Finite-dimensional vertex algebra modules over fixed point commutative subalgebras

TL;DR

This paper investigates the structure of finite-dimensional vertex algebra modules over fixed point subalgebras of commutative algebras with derivations, establishing conditions under which modules over the fixed point algebra extend to twisted modules over the larger algebra.

Contribution

It demonstrates that under certain algebraic conditions, finite-dimensional modules over fixed point subalgebras can be uniquely extended to twisted modules over the entire algebra.

Findings

Finite-dimensional modules over fixed point subalgebras can be extended to twisted modules.

Extension is possible when the algebra is generated by a single element and is a Galois extension.

Provides a framework for understanding module structures in vertex algebra theory.

Abstract

Let $A$ be a connected commutative $\C$-algebra with derivation $D$, $G$ a finite linear automorphism group of $A$ which preserves $D$, and $R=A^G$ the fixed point subalgebra of $A$ under the action of $G$. We show that if $A$ is generated by a single element as an $R$-algebra and is a Galois extension over $R$ in the sense of M. Auslander and O. Goldman, then every finite-dimensional vertex algebra $R$-module has a structure of twisted vertex algebra $A$-module.