Anick resolution for the free unitary quantum group

A resolution P of the counit of the Hopf -algebra O(Uₙ⁺) of representative functions on van Daele and Wang's free unitary quantum group Uₙ⁺ in terms of free O(Uₙ⁺)-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 Uₙ⁺ that enabled the aforementioned authors to find their resolution. Especially, P relies neither on the resolution for Oₙ⁺ obtained by Collins, H\"artel and Thom nor the one for SL₂(q) found by Hadfield and Kr\"ahmer. Rather, as shown in the present article, the recursion defining the Anick resolution of the counit of O(Uₙ⁺) can be solved in closed form. That suggests a potential strategy for determining the cohomologies of arbitrary easy quantum groups.