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 $L$, $\theta$ 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 $\theta$. 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 $\theta$-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 $\theta$.

29 pages. To appear in Journal of Algebra. We divide the article arXiv:1910.07126 into 3 parts for publication. This is the first part

References in corpus (5)

• Z_3 symmetry and W_3 algebra in lattice vertex operator algebras
• $C_2$-cofiniteness of the vertex algebra $V_L^{+}$ when $L$ is a non-degenerate even lattice
• Classification of irreducible modules of the vertex algebra $V_L^{+}$ when $L$ is a nondegenerate even lattice of an arbitrary rank
• Rationality of the vertex algebra $V_L^{+}$ when $L$ is a nondegenerate even lattice of arbitrary rank
• Representations of the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$