Generalized oscillator representations of the twisted Heisenberg-Virasoro algebra

TL;DR

This paper develops a comprehensive framework for constructing and classifying simple modules over the twisted Heisenberg-Virasoro algebra, extending oscillator representations and analyzing module simplicity conditions.

Contribution

It introduces generalized oscillator representations and new criteria for module simplicity over the twisted Heisenberg-Virasoro algebra, advancing the understanding of its module structure.

Findings

Classification of simple modules over the Heisenberg algebra

Conditions for simplicity of Whittaker modules over 

Criteria for tensor products of modules to be simple

Abstract

In this paper, we first obtain a general result on sufficient conditions for tensor product modules to be simple over an arbitrary Lie algebra. We classify simple modules with a nice property over the infinite-dimensional Heisenberg algebra ${\H}$, and then obtain a lot of simple modules over the twisted Heisenberg-Virasoro algebra $\LL$ from generalized oscillator representations of $\LL$ by extending these $\H$-modules. We give the necessary and sufficient conditions for Whittaker modules over $\LL$ (in the more general setting) to be simple. We use the "shifting technique" to determine the necessary and sufficient conditions for the tensor products of highest weight modules and modules of intermediate series over $\LL$ to be simple. At last we establish the "embedding trick" to obtain a lot more simple $\LL$-modules.