}U_q(N) \to \mathcal{S}U_q(n) \quad (n < N). \tag{2.11}
$$

PROPOSITION 2.27. Let $1 \le n < N$. The kernel of $s_{n,N}$ equals the ideal $\mathcal{I}$ generated by the set

$$
S_{n,N} := \{ u_{kj} - \delta_{kj} 1 : 1 \le j, k \le N, \max\{j, k\} > n \}.
$$

Proof. For $m \in \{n, N\}$ let us abbreviate $SU_q(m)$ to $A_m$ and denote its algebra generators by
$u_{jk}^m$ ($1 \le j, k \le m$). We also write $\mathcal{K}$ for the ideal $\ker s_{n,N}$ of $A_N$.

For $\sigma \in \mathcal{S}_N$ and $n < p \le N$, $u_{p,\sigma(p)}^N - \delta_{p,\sigma(p)} 1 \in \mathcal{S}_{n,N} \subset \mathcal{I}$ so

$$
\begin{align*}
1 = D_q([u_{jk}^N]) &\in \sum_{\sigma \in \mathcal{S}_N \text{ s.t. } \sigma(p)=p \text{ for } n<p \le N} (-q)^{i(\sigma)} u_{1,\sigma(1)}^N \cdots u_{n,\sigma(n)}^N + \mathcal{I} \\
&= \sum_{\tau \in \mathcal{S}_n} (-q)^{i(\tau)} u_{1,\tau(1)}^N \cdots u_{n,\tau(n)}^N + \mathcal{I} = D_q([u_{jk}^N]_{1 \le j,k \le n}) + \mathcal{I}.
\end{align*}
$$

It follows that the relation $D_q([u_{jk}^n]) = 1$ in $\mathcal{A}_n$ is preserved by the mapping from the set of generators of $\mathcal{A}_n$ into the quotient algebra $\mathcal{A}_n/\mathcal{I}$ given by $u_{jk}^n \mapsto u_{jk}^N + \mathcal{I} \ (1 \le j, k \le n)$. Since
this clearly also preserves the (remaining defining) relations (2.6), the mapping uniquely extends to an algebra morphism $\phi: \mathcal{A}_n \to \mathcal{A}_N/\mathcal{I}$.

Now the prescription $a+\mathcal{I} \mapsto a+\mathcal{K}$ defines an algebra epimorphism $\psi: \mathcal{A}_N/\mathcal{I} \to \mathcal{A}_N/\mathcal{K}$ (since $\mathcal{I} \subset \mathcal{K}$) and, letting $\tilde{s}_{n,N}$ denote the canonically induced algebra isomorphism $\mathcal{A}_N/\mathcal{K} \to \mathcal{A}_n$,

$$ (\phi \circ \tilde{s}_{n,N} \circ \psi)(u_{jk}^N + \mathcal{I}) = (\phi \circ \tilde{s}_{n,N})(u_{jk}^N) = \begin{cases} u_{jk}^N + \mathcal{I} & \text{if } 1 \le j, k \le n \\ \delta_{jk} 1 + \mathcal{I} & \text{if } \max\{j, k\} > n. \end{cases} $$

Thus, since $u_{jk}^N - \delta_{jk} 1 \in \mathcal{S}_{n,N} \subset \mathcal{I}$ if $\max\{j, k\} > n$, $(\phi \circ \tilde{s}_{n,N} \circ \psi)(u_{jk}^N + \mathcal{I}) = u_{jk}^N + \mathcal{I}$ for all $j$ and $k$ so $\phi \circ \tilde{s}_{n,N} \circ \psi = \text{id}_{\mathcal{A}_N/\mathcal{I}}$. It follows that the algebra epimorphism $\psi$ is injective and thus an isomorphism. Since $\mathcal{I} \subset \mathcal{K}$, this implies that $\mathcal{I} = \mathcal{K}$. $\square$

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

**LEMMA 2.28.** Let $\pi$ be a representation of $SU_q(N)$ and let $\eta$ be a $\pi$-$\varepsilon$-cocycle. For $i < l \le N$ and $j, k < N$,

$$ \eta(u_{il}) = -(I - q\pi(u_{ll}))^{-1}\pi(u_{il})\eta(u_{ll}), \qquad (2.12a) $$

$$ \eta(u_{li}) = -(I - q\pi(u_{ll}))^{-1}\pi(u_{li})\eta(u_{ll}), \qquad (2.12b) $$

$$ \pi(u_{NN}-1)\eta(u_{jk}) = (\pi(u_{jk} - \delta_{jk} 1) - (q^{-1} - q)\pi(1 - q^2 u_{NN})^{-1}\pi(u_{il}u_{li}))\eta(u_{NN}). \qquad (2.12c) $$

In particular, by Remark 2.26, $\eta$ is determined by its value $\eta(u_{NN})$ when $\pi(1-u_{NN})$ is injective.

*Proof.* If $a = u_{il}$ or $a = u_{li}$ where $i < l \le N$, then $a \in \ker \varepsilon$ and, by Identities (2.6a) and (2.6b), $a u_{ll} = q u_{ll} a$. Hence, by the cocycle property, $\pi(a)\eta(u_{ll}) + \eta(a) = q\pi(u_{ll})\eta(a)$. Since $\pi(u_{ll})$ is a contraction, this is equivalent to the identity $\eta(a) = -(I - q\pi(u_{ll}))^{-1}\pi(a)\eta(u_{ll})$.

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

$$ \begin{aligned} \pi(u_{jk})\eta(u_{NN}) + \eta(u_{jk}) &= \eta(u_{jk}u_{NN}) = \eta(u_{NN}u_{jk}) - (q^{-1} - q)\eta(u_{jN}u_{Nk}) \\ &= \pi(u_{NN})\eta(u_{jk}) + \eta(u_{NN})\varepsilon(u_{jk}) - (q^{-1} - q)\pi(u_{jN})\eta(u_{Nk}), \end{aligned} $$

so,

$$ \begin{aligned} \pi(u_{NN}-1)\eta(u_{jk}) &= \pi(u_{jk} - \delta_{jk} 1)\eta(u_{NN}) + (q^{-1} - q)\pi(u_{jN})\eta(u_{Nk}) \\ &= (\pi(u_{jk} - \delta_{jk} 1) - (q^{-1} - q)\pi(u_{jN})(I - q\pi(u_{NN}))^{-1}\pi(u_{Nk}))\eta(u_{NN}) \\ &= (\pi(u_{jk} - \delta_{jk} 1) - (q^{-1} - q)(I - q^2\pi(u_{NN}))^{-1}\pi(u_{jN}u_{Nk}))\eta(u_{NN}). \end{aligned} \qquad \square $$

We end this section by characterising those representations and cocycles on $SU_q(N)$ that live on $SU_q(n)$, for $n < N$.

**PROPOSITION 2.29.** Let $\pi$ be a representation of $SU_q(N)$, let $\eta$ be a $\pi$-$\varepsilon$-cocycle and let $n < N$.

(a) The following are equivalent.

(i) $\pi$ lives on $SU_q(n)$.

(ii) $\pi(u_{kj}) = \delta_{kj} I$ if $\max\{j, k\} > n$.

(iii) $\pi(u_{jj}) = I$ for $n < j \le N$.

(b) Suppose that $\pi$ lives on $SU_q(n)$. Then the following are equivalent.

(i) $\eta$ lives on $SU_q(n)$.

(ii) $\eta(u_{kj}) = 0$ if $\max\{j, k\} > n$.

(iii) $\eta(u_{jj}) = 0$ for $n < j \le N$.
*Proof.* For both parts, the equivalence of (i) and (ii) follows from Proposition 2.27 because (ii) says $\pi$, respectively $\eta$, vanishes on the set $S_{n,N}$ (in the latter case, since cocycles kill the identity element), moreover (ii) obviously implies (iii).

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

$$ \pi(u_{jj})^*\pi(u_{jj}) + \sum_{k \neq j} \pi(u_{kj})^*\pi(u_{kj}) = I = \pi(u_{jj})\pi(u_{jj})^* + \sum_{k \neq j} \pi(u_{jk})\pi(u_{jk})^* $$

so if $\pi(u_{jj}) = I$ then $\pi(u_{kj}) = 0$ for $k \neq j$. Thus (iii) implies (ii).

(b) By Identities (2.12a) and (2.12b), if $\eta(u_{ll}) = 0$ then $\eta(u_{il}) = 0 = \eta(u_{li})$ for $i < l$ and so (iii) implies (ii). $\square$

### 3. CLASSIFICATION OF GAUSSIAN GENERATING FUNCTIONALS

In this Section we investigate the gaussian generating functionals on $SU_q(N)$ and their Schürmann triples. We follow Procedure 2.6 for gaussian representations, that is representations of the form $\iota_h \circ \varepsilon: a \mapsto \varepsilon(a)I_h$. Since gaussian cocycles vanish on $K_2$, we seek a hermitian basis extension $E$ from $K_2$ to $K$ (see Section 2).

