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 ∗∗-algebras of the universal unitary and orthogonal quantum groups U+FU+F and O+FO+F. 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 F∗FF∗F are distinct, we show that these ∗∗-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case F=IdF=Id, we show that they do not have any of these properties. We also compute the second cohomology group of U+dU+d with trivial coefficients — H2(U+d,𝜖ℂ𝜖)≅ℂd2−1 — and construct an explicit basis for the corresponding second cohomology group for O+d (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).
Communicated by M. Bozejko
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