Representations of quantum lattice vertex algebras

Let Q be a non-degenerate even lattice, let $V_Q$ be the lattice vertex algebra associated to Q, and let $V_Q^{\eta }$ be a quantum lattice vertex algebra ([10]). In this paper, we prove 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.