# Two classes of non-weight modules over the twisted Heisenberg-Virasoro algebra

## TL;DR

This paper constructs new classes of non-weight modules over the twisted Heisenberg-Virasoro algebra, establishes their irreducibility conditions, and demonstrates their novelty compared to existing modules.

## Contribution

It introduces two new classes of non-weight modules, provides criteria for their irreducibility and isomorphism, and applies these results to specific induced modules.

## Key findings

- New irreducible non-weight modules constructed
- Necessary and sufficient conditions for irreducibility established
- Application to specific induced modules with irreducibility criteria

## Abstract

In the present paper, we construct two classes of non-weight modules $\Omega(\lambda,\alpha,\beta)\otimes\mathrm{Ind}(M)$ and $\mathcal{M}\big(V,\Omega(\lambda,\alpha,\beta)\big)$ over the twisted Heisenberg-Virasoro algebra, which are both associated with the modules $\Omega(\lambda,\alpha,\beta)$. We present the necessary and sufficient conditions under which modules in these two classes are irreducible and isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules $\mathrm{Ind}_{\underline y,\lambda}(\C_{RS})$ and $\mathrm{Ind}_{\underline z,\lambda}(\C_{PQ})$ over the twisted Heisenberg-Virasoro algebra and then apply the established results to give irreducible conditions for $\mathrm{Ind}_{\underline y,\lambda}(\C_{RS})$ and $\mathrm{Ind}_{\underline z,\lambda}(\C_{PQ})$.