The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1)

Abstract Let V L be the vertex algebra associated to a non-degenerate even lattice L, θ the automorphism of V L induced from the −1-isometry of L, and V L + the fixed point subalgebra of V L under the action of θ. In this series of papers we classify the irreducible weak V L + -modules and show that any irreducible weak V L + -module is isomorphic to a weak submodule of some irreducible weak V L -module or to a submodule of some irreducible θ-twisted V L -module. In this paper (Part 1), we show that when the rank of L is 1, every non-zero weak V L + -module contains a non-zero M ( 1 ) + -module, where M ( 1 ) + is the fixed point subalgebra of the Heisenberg vertex operator algebra M ( 1 ) under the action of θ.