LAUSR

We study the first and second cohomology groups of the ∗-algebras of the universal unitary and orthogonal quantum groups UF+ and OF+. This provides valuable information for constructing and classifying Levy processes on these quantum groups, as pointed out by Schurmann. In the case when all eigenvalues of F∗F are distinct, we show that these ∗-algebras have the properties (GC), (NC) and (LK) introduced by Schurmann and studied recently by Franz, Gerhold and Thom. In the degenerate case F = Id, we show that they do not have any of these properties. We also compute the second cohomology group of Ud+ with trivial coefficients — H2(U d+, ????ℂ????)≅ℂd2−1 — and construct an explicit basis for the corresponding second cohomology group for Od+ (whose dimension was known earlier, thanks to the work of Collins, Hartel and Thom).