Abstract
In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field F, with Char(F)≠2. Let V be a strongly finitely generated vertex operator algebra over F, and M be an irreducible admissible V-module. We prove that every element in EndV(M) is algebraic over F and that EndV(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.
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