On the finite generation of the cohomology of bosonizations

TL;DR

This paper demonstrates that certain bosonizations and smash products of braided Hopf algebras with finite-dimensional Hopf algebras have finitely generated cohomology, extending previous results and providing new examples.

Contribution

It generalizes the finite generation of cohomology to a broader class of bosonizations and smash products of Hopf algebras, including new cases involving Nichols algebras.

Findings

Bosonizations of Nichols algebras have finitely generated cohomology.

Extended results to smash products of Hopf algebras.

Provided new examples of finite generation in cohomology.

Abstract

We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.