Endomorphism property of vertex operator algebras over arbitrary fields

In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field $\mathbb{F}$, with $\text{Char}(\mathbb{F})\mathrm{\ne }2$. 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.