The cohomology ring of some Hopf algebras

TL;DR

This paper studies the algebraic and cohomological structures of certain connected Hopf algebras of dimension p^3 over a field of characteristic p, providing detailed descriptions and classifications.

Contribution

It classifies the cohomology rings of all connected Hopf algebras of dimension p^3, extending the understanding of their algebraic structures.

Findings

Identified the Morita type for each algebra

Determined the cohomology ring structures in most cases

Provided a comprehensive classification of these Hopf algebras

Abstract

Let p be a prime, and k be a field of characteristic p. We investigate the algebra structure and the structure of the cohomology ring for the connected Hopf algebras of dimension p^3, which appear in the classification obtained in [V.C. Nguyen, L.-H. Wang and X.-T. Wang, Classification of connected Hopf algebras of dimension p^3, J. Algebra 424 (2015), 473-505]. The list consists of 23 algebras together with two infinite families. We identify the Morita type of the algebra, and in almost all cases this is sufficient to clarify the structure of the cohomology ring.