Lévy–Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

Abstract

We study the first and second cohomology groups of the [Formula: see text]-algebras of the universal unitary and orthogonal quantum groups [Formula: see text] and [Formula: see text]. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of [Formula: see text] are distinct, we show that these [Formula: see text]-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case [Formula: see text], we show that they do not have any of these properties. We also compute the second cohomology group of [Formula: see text] with trivial coefficients — [Formula: see text] — and construct an explicit basis for the corresponding second cohomology group for [Formula: see text] (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).