Associative Algebras and Intertwining Operators

Abstract

Let V be a vertex operator algebra and $A^{\infty }(V)$ and $A^{N}(V)$ for $N\in {\mathbb {N}}$ the associative algebras introduced by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras. arXiv:2009.00262). For a lower-bounded generalized V-module W, we give W a structure of graded $A^{\infty }(V)$-module and we introduce an $A^{\infty }(V)$-bimodule $A^{\infty }(W)$ and an $A^{N}(V)$-bimodule $A^{N}(W)$. We prove that the space of (logarithmic) intertwining operators of type $\left( {\begin{array}{c}W_{3}\\ W_{1}W_{2}\end{array}}\right)$ for lower-bounded generalized V-modules $W_{1}$, $W_{2}$ and $W_{3}$ is isomorphic to the space $\text{ Hom}_{A^{\infty }(V)}(A^{\infty }(W_{1})\otimes _{A^{\infty }(V)}W_{2}, W_{3})$. Assuming that $W_{2}$ and $W_{3}'$ are equivalent to certain universal lower-bounded generalized V-modules generated by their $A^{N}(V)$-submodules consisting of elements of levels less than or equal to $N\in {\mathbb {N}}$, we also prove that the space of (logarithmic) intertwining operators of type $\left( {\begin{array}{c}W_{3}\\ W_{1}W_{2}\end{array}}\right)$ is isomorphic to the space of $\text{ Hom}_{A^{N}(V)}(A^{N}(W_{1})\otimes _{A^{N}(V)}\Omega _{N}^{0}(W_{2}), \Omega _{N}^{0}(W_{3}))$.