Modular representations of Heisenberg algebras

TL;DR

This paper extends the known minimal dimension of faithful modules for Heisenberg algebras from characteristic zero to prime characteristic fields, and classifies irreducible modules over algebraically closed fields, with applications to matrix theory.

Contribution

It generalizes Burde's result on minimal faithful module dimension to prime characteristic fields and provides classifications of irreducible modules over algebraically closed fields.

Findings

Minimal faithful module dimension is n+2 in prime characteristic, except for (p,n)=(2,1).

Constructs various faithful irreducible modules in prime characteristic.

Classifies irreducible modules over algebraically closed fields.

Abstract

Let $F$ be be an arbitrary field and let $h(n)$ be the Heisenberg algebra of dimension $2n+1$ over $F$. It was shown by Burde that if $F$ has characteristic 0 then the minimum dimension of a faithful $h(n)$-module is $n+2$. We show here that his result remains valid in prime characteristic $p$, as long as $(p,n)\neq (2,1)$. We construct, as well, various families of faithful irreducible $h(n)$-modules if $F$ has prime characteristic, and classify these when $F$ is algebraically closed. Applications to matrix theory are given.