Braided cohomology of quasi-triangular bialgebras and braided Morita invariance

Abstract

We introduce the braided cochain complex and the braided cohomology of braided coalgebras in linear monoidal categories, and compare the braided cohomology of braided coalgebras living in different linear monoidal categories using relative morphisms. The symmetric cohomology was introduced for groups by Staic, and was generalized to cocommutative Hopf algebras by Shiba, Sanada, and the second author. This cohomology involves degreewise actions of the symmetric groups on a cochain complex, which come from the usual symmetric monoidal structure on the category of modules. We generalize this framework by dealing with arbitrary linear monoidal categories, and by replacing symmetries with braidings defined merely on an object. We first give a convenient description of relative morphisms, and apply this result to prove that the braided cochain complex of quasi-triangular bialgebras is a braided Morita invariant under a certain condition, which is automatically satisfied in the finite-dimensional case.