**LEMMA 3.1.** Set $v_j := (u_{jj} - 1) \in K$ and $d_j := (2i)^{-1}(u_{jj} - u_{jj}^*) = (2i)^{-1}(v_j - v_j^*) \in K$. Then the following hold.

(a) $u_{jk} \in K_2$ for $j \neq k$.

(b) $v_j + v_j^* \in K_2$.

(c) $d_1 + \dots + d_N \in K_2$.

(d) $d_j d_k - d_k d_j \in K_3$.

*Proof.* (a) Let $j \neq k$. Combining Relations (2.6a) and (2.6b), one has $u_{jk}u_{ll} = qu_{ll}u_{jk}$ for $j \neq k$ and $l := \max(j, k)$. Therefore, since $u_{ll} - 1, u_{jk} \in K$,

$$ u_{jk} = (1 - q)^{-1}q(u_{ll} - 1)u_{jk} - u_{jk}(u_{ll} - 1) \in K_2. $$

(b) By the unitarity relation (2.7) we see that $1 - u_{jj}u_{jj}^* = \sum_{m \neq j} u_{jm}u_{jm}^* \in K_2$, so

$$ v_j + v_j^* = (u_{jj} - 1) + (u_{jj} - 1)^* = -(1 - u_{jj}u_{jj}^*) - (u_{jj} - 1)(u_{jj} - 1)^* \in K_2. $$

(c) Observe that

$$ u_{11} \cdots u_{NN} = (v_1 + 1) \cdots (v_N + 1) = 1 + (v_1 + \cdots + v_N) + \text{terms in } K_2. $$

Therefore, $v_1 + \cdots + v_N + (1 - u_{11} \cdots u_{NN}) \in K_2$. Since $D_q = 1$, we have

$$ 1 - u_{11} \cdots u_{NN} = \sum_{\sigma \in S_N, \sigma \neq \text{id}} (-q)^{i(\sigma)} u_{1,\sigma(1)} \cdots u_{N,\sigma(N)}. \quad (3.1) $$

Now, for $\sigma \neq \text{id}$ there is at least one $j$ such that $j \neq \sigma(j)$, so, from part (a), the right-hand side of (3.1) is in $K_2$. Thus $v_1 + \cdots + v_N \in K_2$, hence,

$$ d_1 + \cdots + d_N = (2i)^{-1}((v_1 + \cdots + v_N) - (v_1 + \cdots + v_N)^*) \in K_2. $$

(d) This follows from part (a), in view of the relations (2.6d) and (2.8a). $\square$

Now consider the family of characters determined by

$$ \varepsilon_{\theta_2, \dots, \theta_N}(u_{kl}) := e^{i\theta_k} \delta_{k,l} \quad (k, l \in \{1, \dots, N\}), $$
for $\theta_2, \dots, \theta_N \in \mathbb{R}$ and $\theta_1$ given implicitly by $\sum_{k=1}^N \theta_k = 0$. The pointwise defined linear functionals

$$ \varepsilon'_j := \frac{\partial}{\partial \theta_j} \Big|_{\theta_2 = \dots = \theta_N = 0} \varepsilon_{\theta_2, \dots, \theta_N} \quad (j = 2, \dots, N) \qquad (3.2) $$

are drifts because they kill 1 (since each $\varepsilon_{\theta_1, \dots, \theta_d}$ is a character), vanish on $K_2$ (by Leibniz' rule, since $\varepsilon_{0, \dots, 0} = \varepsilon$) and are hermitian (since $d_k^* = d_k$ and $\varepsilon'_j(d_k) = \delta_{jk}$).

**LEMMA 3.2.** Set $E := \{d_2, \dots, d_N\}$. Then the following hold.

(a) $E$ is a hermitian basis extension from $K_2$ to $K$.

(b) $\{\varepsilon'_j : j = 2, \dots, N\}$ is a basis for the real space of drifts on $SU_q(N)$.

*Proof.* The set $E$ is hermitian and it follows from parts (a), (b) and (c) of Lemma 3.1 that $E \cup K_2$ spans $K$. For $j, k = 2, \dots, N$, $\varepsilon'_j(d_k) = \delta_{jk}$ so $E$ is linearly independent, and $\varepsilon'_j$ kills $K_2$ so $E$ and $K_2$ are disjoint. Thus (a) holds, and so does (b) since drifts vanish on $\{1\} \cup K_2$. $\square$

In view of part (d) of Lemma 3.1 and Corollaries 2.13 and 2.12, we deduce the following.

**THEOREM 3.3.** $SU_q(N)$ does not have property (GC) unless $N \le 2$.

This is also proved in [2]. $SU_q(N)$ has Property (AC) if $N = 2$ ([23], [21]).

From now on, we fix the hermitian basis extension $E_N := \{d_2, \dots, d_N\}$ from $K_2$ to $K$, and thereby also the projection in $P_2(SU_q(N))$ as in (2.4), which we denote $P_N$. The resulting family of projections is compatible with the subgroup relations $SU_q(N) \ge SU_q(n)$.

**PROPOSITION 3.4.** $P_n \circ s_{n,N} = s_{n,N} \circ P_N$ for $n < N$.

*Proof.* The epimorphism $s_N$ (see (2.10)) sends $d_N$ to 0 and, for $2 \le n \le N - 1$, sends the $d_n$ of $SU_q(N)$ to the $d_n$ of $SU_q(N - 1)$, so $s_N(E_N) = E_{N-1} \cup \{0\}$. Therefore, by Lemma 2.25, $P_{N-1} \circ s_N = s_N \circ P_N$. By Identity (2.11) this iterates to yield the proposition. $\square$

Note that the $\varepsilon'_j$ obtained in (3.2) coincide with the functionals $\varepsilon'_d$ ($d = d_j$) defined in (2.3) from the basis extension $E_N$. Thus Proposition 2.10 yields the following characterization.

**PROPOSITION 3.5.** The gaussian cocycles on $SU_q(N)$ are precisely the maps of the form

$$ \eta = \sum_{j=2}^{N} \xi_j \varepsilon'_j(\cdot) \qquad (3.3) $$

for a family of vectors $(\xi_j)_{j=2}^N$ in a Hilbert space $\mathfrak{h}$.

We next describe the gaussian generating functionals on $SU_q(N)$. Consider the pointwise defined functionals

$$ \varepsilon''_{jk} := \frac{\partial^2}{\partial \theta_j \partial \theta_k} \Big|_{\theta_2 = \dots = \theta_N = 0} \varepsilon_{\theta_2, \dots, \theta_N} \quad (j, k = 2, \dots, N). $$

**THEOREM 3.6.** Letting $M_n(\mathbb{R})_+$ denote the set of real nonnegative-definite $n \times n$ matrices, the prescription

$$ (r, R) \mapsto \sum_{j=2}^{N} r_j \varepsilon'_j + \frac{1}{2} \sum_{j,k=2}^{N} r_{jk} \varepsilon''_{jk} $$

defines a bijection from $\mathbb{R}^{N-1} \times M_{N-1}(\mathbb{R})_+$ to the set of gaussian generating functionals $\gamma$ on $SU_q(N)$ in which the second sum is the $P_N$-invariant component $\gamma \circ P_N$.
Proof. In view of Lemma 3.2, it suffices to verify that the prescription $[r_{jk}] \mapsto \frac{1}{2} \sum_{j,k=2}^{N} r_{jk} \varepsilon''_{jk}$ defines a bijection from $M_{N-1}(\mathbb{R})_+$ to the set of $P_N$-driftless (i.e. $P_N$-invariant) gaussian generating functionals $\gamma$.

First note that by Leibniz' rule,

$$ \varepsilon''_{jk}(ab) = \varepsilon''_{jk}(a)\varepsilon(b) + \varepsilon'_{j}(a)\varepsilon'_{k}(b) + \varepsilon'_{k}(a)\varepsilon'_{j}(b) + \varepsilon(a)\varepsilon''_{jk}(b) \quad (a, b \in \mathcal{SU}_q(N)). $$

It follows that $\varepsilon''_{jk}$ vanishes on $K_3$ and, by direct computation, $\varepsilon''_{jk}(d_l) = 0$ and $\varepsilon''_{jk}(d_l d_m) = \delta_{jl}\delta_{km} + \delta_{jm}\delta_{kl}$ for $j, k, l, m = 2, \dots, N$. In particular, $\varepsilon''_{jk} \circ P_N = \varepsilon''_{jk}$ and, for all $c \in K$ and $\lambda \in \mathbb{C}^{N-1}$, $\sum \lambda_j \varepsilon''_{jk}(c^*c)\lambda_k = 2|\sum \lambda_k \varepsilon'_{k}(c)|^2 \ge 0$ so, since nonnegative-definiteness is preserved under the Schur product, for any matrix $R = [r_{jk}] \in M_{N-1}(\mathbb{R})_+$ the functional $\frac{1}{2} \sum r_{jk} \varepsilon''_{jk}$ is conditionally positive and therefore a $P_N$-invariant gaussian generating functional.

