Twisted modules for affine vertex algebras over fields of prime characteristic

Abstract

In this paper, twisted modules for modular affine vertex algebras Vgˆ(ℓ,0) and for their quotient vertex algebras Vgˆχ(ℓ,0) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σT=1. It is proved that 1TN-graded irreducible σ-twisted Vgˆ0(ℓ,0)-modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u(g0), where g0 is the subalgebra of σ-fixed points in g. It is also proved that when g=h is abelian, the twisted Heisenberg Lie algebra hˆ[σ] is actually isomorphic to the untwisted Heisenberg Lie algebra hˆ, unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ, irreducible σ-twisted Lhˆ(ℓ,0)-modules are explicitly classified and the complete reducibility of every σ-twisted Lhˆ(ℓ,0)-module is obtained.