Constructing Coherently G-invariant Modules

TL;DR

This paper develops a method to construct G-invariant modules over path algebras with reductive group actions, using the associated quiver of a smash product algebra, and applies it to generate algebraic semi-invariants.

Contribution

It introduces a new construction of G-invariant modules via the quiver of the smash product algebra, expanding tools for studying group actions on quiver representations.

Findings

Describes the quiver of the smash product algebra $kQ\# k[M_G]^*$.

Constructs G-invariant representations of quivers.

Generates algebraic semi-invariants from G-invariant modules.

Abstract

Let $G$ be a reductive group acting on a path algebra $kQ$ as automorphisms. We assume that $G$ admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver $Q_G$ of the smash product algebra $kQ\# k[M_G]^*$, where $M_G$ is the associated algebraic monoid of $G$. We use $Q_G$-representations to construct $G$-invariant representations of $Q$. As an application, we construct algebraic semi-invariants on the quiver representation spaces from those $G$-invariant representations.