Conversely, if $\gamma$ is a gaussian generating functional, its associated cocycle $\eta$ is of the form (3.3) and so, by Corollary 2.13 and part (d) of Lemma 3.1, $\eta$ is hermitian and hence the Gram matrix $[\langle \xi_j, \xi_k \rangle]$ is real and thus in $M_{N-1}(\mathbb{R})_+$. $\square$

REMARKS 3.7. The CQG algebra $\mathcal{T}_{N-1}$ of the torus $\mathbb{T}^{N-1}$ is generated, as unital $*-\text{algebra}$, by a family of commuting unitaries $\{u_j : j = 1, \dots, N\}$ subject to the relation $u_1 \cdots u_N = 1$. The prescription $u_{jk} \mapsto \delta_{jk}u_j$ determines a CQG epimorphism $\tau_N : \mathcal{SU}_q(N) \to \mathcal{T}_{N-1}$ with respect to which the characters $\varepsilon_{\theta_2, \dots, \theta_N}$ of $\mathcal{SU}_q(N)$ live on $\mathbb{T}^{N-1}$. Therefore the gaussian generating functionals of $\mathcal{SU}_q(N)$ live on $\mathcal{T}_{N-1}$. It also follows that, for any compact quantum group $\mathcal{G}$ satisfying $SU_q(N) \ge \mathbb{G} \ge \mathbb{T}^{N-1}$, the projection $P \in P_2(\mathcal{G})$ may be chosen to be compatible with those for $SU_q(N)$ and $\mathcal{T}_{N-1}$, and the gaussian generating functionals of $\mathcal{G}$ correspond to those of $\mathcal{T}_{N-1}$. Application of results on classical compact Lie groups in [24] to $\mathbb{T}^{N-1}$ gives an alternative proof of Theorem 3.6. The original preprint version of our paper has motivated generalisation of the theorem to all $q$-deformations of simply connected semisimple compact Lie groups ([7, Theorem 6.1]).

## 4. DECOMPOSITION

This is the central section of the paper. We decompose an arbitrary representation $\pi$ of $\mathcal{SU}_q(N)$ uniquely into a direct sum $\pi_1 \oplus \cdots \oplus \pi_N$, in which $\pi_1 = \pi_G$, as defined in Proposition 2.16 and, for $2 \le n \le N$, $\pi_n$ lives on $\mathcal{SU}_q(n)$ and $\pi_n(1 - u_{nn})$ is injective. We then show that in the corresponding decomposition $\eta_1 \oplus \cdots \oplus \eta_N$ of a $\pi$-$\varepsilon$-cocycle $\eta$, for $2 \le n \le N$ each cocycle $\eta_n$ is approximately inner and determined by the vector $\eta(u_{nn})$. This implies that $SU_q(N)$ has Property (NAI) and so also (LK). We deduce a Hunt formula for $SU_q(N)$ incorporating full decomposition for generating functionals.

The following elementary lemma plays a key role in the approximation of cocycles (part (a) is well-known, for example in ergodic theory). For bounded operators $T$, we write $\overline{\text{ran}} T$ for $\text{ran} T$.

LEMMA 4.1 (Contraction operator lemma). For any contraction operator $C$ on a Hilbert space,

(a) $\ker(I - C^*) = \ker(I - C)$, so also $\overline{\text{ran}}(I - C) = \ker(I - C)^\perp = \overline{\text{ran}}(I - C^*)$, and

(b) $P(t) := (1 - t)(I - tC)^{-1} \xrightarrow{\text{SOT}} P$ and $P^\perp(t) := (I - tC)^{-1}(I - C) \xrightarrow{\text{SOT}} P^\perp$ as $t \to 1^-$,

where $P := P_{\ker(I-C)}$. In particular, the following four conditions are equivalent.

(i) $I - C$ is injective; (ii) $I - C$ has dense range;

(ii) $(I - tC)^{-1}(I - C) \xrightarrow{\text{SOT}} I$ as $t \to 1^-$; (iii) $(1 - t)(I - tC)^{-1} \xrightarrow{\text{SOT}} 0$ as $t \to 1^-$.
**Proof.** (a) Let $\xi \in \ker(I - C) = \ker(I - C^*)^{\perp} = \ker(C^* - I)^{\perp}$. By symmetry it suffices to prove that $\xi \in \ker(I - C^*)$. This follows by Pythagoras: $\|\xi\|^2 + \|(C^* - I)\xi\|^2 = \|C^*\xi\|^2 \le \|\xi\|^2$.

(b) For $0 < t < 1$, $(1) I - P(t) = tP^{\perp}(t)$, $(2) \|P(t)\| \le 1$, and $(3) P(t)(I - C) = (1 - t)P^{\perp}(t)$; thus $(4) \|P(t)(I - C)\| \le 2(1 - t)/t$. By $(1)$, $P(t) \to I$ on $\ker P^{\perp}(t) = \ker(I - C)$ and, by $(4)$ and $(2)$, $P(t) \to 0$ on $\overline{\ker}(I - C)$. Hence $P(t) \xrightarrow{\text{SOT}} P$ by (a), and so $P^{\perp}(t) \xrightarrow{\text{SOT}} P^{\perp}$ by (1). $\square$

**Decomposition of representations and cocycles.** We start by separating out the maximal subspace on which the operator $\pi(1 - u_{NN})$ acts injectively, for a given representation $\pi$.

**LEMMA 4.2.** Let $\pi$ be a representation of $SU_q(N)$. Then $\pi$ has a unique decomposition $\pi^N \oplus \pi_N$ for which $\pi^N$ lives on $SU_q(N-1)$, equivalently $\pi^N(1 - u_{NN}) = 0$, and $\pi_N(1 - u_{NN})$ is injective. Moreover, $\mathbf{h}^{\pi^N} = \ker \pi(1 - u_{NN})$.

**Proof.** The equivalence is contained in Proposition 2.29. We first show that $k := \ker \pi(1 - u_{NN})$ is an invariant subspace for $\pi$. Since the $u_{jk}$ generate $SU_q(N)$ as an algebra (Remark 2.26), to see this it suffices to fix $\xi \in k$ and $j, k \in \{1, \dots, N\}$, and to verify that $\pi_{jk}\xi \in k$ (in the convenient abbreviation $\pi_{jk} := \pi(u_{jk})$). For $j = k = N$ this is obvious. For $k < N$, applying $\pi$ to Identity (2.7) then the vector functional $\omega_{\xi}$, we see that $\pi_{N_s}^*\xi = 0 = \pi_{sN}\xi$ for $s < N$ so, by Identity (2.8d),

$$ \pi_{Nk}^* \pi_{Nk} \xi = \pi_{Nk} \pi_{Nk}^* \xi + (1 - q^2) \sum_{m<k} \pi_{Nm} \pi_{Nm}^* \xi = 0, $$

thus $\pi_{Nk}\xi = 0$. Lastly, for $j, k < N$, $\pi_{jk}\pi_{NN}\xi = \pi_{NN}\pi_{jk}\xi - (q^{-1} - q)\pi_{jN}\pi_{Nk}\xi$ by Identity (2.6d), so $\pi_{jk}\pi_{NN}\xi = \pi_{NN}\pi_{jk}\xi$, in other words $\pi_{jk}\xi \in k$, as required.

In the resulting decomposition $\pi = \pi^N \oplus \pi_N$, $\pi^N(1 - u_{NN}) = 0$ and $\pi_N(1 - u_{NN})$ is injective. It remains to prove uniqueness. Thus let $\rho \oplus \sigma$ be another such decomposition of $\pi$; we must show that $\mathbf{h}^{\rho} = k$. This follows from Lemma 4.1:

$$ \mathbf{h}^{\rho} = \ker \rho(1 - u_{NN}) \subset k = \ker \pi(1 - u_{NN})^{\perp} \subset \ker \sigma(1 - u_{NN})^{\perp} = (\mathbf{h}^{\sigma})^{\perp} = \mathbf{h}^{\rho}. \quad \square $$

**DEFINITION 4.3.** A decomposition $\pi_1 \oplus \dots \oplus \pi_N$ of a representation of $SU_q(N)$ is full if

(1) for $1 \le n < N$, there is a representation $\tilde{\pi}_n$ of $SU_q(n)$ such that $\pi_n = \tilde{\pi}_n \circ s_{n,N}$ and,

(2) for $n \ge 2$, $\pi_n(1 - u_{nn})$ is injective.

