Anick resolution for the free unitary quantum group

Abstract

A resolution $P$ of the counit of the Hopf $\ast$-algebra $\mathcal{O}(U_n^+)$ of representative functions on van Daele and Wang's free unitary quantum group $U_n^+$ in terms of free $\mathcal{O}(U_n^+)$-modules is computed for arbitrary $n$. A different such resolution was recently found by Baraquin, Franz, Gerhold, Kula and Tobolski. While theirs has desirable properties which $P$ lacks, $P$ is still good enough to compute the (previously known) quantum group cohomology and comes instead with an important advantage: $P$ can be arrived at without the clever combination of certain results potentially very particular to $U_n^+$ that enabled the aforementioned authors to find their resolution. Especially, $P$ relies neither on the resolution for $O_n^+$ obtained by Collins, H\"artel and Thom nor the one for $SL_2(q)$ found by Hadfield and Kr\"ahmer. Rather, as shown in the present article, the recursion defining the Anick resolution of the counit of $\mathcal{O}(U_n^+)$ can be solved in closed form. That suggests a potential strategy for determining the cohomologies of arbitrary easy quantum groups.