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We study the first and second cohomology groups of the $^*$-algebras of theuniversal unitary and orthogonal quantum groups $U_F^+$ and $O_F^+$. Thisprovides valuable information for constructing and classifying L\'evy processeson these quantum groups, as pointed out by Sch\"urmann. In the case when alleigenvalues of $F^*F$ are distinct, we show that these $^*$-algebras have theproperties (GC), (NC), and (LK) introduced by Sch\"urmann and studied recentlyby Franz, Gerhold and Thom. In the degenerate case $F=I_d$, we show that theydo not have any of these properties. We also compute the second cohomologygroup of $U_d^+$ with trivial coefficients --$H^2(U_d^+,{}_\epsilon\Bbb{C}_\epsilon)\cong \Bbb{C}^{d^2-1}$ -- and constructan explicit basis for the corresponding second cohomology group for $O_d^+$(whose dimension was known earlier thanks to the work of Collins, H\"artel andThom).

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