Second cohomology groups of the Hopf$^*$-algebras associated to universal unitary quantum groups

TL;DR

This paper computes the first and second cohomology groups of $^*$-algebras linked to universal quantum unitary groups, extending previous results to more general cases using infinite-dimensional representations.

Contribution

It extends the computation of cohomology groups to universal quantum unitary groups of not necessarily Kac type, utilizing infinite-dimensional representations.

Findings

Computed first and second cohomology groups for these quantum groups

Extended previous results to broader class of quantum groups

Used infinite-dimensional representations to construct cocycles

Abstract

We compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to use infinite-dimensional representations to construct the cocycles.