Classification of Harish-Chandra modules over some Lie algebras related to the Virasoro algebra

TL;DR

This paper develops a uniform classification method for Harish-Chandra modules over certain Lie algebras related to the Virasoro algebra, including new results for specific algebra classes and potential extensions to superconformal algebras.

Contribution

It introduces a comprehensive classification approach for Harish-Chandra modules over Lie algebras related to the Virasoro algebra, covering previously unclassified algebra types.

Findings

Classified modules over the Lie algebra W(ρ)[s] for s=0, 1/2

Extended classification method to Schrödinger-Virasoro Lie algebras

Provided a framework for future classifications of related superconformal algebras

Abstract

In this paper, we provide a uniform method to thoroughly classify all Harish-Chandra modules over some Lie algebras related to the Virasoro algebras. We first classify such modules over the Lie algebra $W(\varrho)[s]$ for $s=0,\frac12$. With this result and method, we can also do such works for some Lie algebras and superconformal algebras related to the Virasoro algebra, including the several kinds of Schr\"odinger-Virasoro Lie algebras, which are open up to now.