Twisted bimodules and universal enveloping algebras associated to VOAs

For any vertex operator algebra V, finite automorphism g of V of order T and $m,n\in (1/T)Z_{+}$, we construct a family of associative algebras $A_{g,n}\left(V\right)$ and $A_{g,n}\left(V\right)-A_{g,m}\left(V\right)$-bimodules $A_{g,n,m}\left(V\right)$ from the point of view of representation theory. We prove that the algebra $A_{g,n}\left(V\right)$ is identical to the algebra $A_{g,n}\left(V\right)$ constructed by Dong, Li and Mason, and that the bimodule $A_{g,n,m}\left(V\right)$ is identical to $A_{g,n,m}\left(V\right)$ which was constructed by Dong and Jiang. We also prove that the $A_{g,n}\left(V\right)-A_{g,m}\left(V\right)$-bimodule $A_{g,n,m}\left(V\right)$ is isomorphic to $U{\left(V\right[g\left]\right)}_{n-m}/U{\left(V\right[g\left]\right)}_{n-m}^{-m-1/T}$, where $U{\left(V\right[g\left]\right)}_k$ is the subspace of degree k of the $(1/T)Z$-graded universal enveloping algebra $U\left(V\right[g\left]\right)$ of V with respect to g and $U{\left(V\right[g\left]\right)}_k^l$ is some subspace of $U{\left(V\right[g\left]\right)}_k$. And we show that all these bimodules $A_{g,n,m}\left(V\right)$ can be defined in a simpler way.