L\'evy-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 $U_F^+$ 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 $F^*F$ are distinct, we show that these $^*$-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 -- $H^2(U_d^+,{}_\epsilon\Bbb{C}_\epsilon)\cong \Bbb{C}^{d^2-1}$ -- 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).