Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras
Isabelle Baraquin, Uwe Franz, Malte Gerhold, Anna Kula, Mariusz, Tobolski

TL;DR
This paper constructs finite free resolutions for certain free quantum groups and Hopf algebras, enabling the computation of Hochschild and bialgebra cohomologies, and extends known resolutions using algebraic and categorical techniques.
Contribution
It introduces a new finite free resolution for free unitary quantum groups and universal cosovereign Hopf algebras, generalizing previous results and applying categorical methods.
Findings
Computed Hochschild cohomology for these Hopf algebras.
Extended resolutions to a broader class of quantum groups.
Connected resolutions with monoidal equivalences and Galois objects.
Abstract
We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with 1-dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter-Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, H\"artel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then we make use of the fact that the free unitary quantum groups and some of its non-Kac versions can be realized as a glued free product of a (non-Kac)…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research
