Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem

Abstract

We introduce Kleisli convolution representations for groups, rings, and algebras. We show that their representation categories are equivalent to the usual ones for groups and finite-dimensional algebras, but for rings only recover modules whose underlying Abelian groups are free of finite rank. We also apply the Kleisli convolution representation to provide a partial answer to the Margolis--Sakurai version of the modular isomorphism problem: If $G$ is a finite $p$-group with $D_3(G)=1$ and $\gcd (m,c_G!)=1$, then ${\mathbb{F}}_{p^m}G\cong {\mathbb{F}}_{p^m}H$ implies $G\cong H$, where $c_G$ is the number of geometric connected components of $\mathrm{Aut}({\overline{\mathbb{F}}}_{p}G)$.

arXiv comment58 pages; some typos have been corrected