A Schur-Weyl type duality for twisted weak modules over a vertex algebra

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${\rm Aut} V$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${\mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${\mathscr A}_{\alpha}(G,{\mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${\mathscr S}$ where ${\mathscr A}_{\alpha}(G,{\mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{\mathscr S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on ${\mathscr S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module.Comment: 13 page

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 22 (Part 22)

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$ symmetry 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. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$. In this paper (Part $2$), we show that there exists an irreducible $M(1)^{+}$-submodule in any non-zero weak $V_{L}^{+}$-module and we compute extension groups for $M(1)^{+}$.Comment: 42 pages. To appear in Journal of the Mathematical Society of Japan. We divide the article arXiv:1910.07126 into 3 parts for publication. This is the 2nd par