Irreducible actions and compressible modules

TL;DR

This paper explores the structure of modules under linear operators, extending known results on irreducible actions, and provides criteria for critical compressibility in various algebraic contexts such as group and Lie actions.

Contribution

It extends results on irreducible Hopf actions to more general operator algebras and establishes conditions for critical compressibility in group, grading, and Lie actions.

Findings

Bound on the dimension of the self-injective hull of a critically compressible module

Extension of irreducible Hopf action results to weak Hopf actions

Necessary and sufficient conditions for critical compressibility under various algebraic actions

Abstract

Any finite set of linear operators on an algebra $A$ yields an operator algebra $B$ and a module structure on A, whose endomorphism ring is isomorphic to a subring $A^B$ of certain invariant elements of $A$. We show that if $A$ is a critically compressible left $B$-module, then the dimension of its self-injective hull $A$ over the ring of fractions of $A^B$ is bounded by the uniform dimension of $A$ and the number of linear operators generating $B$. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions.