Cohomology and Coquasi-bialgebras in the category of Yetter-Drinfeld Modules

TL;DR

This paper establishes conditions under which finite-dimensional Hopf algebras with the dual Chevalley Property are quasi-isomorphic to Radford-Majid bosonizations, linking cohomology vanishing to algebraic structure.

Contribution

It proves that the vanishing of the third Hochschild cohomology in Yetter-Drinfeld modules implies a quasi-isomorphism to Radford-Majid bosonization for certain Hopf algebras.

Findings

Vanishing of third Hochschild cohomology in Yetter-Drinfeld modules occurs in key examples.

Finite-dimensional Hopf algebras with the dual Chevalley Property relate to Nichols algebras.

Conditions for quasi-isomorphism are characterized via cohomology vanishing.

Abstract

We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.