Jet modules for the centerless Virasoro-like algebra

TL;DR

This paper classifies and characterizes jet modules over the centerless Virasoro-like algebra, connecting them to modules over related infinite-dimensional Lie algebras and using polynomial module techniques.

Contribution

It provides a detailed classification of irreducible and indecomposable jet modules for the algebra, linking them to modules over Block type algebras.

Findings

Irreducible jet modules are characterized by finite-dimensional modules over sl_2.

Indecomposable jet modules are described via modules over the algebra B_+.

Reduction of jet modules to modules over infinite-dimensional Lie algebras.

Abstract

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a $2$-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over $\mathfrak{sl}_2$. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of \cite{BB, BZ}. Consequently, indecomposable jet modules are described using modules over the algebra $\BB_+$, which is the "positive part" of a Block type algebra studied first by \cite{DZ} and recently by \cite{IM, I}).