Prehomogeneous modules of commutative linear algebraic groups

TL;DR

This paper classifies prehomogeneous modules of commutative linear algebraic groups over algebraically closed fields of characteristic zero, linking them to invertible elements of finite-dimensional commutative algebras and establishing finiteness conditions.

Contribution

It demonstrates that all prehomogeneous modules of such groups arise from invertible elements of commutative algebras and characterizes when the classification is finite.

Findings

Every prehomogeneous module of a commutative linear algebraic group is associated with an algebra of invertible elements.

The number of equivalence classes of these modules is finite if and only if the group's corank is at most 5.

Provides a complete description of the structure of prehomogeneous modules in this setting.

Abstract

Let $A$ be a finite dimensional commutative associative algebra with unit over an algebraically closed field of characteristic zero. The group $G(A)$ of invertible elements is open in $A$ and thus $A$ has a structure of a prehomogeneous $G(A)$-module. We show that every prehomogeneous module of a commutative linear algebraic group appears this way. In particular, the number of equivalence classes of prehomogeneous $G$-modules is finite if and only if the corank of $G$ is at most $5$.