Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra

For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module $\widehat{M}^{[g]}_{B}$ constructed by the author in \cite{H-const-twisted-mod}. We prove that there exist irreducible lower-bounded generalized $g$-twisted $V$-modules by showing that there exists a maximal proper submodule of $\widehat{M}^{[g]}_{B}$ for a one-dimensional space $M$. We then give several spanning sets of $\widehat{M}^{[g]}_{B}$ and discuss the relations among elements of the spanning sets. Assuming that $V$ is a M\"{o}bius vertex superalgebra (to make sure that lowest weights make sense) and that $P(V)$ (the set of all numbers of the form $\Re(\alpha)\in [0, 1)$ for $\alpha\in \C$ such that $e^{2\pi i \alpha}$ is an eigenvalue of $g$) has no accumulation point in $\R$ (to make sure that irreducible lower-bounded generalized $g$-twisted $V$-modules have lowest weights). Under suitable additional conditions, which hold when the twisted zero-mode algebra or the twisted Zhu's algebra is finite dimensional, we prove that there exists an irreducible grading-restricted generalized $g$-twisted $V$-module, which is in fact an irreducible ordinary $g$-twisted $V$-module when $g$ is of finite order. We also prove that every lower-bounded generalized module with an action of $g$ for the fixed-point subalgebra $V^{g}$ of $V$ under $g$ can be extended to a lower-bounded generalized $g$-twisted $V$-module.