The cohomology of certain Hopf algebras associated with $p$-groups

Abstract:

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