The cohomology of certain Hopf algebras associated with 𝑝-groups

Abstract

We study the cohomology H ∗ ( A ) = Ext A ∗ ⁡ ( k , k ) H^*(A)=\operatorname {Ext}_A^*(k,k) of a locally finite, connected, cocommutative Hopf algebra A A over k = F p k=\mathbb {F}_p . Specifically, we are interested in those algebras A A for which H ∗ ( A ) H^*(A) is generated as an algebra by H 1 ( A ) H^1(A) and H 2 ( A ) H^2(A) . We shall call such algebras semi-Koszul . Given a central extension of Hopf algebras F → A → B F\rightarrow A\rightarrow B with F F monogenic and B B semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for A A to be semi-Koszul. Special attention is given to the case in which A A is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- p p lower central series of a p p -group. We show that the algebras arising in this way from extensions by Z / ( p ) \mathbb {Z}/(p) of an abelian p p -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 p p -groups, and it is shown that these are all semi-Koszul for p ≥ 5 p\geq 5 .