2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69174The lattice vertex operator algebra $V_L$ associated to a positive definite even lattice $L$ has an automorphism of order 2 lifted from -1-isometry of $L$. We prove that for the fixed point vertex operator algebra $V_L^+$, any $\Z_{\geq0}$-graded weak modules is completely reducible.30 pages, No figureQuantum AlgebraRepresentation Theorytext