Associative algebras and intertwining operators

Let $V$ be a vertex operator algebra and $A^{\mathrm{\infty }}(V)$ and $A^N(V)$ for $N\in \mathbb{N}$ the associative algebras introduced by the author in [H5]. For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of graded $A^{\mathrm{\infty }}(V)$-module and we introduce an $A^{\mathrm{\infty }}(V)$-bimodule $A^{\mathrm{\infty }}(W)$ and an $A^N(V)$-bimodule $A^N(W)$. We prove that the space of (logarithmic) intertwining operators of type $\left(\genfrac{}{}{0px}{}{W_3}{W_1{W}_2}\right)$ for lower-bounded generalized $V$-modules $W_1$, $W_2$ and $W_3$ is isomorphic to the space ${\mathrm{hom}}_{A^{\mathrm{\infty }}(V)}(A^{\mathrm{\infty }}(W_1){\otimes }_{A^{\mathrm{\infty }}(V)}{W}_2,W_3)$. Assuming that $W_2$ and $W_3^{\mathrm{\prime }}$ 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 $\left(\genfrac{}{}{0px}{}{W_3}{W_1{W}_2}\right)$ is isomorphic to the space of $\hom_{A^{N}(V)}(A^{N}(W_{1})\otimes_{A^{N}(V)}\Omega_{N}^{0}(W_{2}), \Omega_{N}^{0}(W_{3}))$.