Representations of quantum lattice vertex algebras

Let $Q$ be a non-degenerated even lattice, let $V_Q$ be the lattice vertex algebra associated to $Q$, and let $V_Q^\eta$ be a quantum lattice vertex algebra. In this paper, we prove the equivalence between the category $V_Q$-modules and the category of $V_Q^\eta$-modules. As a consequence, we show that every $V_Q^\eta$-module is completely reducible, and the set of simple $V_Q^\eta$-modules are in one-to-one correspondence with the set of cosets of $Q$ in its dual lattice.