Cohomology of finite dimensional pointed Hopf algebras

TL;DR

This paper proves finite generation of the cohomology ring for finite dimensional pointed Hopf algebras with abelian grouplike elements, using recent classification results, and explores the braided commutativity of cohomology in braided categories.

Contribution

It establishes finite generation of cohomology rings for a broad class of pointed Hopf algebras and demonstrates braided commutativity in their cohomology.

Findings

Finite generation of cohomology rings for pointed Hopf algebras with abelian grouplike groups.

Examples include Lusztig's small quantum groups and new pointed Hopf algebras.

Cohomology rings in braided categories are braided commutative.

Abstract

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig's small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra.