L\'evy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

We study the first and second cohomology groups of the ${}^{\ast }$-algebras of the universal unitary and orthogonal quantum groups $U_F^{+}$ and $O_F^{+}$. This provides valuable information for constructing and classifying L'evy processes on these quantum groups, as pointed out by Sch"urmann. In the case when all eigenvalues of $F^{\ast }F$ are distinct, we show that these ${}^{\ast }$-algebras have the properties (GC), (NC), and (LK) introduced by Sch"urmann 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 – $H^2(U_d^{+},{}_{ϵ}{\mathbb{C}}_{ϵ})\cong {\mathbb{C}}^{d^2-1}$ – 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, H"artel and Thom).