Levy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

We study the first and second cohomology groups of the *-algebras of the universal unitary and orthogonal quantum groups U-F(+) and O-F(+). 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 = I-d, we show that they do not have any of these properties. We also compute the second cohomology group of U-d(+) with trivial coefficients - H-2(U-d(+), C-is an element of(is an element of)) congruent to 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, Hartel and Thom).