Twisted representations of ℕ-graded vertex algebras over a good field

Abstract

Let [Formula: see text] be an [Formula: see text]-graded vertex algebra over a field [Formula: see text], and let [Formula: see text] be an automorphism of [Formula: see text] with a finite order [Formula: see text]. Assume that the field [Formula: see text] contains a [Formula: see text]th primitive root of unity. We construct two associative algebras, [Formula: see text] and [Formula: see text], and investigate their properties and interconnections. As applications, we establish the following results: (1) If [Formula: see text] is finite-dimensional, then [Formula: see text] is also finite-dimensional; (2) If [Formula: see text] is strongly finitely generated, then [Formula: see text] is stably noetherian; (3) If [Formula: see text] is strongly finitely generated and [Formula: see text] is an irreducible admissible [Formula: see text]-twisted [Formula: see text]-module, then the endomorphism algebra [Formula: see text] is finite-dimensional.