Partial classification of cuspidal simple modules for Virasoro-like algebra

TL;DR

This paper classifies certain simple modules over the Virasoro-like algebra, showing they originate from a known construction, thereby advancing understanding of its representation theory.

Contribution

It proves that all quasi-finite simple modules of the Virasoro-like algebra are derived from Larsson-Shen's construction, using a backward induction approach.

Findings

All quasi-finite simple modules are from Larsson-Shen's construction.

The proof employs backward induction strategy.

Provides a classification of modules for the Virasoro-like algebra.

Abstract

Let $\mathfrak P$ be the Lie algebra of Hamiltonian vector fields on the torus, which is also known as the Virasoro-like algebra, a special kind of the so-called Block type Lie algebra. And let $\mathfrak A$ be the Laurent polynomial algebra in two variables. In this paper, by following S.E. Rao's strategy of "backward induction", we prove that any quasi-finite simple $(\mathfrak A,\mathfrak P)$-module has to come from Larsson-Shen's construction.