Locally Frobenius algebras and Hopf algebras

TL;DR

This paper introduces the concept of locally Frobenius algebras, explores their module categories, and discusses their properties and examples, extending the theory of Frobenius and nearly Frobenius algebras to more general algebraic structures.

Contribution

It develops a new framework for locally Frobenius algebras, analyzing their modules, minimal ideals, and monoidal structures, and connects to existing theories of Frobenius and Hopf algebras.

Findings

Category of coherent modules is abelian with enough projectives and injectives.

Finite dimensional modules form an abelian category but are not coherent.

Examples include group algebras of locally finite groups.

Abstract

We develop a theory of \emph{locally Frobenius algebras} which are colimits of certain directed systems of Frobenius algebras. A major goal is to obtain analogues of the work of Moore \& Peterson and Margolis on \emph{nearly Frobenius algebras} and \emph{$P$-algebras} which was applied to graded Hopf algebras such as the Steenrod algebra for a prime. Such locally Frobenius algebras are coherent and in studying their modules we are naturally led to focus on coherent and finite dimensional modules. Indeed, the category of coherent modules over locally Frobenius algebra $A$ is abelian with enough projectives and injectives since $A$ is injective relative to the coherent modules; however it only has finite limits and colimits. The finite dimensional modules also form an abelian category but finite dimensional modules are never coherent. The minimal ideals of a locally Frobenius algebra are precisely the ones which are isomorphic to coherent simple modules; in particular it does not contain a copy of any finite dimensional simple module so it is not a Kasch algebra. We discuss possible versions of stable module categories for such algebras. We also discuss possible monoidal structures on module categories of a locally Frobenius Hopf algebra: for example tensor products of coherent modules turn out to be pseudo-coherent. Examples of locally Frobenius Hopf algebras include group algebras of locally finite groups, already intensively studied in the literature.