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)

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