The Cohomology of certain Hopf Algebras Associated with p-Groups

Justin Mauger

AMS Classification numbers: 16E40, 16S37

2033 Sheridan Road
Northwestern University 
Evanston, IL 60208
justin@math.northwestern.edu


In this paper, we study the cohomology H^*(A)=Ext_A^*(k,k) of a locally
finite, connected, cocommutative Hopf algebra A over k=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 \emph{semi-Koszul}.  Given a central
extension of Hopf algebras $F\lra A\lra B$ 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 $\z$ of an abelian $p$-group are semi-Koszul.  Explicit
calculations are carried out for algebras arising from rank 2 $p$-groups.