For $n = 1$, $(1)$ says that $\pi_n$ is gaussian, and for $n \ge 2$, $\pi_n(1 - u_{nn}) = \tilde{\pi}_n(1 - u_{nn}^n)$ where $u_{nn}^n$ denotes $u_{nn}$ in $SU_q(n)$; $(2)$ is equivalent to $\pi(1 - u_{nn})$ having dense range for $n \ge 2$.

This superscript convention, indicating which quantum subgroup is being referred to, continues below.

**THEOREM 4.4.** Every representation of $SU_q(N)$ has a unique full decomposition.

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

**Existence.** By Lemma 4.2, $\pi = \pi^K \oplus \pi_K$ where $\pi_K(1 - u_{KK})$ is injective and $\pi^K = \tilde{\pi} \circ s_K$ for a representation $\tilde{\pi}$ of $SU_q(K - 1)$. By the induction hypothesis, $\tilde{\pi} = \rho_1 \oplus \dots \oplus \rho_{K-1}$ where $\rho_1$ is gaussian and, for $k = 2, \dots, K - 1$, $\rho_k(1 - u_{kk}^{K-1})$ is injective and $\rho_k = \tilde{\rho}_k \circ s_{k,K-1}$, for some representation $\tilde{\rho}_k$ of $SU_q(k)$. Set $\pi_k := \rho_k \circ s_K$ for $k = 1, \dots, K - 1$. Then $\pi = \pi_1 \oplus \dots \oplus \pi_K$, where $\pi_1$ is gaussian, $\pi_K(1 - u_{KK})$ is injective and, for $k = 2, \dots, K - 1$, $\pi_k(1 - u_{kk})$ equals $\rho_k(1 - u_{kk}^{K-1})$ and so is injective, and $\pi_k = \tilde{\rho}_k \circ s_{k,K-1} \circ s_K = \tilde{\rho}_k \circ s_{k,K}$, so $\pi_k$ lives on $SU_q(k)$.
Uniqueness. Suppose that $\pi = \rho_1 \oplus \cdots \oplus \rho_K$ is another such decomposition. Then, by the uniqueness part of Lemma 4.2, $\rho_K = \pi_K$ and $\rho_1 \oplus \cdots \oplus \rho_{K-1} = \pi_1 \oplus \cdots \oplus \pi_{K-1}$. Now, for $k = 1, \dots, K-1$, $\pi_k = \tilde{\pi}_k \circ s_K$ and $\rho_k = \tilde{\rho}_k \circ s_K$ for representations $\tilde{\pi}_1, \dots, \tilde{\rho}_{K-1}$ of $SU_q(K-1)$ and, by the surjectivity of $s_K$, $\tilde{\pi}_1 \oplus \cdots \oplus \tilde{\pi}_{K-1} = \tilde{\rho}_1 \oplus \cdots \oplus \tilde{\rho}_{K-1}$. Since $\tilde{\pi}_1$ and $\tilde{\rho}_1$ are gaussian and, for $k = 2, \dots, K-1$, $\tilde{\pi}_k$ and $\tilde{\rho}_k$ live on $SU_q(k)$ and $\tilde{\pi}_k(1 - u_{kk}^{K-1})$ and $\tilde{\rho}_k(1 - u_{kk}^{K-1})$ are injective, it follows from the induction hypothesis that $\tilde{\pi}_k = \tilde{\rho}_k$ for $k = 1, \dots, K-1$. Therefore $\pi_k = \rho_k$ for $k = 1, \dots, K$, as required. $\square$

THEOREM 4.5. Let $\pi_1 \oplus \cdots \oplus \pi_N$ be the full decomposition of a representation $\pi$ of $SU_q(N)$ and let $\eta_1 \oplus \cdots \oplus \eta_N$ be the induced decomposition of a $\pi$-$\varepsilon$-cocycle $\eta$. Then $\eta_1$ is gaussian and, for $n \ge 2$, $\eta_n$ lives on $SU_q(n)$.

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

$$ \begin{aligned} \pi_n(u_{nn})\eta_n(u_{mm}) + \eta_n(u_{nn}) &= \pi_n(u_{mm})\eta_n(u_{nn}) + \eta_n(u_{mm}) - (q^{-1} - q)\pi_n(u_{nm})\eta_n(u_{mn}) \\ &= \eta_n(u_{nn}) + \eta_n(u_{mm}), \end{aligned} $$

so $\eta_n(u_{mm}) \in \ker \pi_n(1 - u_{nn}) = \{0\}$ thus, by part (b) of Proposition 2.29, $\eta_n$ lives on $SU_q(n)$. $\square$

Approximation of cocycles and (NAI) for $SU_q(N)$. We now show that each of the cocycles $\eta_n$ ($n \ge 2$) in Theorem 4.5 is approximately inner.

PROPOSITION 4.6. Let $\eta$ be a cocycle of a representation $\pi$ of $SU_q(N)$ such that $\pi(1 - u_{NN})$ is injective. Then

$$ \eta = \text{pw-lim}_{t \to 1^{-}} \eta_{\pi, \zeta(t)} \quad \text{where } \zeta(t) := -\pi(1 - tu_{NN})^{-1}\eta(u_{NN}). $$

Proof. In view of the cocycle relations and Remark 2.26, it suffices to prove that, for each of the algebra generators $a = u_{jk}$, $\eta(a)$ is the pointwise limit as $t \to 1^{-}$ of the following expression

$$ -\pi(a - \varepsilon(a))\pi(1 - tu_{NN})^{-1}\eta(u_{NN}). \quad (4.1) $$

We prove this using Lemma 4.1 (the contraction operator lemma) and Lemma 2.28.

Case $a = u_{NN}$. Lemma 4.1 implies that $\pi(1 - tu_{NN})^{-1}\pi(1 - u_{NN})\eta(u_{NN}) \to \eta(u_{NN})$.

Case $a = u_{kN}$ or $a = u_{Nk}$ ($k < N$). Then $a \in \ker \varepsilon$ so $\pi(a) = \pi(a - \varepsilon(a)1)$. Thus, using Relations (2.12a)-(2.12b), Lemma 4.1 implies that $\eta(a)$ equals

$$ \begin{aligned} -\pi(1 - qu_{NN})^{-1}\pi(a)\eta(u_{NN}) &= -\lim_{t \to 1^{-}} \pi(1 - qu_{NN})^{-1}\pi(a)\pi(1 - tu_{NN})^{-1}\pi(1 - u_{NN})\eta(u_{NN}) \\ &= -\lim_{t \to 1^{-}} \pi(a - \varepsilon(a))\pi(1 - tu_{NN})^{-1}\eta(u_{NN}). \end{aligned} $$

Case $a = u_{jk}$ ($j, k < N$). We must show that $-\pi(u_{jk} - \delta_{jk}1)\pi(1 - tu_{NN})^{-1}\eta(u_{NN}) \to \eta(u_{jk})$. By the contraction operator lemma $-\pi(1 - tu_{NN})^{-1}\pi(u_{NN} - 1)\eta(u_{jk}) \to \eta(u_{jk})$. It therefore suffices to show that

$$ -\pi(1 - tu_{NN})^{-1}\pi(u_{NN} - 1)\eta(u_{jk}) + \pi(u_{jk} - \delta_{jk}1)\pi(1 - tu_{NN})^{-1}\eta(u_{NN}) \to 0. $$

By Identity (2.12c) the first term equals

$$ -\pi(1 - tu_{NN})^{-1}(\pi(u_{jk} - \delta_{jk}1) - (q^{-1} - q)\pi(1 - q^2 u_{NN})^{-1}\pi(u_{jN}u_{Nk})\eta(u_{NN}) $$
and so, since the operators $\pi(1 - q^2u_{NN})^{-1}$ and $\pi(1 - tu_{NN})^{-1}$ commute, after cancellation of the $\delta_{jk}$ terms and multiplication through by the invertible operator $\pi(1 - q^2u_{NN})$ we see that the task is equivalent to showing that the following converges to 0 on the vector $\eta(u_{NN})$:

$$ \pi(1 - q^2 u_{NN})[\pi(u_{jk}), \pi(1 - tu_{NN})^{-1}] + (q^{-1} - q)\pi(1 - tu_{NN})^{-1}\pi(u_{jN}u_{Nk}) \quad (4.2) $$

