Bimodules and universal enveloping algebras associated to VOAs

Abstract

Given a vertex operator algebra V, we construct a family of associative algebras ${{\cal A}_n}(V)$ and a family of ${{\cal A}_n}(V) - {{\cal A}_m}(V)$-bimodules ${{\cal A}_{n,m}}(V)$ for m, n ∈ ℤ₊. We prove that the algebra ${{\cal A}_n}(V)$ is identical to the algebra A_n(V) constructed by Dong, Li and Mason, and that the bimodule ${{\cal A}_{n,m}}(V)$ is identical to the bimodule A_n,m(V) constructed by Dong and Jiang. And we also prove that the A_n(V) − A_m(V)-bimodule A_n,m(V) is isomorphic to U(V)_n−m/U(V) ^−m−1_n−m , where U(V)_k is the subspace of degree k of the ℤ-graded universal enveloping algebra U(V) associated to V and U(V) ^l_k is some subspace of U(V)_k.