On the Lévy-Khinchin decomposition of generating functionals

Abstract

We study several sufficient conditions for the existence of a Lévy-Khinchin decomposition of generating functionals on unital involutive algebras with a fixed character. We show that none of these conditions are equivalent and we show that such a decomposition does not always exist.2000 Mathematics Subject Classification. 43A35,81R50,60E99.

One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

“…stated for group algebras in [12,Lemma5.6], but holding more generally for unital * -algebras with a character, as shown in [5] (recall that K 1 ⊗ A K 1 denotes the tensor product of Amodules over A). Earlier remarks and simple observations imply that a functional L : A → C for which η yields the coboundary exists if and only if ϕ vanishes on the kernel of the multiplication map from…”

“…The decomposition exists for all quantum Lévy processes on classical groups and for SU q (2) ( [15]) or, more generally SU q (N ) for arbitrary N ∈ N ( [6]). For the non-existence examples we refer to the forthcoming work [5]; here we will prove, using the results of Sect. 2, that the decomposition exists in presence of α-symmetry.…”

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

“…It is noteworthy that not all quantum groups allow to decompose every generating functional into 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 problem has a solution or not, depends on the example under consideration.…”

Homological properties of quantum permutation algebras

“…Notice that Theorem 1.1 states in particular that the second cohomology group H 2 (A s (n), C) vanishes, and hence in particular A s (n) has the property called AC in [20]. The AC property is of particular interest in quantum probability and the study of Lévy processes on quantum groups: it means that all cocycles can be completed to a Schürmann triple.…”

“…The AC property is of particular interest in quantum probability and the study of Lévy processes on quantum groups: it means that all cocycles can be completed to a Schürmann triple. See [20] for details, and the recent survey [21] on these questions, where the AC property was shown for A s (n). In fact we will show that the vanishing result for the second cohomology (and hence the AC property) holds for a large class of quantum permutation algebras (quotients of A s (n)), including those representing quantum symmetry groups of finite graphs, in the sense of [4]: see Theorem 5.2.…”