— we show that it converges to 0 strongly. Let us abbreviate $\pi(u_{il})$ to $\pi_{il}$ for each $i$ and $l$. It follows from Identity (2.6d) that

$$ [\pi_{jk}, \pi_{NN}^{\alpha}] = -(q^{-1} - q) \left( \sum_{\nu=0}^{\alpha-1} q^{2\nu} \right) \pi_{NN}^{\alpha-1} \pi_{jN} \pi_{Nk} \quad (\alpha \in \mathbb{Z}_+), $$

thus, taking the Neumann series for $(I - t\pi_{NN})^{-1}$, which is valid since $t\pi_{NN}$ is a strict contraction,

$$ [\pi_{jk}, (I - t\pi_{NN})^{-1}] = -(q^{-1} - q) \sum_{\alpha=1}^{\infty} \sum_{\nu=0}^{\alpha-1} q^{2\nu} t^{\alpha} \pi_{NN}^{\alpha-1} \pi_{jN} \pi_{Nk}. $$

Substituting this into (4.2) then gives the following operator composed with $(q^{-1} - q)\pi_{jN}\pi_{Nk}$:

$$ \begin{aligned} & -(I - q^2\pi_{NN}) \sum_{\alpha=1}^{\infty} \sum_{\nu=0}^{\alpha-1} q^{2\nu} t^{\alpha} \pi_{NN}^{\alpha-1} + (I - tu_{NN})^{-1} \\ &= \sum_{\nu=0}^{\infty} \sum_{\alpha=\nu+1}^{\infty} (q^{2(\nu+1)}(t\pi_{NN}^{\alpha} - tq^{2\nu}(t\pi_{NN})^{\alpha-1}) + (I - tu_{NN})^{-1} \\ &= \sum_{\nu=0}^{\infty} \left( (q^2t\pi_{NN})^{\nu+1} \sum_{\beta=0}^{\infty} (t\pi_{NN})^{\beta} - t(q^2t\pi_{NN})^{\nu} \sum_{\beta=0}^{\infty} (t\pi_{NN})^{\beta} \right) + (I - tu_{NN})^{-1} \\ &= (I - q^2tu_{NN})^{-1}(q^2t\pi u_{NN} - tI + I - q^2tu_{NN})I - tu_{NN})^{-1} \\ &= (I - q^2tu_{NN})^{-1}(1 - t)(I - tu_{NN})^{-1} \end{aligned} $$

so the required convergence follows from Lemma 4.1. □

**THEOREM 4.7.** Let $\pi_1 \oplus \cdots \oplus \pi_N$ be the full decomposition of a representation $\pi$ of $SU_q(N)$ and let $\eta_1 \oplus \cdots \oplus \eta_N$ be the induced decomposition of a $\pi$-$\varepsilon$-cocycle $\eta$. Then, for $n \ge 2$, $\eta_n = \text{pw-lim}_{t \to 1^-} \eta_{\pi_n, \xi(n,t)}$ where $\xi(n,t) := -\pi_n(1 - tu_{nn})^{-1}\eta_n(u_{nn})$.

Thus, in terms of the decomposition $h^\pi = h^{\pi_G} \oplus h^{\pi_R}$,

$$ \eta = \text{pw-lim}_{t \to 1^-} \eta_G \oplus \eta_{\pi_R, \xi(t)} \quad \text{where} \quad \xi(t) := -\pi_2(1 - tu_{22})^{-1}\eta_2(u_{22}) \oplus \cdots \oplus \pi_N(1 - tu_{NN})^{-1}\eta_N(u_{NN}). \quad (4.3) $$

*Proof.* Let $n \ge 2$. By Theorem 4.5, $\eta_n = \tilde{\eta}_n \circ s_{n,N}$ for a cocycle $\tilde{\eta}_n$ on $SU_q(n)$ and, by Lemma 2.22, it suffices to prove that $\eta_{\tilde{\pi}_n, \xi(n,t)}$ converges pointwise to $\tilde{\eta}_n$. Now $\pi_n(1 - u_{nn})$ is injective (by Theorem 4.4), $\tilde{\pi}_n(1 - tu_{nn}^n) = \pi_n(1 - tu_{nn})$ for all $t \in [0, 1]$ and $\tilde{\eta}_n(u_{nn}^n) = \eta_n(u_{nn})$ so $\tilde{\pi}_n(1 - u_{nn}^n)$ is injective and $\xi(n,t) = -\tilde{\pi}_n(1 - tu_{nn}^n)\tilde{\eta}_n(u_{nn}^n)$. The theorem therefore follows by applying Proposition 4.6 with $N = n$. □

Noting that if $\pi$ is completely non-gaussian, so $h^{\pi_G} = \{0\}$, then (4.3) simplifies to the pointwise convergence $\eta_{\pi, \xi(t)} \to \eta$ as $t \to 1^-$, we draw the following immediate corollary.

**THEOREM 4.8.** $SU_q(N)$ has property (NAI), and thus also (LK).

**Decomposition of generating functionals and Hunt formula for $SU_q(N)$**

**LEMMA 4.9.** Let $(\pi', \eta')$ and $(\pi'', \eta'')$ be cyclic representation-cocycle pairs on $SU_q(N)$ such that $(\pi', \eta')$ lives on $SU_q(N-1)$ and $\pi''(1 - u_{NN})$ is injective. Then the following hold.

(a) The cocycle $\eta'$ vanishes on $(1 - u_{NN})K$.
(b) The set $η''((1 - u_{NN})K) = π''(1 - u_{NN})η''(K)$ is dense in $h^{π''}$.

(c) The cocycle $η' ⊕ η''$ is cyclic.

*Proof.* (a) This follows since $π'(1 - u_{NN}) = 0$ because $π'$ lives on $SU_q(N-1)$ and $1 - u_{NN} ∈ ker s_N$.

(b) By Lemma 4.1, $π''(1 - u_{NN})$ has dense range so this follows from the cyclicity of $η''$.

(c) The cyclicity of $η' ⊕ η''$ follows from that of $η'$ and $η''$ since, for $c_1, c_2 ∈ K$, by part (b) there is a sequence ($d_p$) in $K$ such that $η''((1 - u_{NN})d_p) → η''(c_2 - c_1)$, and by part (a) $η'((1 - u_{NN})d_p) = 0$ for all $p$ so

$$ (η' ⊕ η'')(c_1 + (1 - u_{NN})d_p) = \begin{pmatrix} η'(c_1) \\ η''(c_1) + η''((1 - u_{NN})d_p) \end{pmatrix} → \begin{pmatrix} η'(c_1) \\ η''(c_2) \end{pmatrix} \quad \text{as } p → ∞. \quad \square $$

**DEFINITION 4.10.** Let $N ≥ 2$. We say that a completely non-gaussian generating functional $γ$ on $SU_q(N)$ is *gf-irreducible* if the following holds: for any generating functional decomposition $γ = γ' + γ''$, if $γ'$ lives on $SU_q(N-1)$ then it is a drift.

**PROPOSITION 4.11.** Let $γ$ be a generating functional on $SU_q(N)$ for $N ≥ 2$, and let $(π, η, γ)$ be its Schürmann triple. Then $γ$ is gf-irreducible if and only if $π(1 - u_{NN})$ is injective.

*Proof.* Suppose first that $γ$ is gf-irreducible. By Theorems 4.4 and 4.5 and Propositions 4.6 and 2.8, $π$ and $η$ decompose as $π^N ⊕ π_N$ and $η^N ⊕ η_N$, where $π_N(1 - u_{NN})$ is injective, $η^N$ lives on $SU_q(N-1)$ and $η_N$ is approximately inner and so completable by a $P_N$-invariant generating functional $γ_N$. The normalised hermitian functional $γ^N := γ - γ_N$ satisfies $γ^N(c^*c) = ||η(c)||^2 - ||η_N(c)||^2 = ||η^N(c)||^2$ for all $c ∈ K$ and so is a generating functional which completes $η^N$ and thus also lives on $SU_q(N-1)$, and satisfies $γ^N + γ_N = γ$. Thus $γ^N$ is a drift and so $η^N = 0$. But $η^N$ is cyclic (since $η$ is) and so $h^{π^N} = \{0\}$ thus $π = π_N$ and so $π(1 - u_{NN})$ is injective.

