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

Let V be an N-graded vertex algebra over a field F, and let g be an automorphism of V with a finite order T. Assume that the field F contains a Tth primitive root of unity. We construct two associative algebras, A(g)(V ) and R-g(V ), and investigate their properties and interconnections. As applications, we establish the following results: (1) If R-g(V ) is finite-dimensional, then A(g)(V ) is also finite-dimensional; (2) If V is strongly finitely generated, then A(g)(V ) is stably noetherian; (3) If V is strongly finitely generated and M is an irreducible admissible g-twisted V-module, then the endomorphism algebra End(V)(M) is finite-dimensional.