The cohomology ring of some 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.