Suppose conversely that $π(1 - u_{NN})$ is injective, and let $γ' + γ''$ be a generating functional decomposition of $γ$ such that $γ'$ lives on $SU_q(N-1)$. Let $(π', η', γ')$ and $(π'', η'', γ'')$ be the Schürmann triples of $γ'$ and $γ''$. Then $(π', η', γ')$ lives on $SU_q(N-1)$, so $η'$ vanishes on $(1 - u_{NN})K$ by part (a) of Lemma 4.9, also $(π' ⊕ π'', η' ⊕ η'', γ)$ is a Schürmann triple so there is an isometry $V ∈ B(h^π; h^{π'} ⊕ h^{π''})$ such that $(η^{(c)}_{η''(c)}) = Vη(c)$ for all $c ∈ K$. In view of part (b) of Lemma 4.9, these together imply that $η' = 0$, so $γ'$ is a drift. Therefore $γ$ is gf-irreducible. $\square$

**DEFINITION 4.12.** A generating functional decomposition $γ = γ_1 + \dots + γ_N$ on $SU_q(N)$ is *full* if

(1) for $1 ≤ n < N$, $γ_n = \tilde{γ}_n ◦ s_{n,N}$ for a generating functional $\tilde{γ}_n$ on $SU_q(n)$, and

(2) for $n ≥ 2$, $\tilde{γ}_n$ is gf-irreducible and $P_n$-invariant.

For $n = 1$, (1) says that $γ_n$ is gaussian. Given (1), letting $(\tilde{π}_n, \tilde{η}_n, \tilde{γ}_n)$ be $\tilde{γ}_n$'s Schürmann triple, so that $(π_n := \tilde{π}_n ◦ s_{n,N}, η_n := \tilde{η}_n ◦ s_{n,N}, γ_n := \tilde{γ}_n ◦ s_{n,N})$ is $γ_n$'s Schürmann triple, the condition (2) is equivalent to (2)'. $π_n(1 - u_{nn})$ is injective and $γ_n$ is $P_N$-invariant, by Proposition 4.11 (since $π_n(1 - u_{nn}) = \tilde{π}_n(1 - u_{nn}^n)$), and the compatibility of the family of projections (Proposition 3.4).

**LEMMA 4.13.** If a generating functional $γ$ on $SU_q(N)$ has a full decomposition $γ_1 + \dots + γ_N$ then, in terms of each $γ_n$'s Schürmann triple $(π_n, η_n, γ_n)$,

(a) $π_1 ⊕ \dots ⊕ π_N$ is a full (representation) decomposition, and

(b) the cocycle $η_1 ⊕ \dots ⊕ η_N$ is cyclic.
**Proof.** Let $\gamma = \gamma_1 + \cdots + \gamma_N$ be such a decomposition. For each $n$ denote by $(\tilde{\pi}_n, \tilde{\eta}_n, \tilde{\gamma}_n)$ the induced Schürmann triple on $SU_q(n)$, noting that for $n = 2, \dots, N$, $\tilde{\gamma}_n$ is gf-irreducible and, by (2)', the operator $\pi_n(1 - u_{nn})$ is injective and $\gamma_n$ is $P_N$-invariant, in particular (a) holds.

(b) For $N=1$ there is nothing to prove. Suppose therefore that the proposition holds for $N=K-1$ where $K \ge 2$, and that a generating functional $\gamma$ on $SU_q(K)$ has a full decomposition $\gamma = \gamma_1 + \cdots + \gamma_K$. In the above tilde notations, note that for $k=1, \cdots, K-1$, $(\tilde{\pi}_k := \tilde{\pi}_k \circ s_{k,K-1}, \tilde{\eta}_k := \tilde{\eta}_k \circ s_{k,K-1}, \tilde{\gamma}_k := \tilde{\gamma}_k \circ s_{k,K-1})$ is a cyclic Schürmann triple (since $(\pi_k, \eta_k, \gamma_k)$ is) and set $\tilde{\gamma}^K := \tilde{\gamma}_1 + \cdots + \tilde{\gamma}_{K-1}$, noting that this generating functional decomposition is full because $\tilde{\gamma}_k = \tilde{\gamma}_k \circ s_{k,K-1}$ for each $k$ and, for $k=2, \cdots, K-1$, $\tilde{\gamma}_k$ is gf-irreducible and $P_k$-invariant. Therefore, by the induction hypothesis, $\tilde{\eta}_1 + \cdots + \tilde{\eta}_{K-1}$ is cyclic which means that $\eta_1 \oplus \cdots \oplus \eta_{K-1}$ is cyclic and so, by part (c) of Lemma 4.9, $\eta_1 \oplus \cdots \oplus \eta_K = (\eta_1 \oplus \cdots \oplus \eta_{K-1}) \oplus \eta_K$ is too. Hence (b) follows by induction. $\square$

**THEOREM 4.14.** Every generating functional $\gamma$ on $SU_q(N)$ has a unique full decomposition.

**Proof. Existence.** Let $\gamma$ be a generating functional on $SU_q(N)$ and let $(\pi, \eta, \gamma)$ be its Schürmann triple. By Theorem 4.4, $\pi$ has a full decomposition $\pi_1 \oplus \cdots \oplus \pi_N$; let $\eta_1 \oplus \cdots \oplus \eta_N$ be the corresponding decomposition of $\eta$. By Theorems 4.5 and 4.7, $\eta_n$ lives on $SU_q(n)$ for each $n$ and, for $n=2, \cdots, N$, $\eta_n$ is approximately inner and thus completable by a $P_N$-invariant generating functional $\gamma_n$, so $\gamma_n$ also lives on $SU_q(n)$. Moreover, letting $(\tilde{\pi}, \tilde{\eta}, \tilde{\gamma})$ be the induced Schürmann triple on $SU_q(n)$, $\tilde{\pi}_n(1-u_{nn}^n)$ equals $\pi_n(1-u_{nn})$ and so is injective, thus $\tilde{\gamma}_n$ is gf-irreducible by Proposition 4.11. Now the functional $\gamma_1 := \gamma - (\gamma_2 + \cdots + \gamma_N)$ is hermitian and normalised, and satisfies $\gamma_1(c^*c) = \|\eta(c)\|^2 - (\|\eta_2(c)\|^2 + \cdots + \|\eta_N(c)\|^2) = \|\eta_1(c)\|^2$ for all $c \in K$ and so is a generating functional which completes $\eta_1$; moreover it is gaussian because $\pi_1$ is. It follows that $\gamma_1 + \cdots + \gamma_N$ is a full decomposition of $\gamma$.

**Uniqueness.** Let $\gamma_1 + \cdots + \gamma_N$ and $\gamma'_1 + \cdots + \gamma'_N$ be full decompositions of a generating functional $\gamma$ on $SU_q(N)$. Set $\pi := \pi_1 \oplus \cdots \oplus \pi_N$ and $\eta := \eta_1 \oplus \cdots \oplus \eta_N$ where, for each $n$, $(\pi_n, \eta_n, \gamma_n)$ is $\gamma_n$'s Schürmann triple — and do likewise for $\gamma'_1, \cdots, \gamma'_N$. Since $\gamma_1 = \gamma - (\gamma_2 + \cdots + \gamma_N)$ and for $n \ge 2$, $\gamma_n \circ P_N = \gamma_n$ and $\gamma_n(c^*c) = \|\eta_n(c)\|^2$ for $c \in K$, and likewise for $\gamma'_1, \cdots, \gamma'_N$, uniqueness follows once it is verified that $\|\eta_n(\cdot)\| = \|\eta'_n(\cdot)\|$ for $n \ge 2$. By Lemma 4.13, $\pi := \pi_1 \oplus \cdots \oplus \pi_N$ and $\pi' := \pi'_1 \oplus \cdots \oplus \pi'_N$ are full (representation) decompositions and $(\pi, \eta, \gamma)$ and $(\pi', \eta', \gamma)$ are cyclic Schürmann triples. Therefore there is a unitary operator $U \in B(h^\pi; h^{\pi'})$ such that $\eta' = U\eta(\cdot)$ and $\pi' = U\pi(\cdot)U^*$. The full decomposition $\pi = \pi_1 \oplus \cdots \oplus \pi_N$ evidently induces a full decomposition, say $\pi_1^U \oplus \cdots \oplus \pi_N^U$, of $\pi'$; the resulting decomposition $\eta' = \eta_1^U \oplus \cdots \oplus \eta_N^U$ satisfies $\|\eta_n^U(\cdot)\| = \|\eta_n(\cdot)\|$ for each $n$. Thus, by the uniqueness part of Theorem 4.4, for each $n$, $\pi_n^U = \pi'_n$ so $\eta_n^U = \eta'_n$, thus $\|\eta'_n(\cdot)\| = \|\eta_n(\cdot)\|$ as required. $\square$

Combining the theorems of this section with Theorem 3.6 and Remarks 2.18 and 2.4, we deduce our main result.

