Simple restricted modules of non-zero level over a deformed Heisenberg-Virasoro algebra

TL;DR

This paper classifies simple restricted modules of a deformed Heisenberg-Virasoro algebra at non-zero level, revealing they are induced from modules of finite-dimensional solvable Lie algebras, despite the algebra lacking a triangular decomposition.

Contribution

It provides a classification of simple restricted modules for a non-triangular deformed Heisenberg-Virasoro algebra at non-zero level, using induction from finite-dimensional solvable Lie algebra modules.

Findings

All simple restricted modules of non-zero level are induced from finite-dimensional solvable Lie algebra modules.

The algebra's $ ext{Z}$-gradation enables classification despite lacking a triangular decomposition.

The modules are explicitly constructed via induction methods.

Abstract

We study representations of a deformed Heisenberg-Virasoro algebra that does not admit a triangular decomposition. Despite this, its $\mathbb{Z}$-gradation allows the classification of simple restricted modules. We show that all such modules of non-zero level arise via induction from simple modules of finite-dimensional solvable Lie algebras.