Bimodules and universal enveloping algebras associated to SVOAs

For a vertex operator superalgebra V and $n,m\in (1/2)Z_{+}$, let $A_n\left(V\right):=V/O_n\left(V\right)$ denote the associative algebra, and $A_{n,m}\left(V\right):=V/O_{n,m}\left(V\right)$ denote the $A_n\left(V\right)-A_m\left(V\right)$-bimodule, as constructed by W. Jiang and C. Jiang [10], where $O_n\left(V\right)$ and $O_{n,m}\left(V\right)$ are specific subspaces of V. We introduce a novel representation-theoretic method for constructing subspaces $O_{n,m}\left(V\right)$ of V, similar to our previous work [8], and set $O_n\left(V\right)=O_{n,n}\left(V\right)$. We demonstrate that $O_{n,m}\left(V\right)=O_{n,m}\left(V\right)$ and $O_n\left(V\right)=O_n\left(V\right)$ through a method that is notably simpler and more straightforward compared to the approach detailed in [6] (also see [8]). Moreover, we offer a simpler definition for the bimodules $A_{n,m}\left(V\right)$, contributing towards the resolution of a conjecture proposed by Dong and Jiang [2] regarding superalgebras. Additionally, we demonstrate that the $A_n\left(V\right)-A_m\left(V\right)$-bimodule $A_{n,m}\left(V\right)$ is a quotient of $U{\left(V\right)}_{n-m}$, where $U\left(V\right)$ denotes the universal enveloping algebra of V, employing a method distinct from [6] (see also [8]), which is unified and simpler.