**THEOREM 4.15 (Hunt formula for $SU_q(N)$).** Let $\gamma$ be a generating functional on $SU_q(N)$. Then there is a unique decomposition $\gamma = \gamma_D + \gamma_G + \gamma_{NG}$, in which $\gamma_D$ is a drift, and $\gamma_G$ and $\gamma_{NG}$ are $P_N$-invariant generating functionals which are respectively gaussian and completely non-gaussian. Moreover, the following hold.

(1) $\gamma_G$ and $\gamma_D$ are uniquely parameterised by a matrix in $M_{N-1}(\mathbb{R})_+$ and vector in $\mathbb{R}^{N-1}$.

(2) $\gamma$ has a unique full decomposition $\gamma_1 + \cdots + \gamma_N$, and if $(\pi_n, \eta_n, \gamma_n)$ is $\gamma_n$'s Schürmann triple for each $n$ then $(\pi := \pi_1 \oplus \cdots \oplus \pi_N, \eta := \eta_1 \oplus \cdots \oplus \eta_N, \gamma)$ is $\gamma$'s Schürmann triple.
(3) $\gamma_{NG} = \text{pw-lim}_{t \to 1^-} \omega_{\xi(t)} \circ \pi_R \circ P_N$ where $\pi_R$ is the non-gaussian remainder of $\pi$ and $\xi(t) := -\pi_2(1 - tu_{22})^{-1}\eta_2(u_{22}) \oplus \cdots \oplus \pi_N(1 - tu_{NN})^{-1}\eta_N(u_{NN})$.

The realisation of $\gamma_{NG}$ in (3) is analogous to that of $\gamma_L$ in the classical Hunt formula (1.1) given in the remark following Proposition 2.8.

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

### 5. FROM PARAMETRIZATION BY $h^\pi$ TO QUASI-INNERNESS

Given a gf-irreducible generating functional on $SU_q(N)$, with Schürmann triple $(\pi, \eta, \gamma)$, by Proposition 4.11 and Lemma 2.28 we know that $\pi(1 - u_{NN})$ is injective and so $\eta$ is determined by its value $\eta(u_{NN})$. One may therefore ask which vectors of the representation space $h^\pi$ arise in this way. In case $N = 2$ every vector does, so the cocyles are parameterised by $h^\pi$ ([23, Theorem 2.8]; [21, Theorem 3.3]). We now show this to be false for $N = 3$; the argument extends to higher values of $N$. The section ends with an indication of a positive counterpart to this, namely a quasi-innerness property of completely non-gaussian cocycles/$\pi$-$\varepsilon$-derivations.

**PROPOSITION 5.1.** There is a representation $\pi$ of $SU_q(3)$ and vector $\xi$ in $h^\pi$ such that $\pi(1 - u_{33})$ is injective but $\eta(u_{33}) \neq \xi$ for every $\pi$-$\varepsilon$-cocycle $\eta$.

*Proof.* Following Woronowicz, we write the generators $u_{jk}$ of $SU_q(2)$ as

$$ \begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \end{bmatrix} = \begin{bmatrix} \alpha & -q\gamma^* \\ \gamma & \alpha^* \end{bmatrix}. $$

Let $\rho$ be the irreducible representation of $SU_q(2)$ on $\ell^2(\mathbb{Z}_+)$ defined, in terms of the standard orthonormal basis $(e_n)_{n \ge 0}$ by

$$ \rho(\alpha) : e_n \mapsto \sqrt{1 - q^{2n}} e_{n-1} \quad \text{and} \quad \rho(\gamma) : e_n \mapsto q^n e_n $$

(where $e_{-1} := 0$). For $k = 1, 2$, set $\rho_k := \rho \circ r_k$ for the CQG epimorphisms $r_k : SU_q(3) \to SU_q(2)$ given by

$$ r_1 : [u_{jk}] \mapsto \begin{bmatrix} \alpha & -q\gamma^* & 0 \\ \gamma & \alpha^* & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad r_2 : [u_{jk}] \mapsto \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & -q\gamma^* \\ 0 & \gamma & \alpha^* \end{bmatrix} $$

(so $r_1 = s_3$). Then $\rho_1$ and $\rho_2$ are representations of $SU_q(3)$ and so, setting $\pi := \rho_1 \star \rho_2$,

$$ [\pi(u_{jk})]_{j,k} = \left[ \sum_{i=1}^{3} \rho_1(u_{ji}) \otimes \rho_2(u_{ik}) \right]_{j,k} = \begin{bmatrix} \rho(\alpha) \otimes I & -q\rho(\gamma)^* \otimes \rho(\alpha) & q^2\rho(\gamma)^* \otimes \rho(\gamma)^* \\ \rho(\gamma) \otimes I & \rho(\alpha)^* \otimes \rho(\alpha) & -q\rho(\alpha)^* \otimes \rho(\gamma)^* \\ 0 & I \otimes \rho(\gamma) & I \otimes \rho(\alpha)^* \end{bmatrix}. $$

Now $\pi(1 - u_{33}) = I \otimes \rho(1 - \alpha^*)$ is injective because $\rho(1 - \alpha^*)$ is. Suppose for a contradiction that there is a $\pi$-$\varepsilon$-cocycle $\eta$ satisfying $\eta(u_{33}) = e_0 \otimes e_0$. Since $\pi(u_{31}) = 0$ and $\rho(\alpha)e_0 = 0$, Relation (2.12c) for $j = 1 = k$ implies that

$$ (I \otimes \rho(1 - \alpha^*))\eta(u_{11}) = (I - \pi(u_{33}))\eta(u_{11}) = (I - \pi(u_{11}))\eta(u_{33}) = \rho(1 - \alpha)e_0 \otimes e_0 = e_0 \otimes e_0. $$
For $n \ge 0$, set $a_n := \langle e_0 \otimes e_n, \eta(u_{11}) \rangle$. Then, since $\rho(\alpha^*)e_n = \sqrt{1-q^{2(n+1)}}e_{n+1}$,

$$e_0 = \sum_{n \ge 0} a_n (I - \rho(\alpha^*)) e_n = \sum_{n \ge 0} a_n (e_n - \sqrt{1 - q^{2(n+1)}} e_{n+1}) \\ = a_0 e_0 + \sum_{n \ge 1} (a_n - a_{n-1} \sqrt{1 - q^{2n}}) e_n.$$

Thus $a_0 = 1$ and, for $n \ge 1$, $|a_n|^2 = \prod_{k=1}^n (1 - q^{2k})$. Therefore, since $\sum |a_n|^2 \le \|\eta(u_{11})\|^2 < \infty$, $\prod_{k=1}^n (1 - q^{2k}) \to 0$ as $n \to \infty$ so $\sum q^{2k}$ diverges and we have our contradiction. $\square$

This leaves us with the question, which vectors in $\mathfrak{h}^\pi$ may occur as values $\eta(u_{NN})$ for a cocycle $\eta$. Every element in the dense subspace $\text{ran } \pi(1 - u_{NN})$ occurs; and the collection of cocycles determined by them is precisely the set of coboundaries. Indeed, for $\xi' = -\pi(1 - u_{NN})\xi$, by the contraction operator lemma we have (see Theorem 4.7) the following pointwise convergence:

$$-\pi \circ (\text{id} - \iota \circ \varepsilon)(\cdot)\pi(1 - t u_{NN})^{-1}\xi' \to \pi \circ (\text{id} - \iota \circ \varepsilon)(\cdot)\xi = \eta_{\pi, \xi} \quad \text{as } t \to 1^{-},$$

and the identity $\eta_{\pi, \xi}(u_{NN}) = \pi(u_{NN} - 1)\xi = \xi'$.

**PROPOSITION 5.2.** Let $(\xi(\lambda))$ be a net in $\mathfrak{h}^\pi$. Then the net of coboundaries $(\eta_{\pi, \xi(\lambda)})$ converges pointwise on $SU_q(N)$ provided that it converges on $u_{jj}$ for $1 \le j \le N$.

*Proof.* This follows from Remark 2.26 since, for $j \ne k$, setting $l := \max(j, k)$. Relations (2.6a) or (2.6b) imply that

$$\begin{align*}
\eta_{\pi, \xi(\lambda)}(u_{jk}) &= \pi(u_{jk})\xi(\lambda) = \pi(1 - qu_{ll})^{-1}\pi(1 - qu_{ll})\pi(u_{jk})\xi(\lambda) \\
&= -\pi(1 - qu_{ll})^{-1}\pi(u_{jk})\pi(u_{ll} - 1)\xi(\lambda) = -\pi(1 - qu_{ll})^{-1}\pi(u_{jk})\eta_{\pi, \xi(\lambda)}(u_{ll}). \quad \square
\end{align*}$$

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

**THEOREM 5.3.** Let $\pi$ be a completely non-gaussian representation of $SU_q(N)$, and let $(\overline{\mathfrak{h}^\pi}, J)$ denote the completion of $\mathfrak{h}^\pi$ with respect to the norm $\|\cdot\| : \xi \mapsto (\sum_{j=1}^N \|\pi(1 - u_{jj})\xi\|^2)^{1/2}$. Then a net $(\xi(\lambda))$ in $\mathfrak{h}^\pi$ is $\|\cdot\|$-Cauchy if and only if the corresponding net of $\pi$-\varepsilon-coboundaries $(\eta_{\pi, \xi(\lambda)})$ converges pointwise. Moreover, the following hold.

