Twisted bimodules and associative algebras associated to VOAs

Let $V$ be a vertex operator algebra, $g$ be an automorphism of $V$ of order $T$, and $m, n \in (1/T)\mathbb{N}$. In~\cite{HX2} and~\cite{HXX1}, it was shown respectively that the associative algebra $A_{g,n}(V)$ constructed by Dong, Li, and Mason~\cite{DLM3}, and the $A_{g,n}(V)\!-\!A_{g,m}(V)$-bimodule $A_{g,n,m}(V)$ constructed by Dong and Jiang~\cite{DJ2}, are both isomorphic to certain subquotients of $U(V[g])$, where $U(V[g])$ denotes the universal enveloping algebra of $V$ with respect to $g$. In this paper, we give a unified and concise proof of these isomorphisms.