Endomorphism property of vertex operator algebras over arbitrary fields

TL;DR

This paper investigates the endomorphism structures of vertex operator algebras over arbitrary fields, establishing algebraic and finite-dimensional properties of endomorphisms and extending classical results like Schur's lemma.

Contribution

It proves that endomorphisms of irreducible modules are algebraic and finite-dimensional over any field, generalizing key properties of vertex operator algebras beyond algebraically closed fields.

Findings

Endomorphisms are algebraic over the base field.

Endomorphism spaces are finite-dimensional.

Schur's lemma holds for these algebras over arbitrary fields.

Abstract

In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field $\mathbb{F}$, with $\text{Char}(\mathbb{F}) \neq 2$. Let $V$ be a strongly finitely generated vertex operator algebra over $\mathbb{F}$, and $M$ be an irreducible admissible $V$-module. We prove that every element in $\text{End}_V(M)$ is algebraic over $\mathbb{F}$ and that $\text{End}_V(M)$ is also finite-dimensional. As an application, we prove Schur's lemma for strongly finitely generated vertex operator algebras over arbitrary algebraically closed fields, and we give a test for absolute irreducibility of $V$-modules.