(1) There is a unique operator $\overline{\pi} : K \to B(\overline{\mathfrak{h}^\pi}; \mathfrak{h}^\pi)$ which 'extends' the representation $\pi$ in the sense that it satisfies

$$\overline{\pi}(ac) = \pi(a)\overline{\pi}(c) \quad \text{and} \quad \overline{\pi}(c)J = \pi(c) \quad (a \in SU_q(N), c \in K).$$

(2) The prescription $\chi \mapsto \eta_{\overline{\pi}, \chi} := (a \mapsto \overline{\pi}(a - \varepsilon(a)1)\chi)$ defines a linear isomorphism from $\overline{\mathfrak{h}^\pi}$ to the space of $\pi$-\varepsilon-cocycles.

There is also a unique operator $\overline{\pi} : K_2 \to B(\overline{\mathfrak{h}^\pi})$ such that

$$\overline{\pi}(c^*c) = \overline{\pi}(c)^*\overline{\pi}(c) \quad \text{and} \quad J^*\overline{\pi}(e)J = \pi(e) \quad (c \in K, e \in K_2).$$

This has the property: for all $\chi \in \overline{\mathfrak{h}^\pi}$, the generating functional $\omega_\chi \circ \overline{\pi} \circ P_N$ completes $(\pi, \eta_{\overline{\pi}, \chi})$.
## 6. THE CASE OF $U_q(N)$

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

Since $SU_q(N+1) \ge U_q(N) \ge \mathbb{T}^N$, it follows from Remarks 3.7 that $U_q(N)$ has the same gaussian generating functionals as $SU_q(N+1)$ and a hermitian projection $P'$ for $U_q(N)$ compatible with that of $SU_q(N+1)$ is the one corresponding to the following choice of basis extension:

$$E' = \{t_N(d_n) : 2 \le n \le N\} \cup \{t_N(d_{N+1}) = (2i)^{-1}(D^{-1} - D^{-1^*})\}.$$

**THEOREM 6.1.** $U_q(N)$ does not have property (GC), unless $N=1$.

*Proof.* The reasoning used in the proof of the $SU_q(N)$ counterpart (Theorem 3.3) applies. By part (d) of Lemma 3.1, the basis extension $E'$ again consists of elements whose commutators lie in $K_3$, and $\dim K/K_2 = N \ge 2$ unless $N=1$ so Corollaries 2.13 and 2.12 again apply. □

Since the (NAI) property is hereditary (Proposition 2.23) and $SU_q(N+1)$ has it (Theorem 4.8), $U_q(N)$ does too.

**THEOREM 6.2.** $U_q(N)$ has property (NAI), and thus also (LK).

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

## REFERENCES

[1] D. Applebaum, *Lévy Processes and Stochastic Calculus*, Cambridge University Press, Cambridge 2004.

[2] B. Das, U. Franz, A. Kula, A. Skalski, Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups, *Infin. Dimens. Anal. Quantum Probab. Relat. Top.* **281** (2018) no. 3, 1850017, 36 pp.

[3] S. Dascalescu, C. Nastasescu, S. Raianu, *Hopf algebras. An introduction*, Pure and Applied Mathematics **235** Marcel Dekker, New York 2001.

[4] M.S. Dijkhuizen, T.H. Koornwinder, CQG algebras: A direct algebraic approach to compact quantum groups, *Lett. Math. Phys.* **32** (1994), 315–330.

[5] U. Franz, Lévy processes on quantum groups and dual groups, In: *Quantum independent increment processes. II*, pp. 161–257, Lect. Notes Math. **1866**, Springer, Berlin 2006.

[6] — — —, Independence and Lévy processes in quantum probability, In: U. Franz, A. Skalski, *Noncommutative Mathematics for Quantum Systems*, Cambridge University Press, 2016.

[7] U. Franz, A. Freslon, A. Skalski, Connectedness and Gaussian parts for compact quantum groups, J. Geom. Phys. **184** (2023), Paper No. 104710, 18 pp.
[8] U. Franz, M. Gerhold, A. Thom, On the Lévy-Khinchin decomposition of generating functionals, *Commun. Stoch. Anal.* **9** (2015) no. 4, 529–544.

[9] U. Franz, A. Kula, J.M. Lindsay, M. Skeide, Hunt's formula for $SU_q(N)$ and $U_q(N)$, *Preprint*, arXiv:2009.06323 [math.OA]

[10] E. Hewitt, K. Ross, *Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact abelian groups*, Die Grundlehren der mathematischen Wissenschaften, Band **152** Springer Verlag, New York-Berlin 1970.

[11] G.A. Hunt, Semigroups of measures on Lie groups, *Trans. Amer. Math. Soc.* **81** (1956), 264–293.

[12] A. Klimyk, K. Schmüdgen, *Quantum groups and their representations*, Springer, 1997.

[13] H.T. Koelink, On $^*$-representations of the Hopf $^*$-algebra associated with the quantum group $U_q(n)$, *Compositio Math.* **77**(2) (1991), 199–231.

[14] J.M. Lindsay, A.G. Skalski, Quantum stochastic convolution cocycles I, *Ann. Inst. H. Poincaré, Probab. Statist.* **41** (2005) no. 3, 581–604.

[15] P.-A. Meyer, *Quantum probability for probabilists*, Lect. Notes Math. **1538** Springer-Verlag, Berlin 1993.

[16] S. Neshveyev, L. Tuset, *Compact quantum groups and their representation categories*, Société Mathématique de France, Paris 2013.

[17] M. Rosso, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif, *Duke Math. J.* **61** (1990) no. 1, 11–40.

[18] K. Sato, *Lévy Processes and Infinitely Divisible Distributions*, Cambridge University Press, Cambridge 1999.

[19] M. Schürmann, Gaussian states on bialgebras, In: L. Accardi and W. von Waldenfels, eds., *Quantum Probability and Applications V*, pp. 347–367, Lect. Notes Math. **1442** Springer, 1990.

[20] — — —, *White noise on bialgebras*, Lect. Notes Math. **1544**, Springer, 1993.

[21] M. Schürmann, M. Skeide, Infinitesimal generators on the quantum group $SU_q(2)$, *Infin. Dimens. Anal. Quantum Probab. Relat. Top.* **1** (1998), 573–598.

[22] M. Schürmann, M. Skeide, S. Volkwardt, Transformations of quantum Lévy processes on Hopf algebras, *Commun. Stoch. Anal.* **4** (2010), 553–577.

[23] M. Skeide, *The Lévy-Khinchine formula for the quantum group $SU_q(2)$*, PhD thesis, Heidelberg 1994. Available at http://web.unimol.it/skeide/.

[24] — — —, Hunt's formula for $SU_q(2)$ – a unified view, *Open Syst. Inf. Dyn.* **6** (1999) no. 1, 1–27.

[25] T. Timmermann, *An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond*, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich 2008.

[26] S.L. Woronowicz, Compact matrix pseudogroups, *Commun. Math. Phys.* **111** (1987), 613–665.

[27] — — —, Twisted $SU(2)$ group. An example of a non-commutative differential calculus, *Publ. Res. Inst. Math. Sci.* **23** (1987), 117–181.

[28] — — —, Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups, *Invent. Math.* **93** (1988) no. 1, 35–76.

[29] — — —, *Compact quantum groups*, Symétries quantiques (Les Houches, 1995), pp. 845–884, North-Holland, Amsterdam 1998.

UNIVERSITÉ DE FRANCHE-COMTÉ, CNRS, LMB, BESANÇON, FRANCE

Email address: uwe.franz@univ-fcomte.fr

INSTITUTE OF MATHEMATICS, UNIVERSITY OF WROCLAW, POLAND

Email address: anna.kula@math.uni.wroc.pl

DEPARTMENT OF MATHEMATICS AND STATISTICS, LANCASTER UNIVERSITY, UK

Email address: j.m.lindsay@lancaster.ac.uk

DIPARTIMENTO DI ECONOMIA, UNIVERSITÀ DEGLI STUDI DEL MOLISE, CAMPOBASSO, ITALY

Email address: skeide@unimol.it