Classification of irreducible Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras

TL;DR

This paper classifies irreducible Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras, extending previous classifications and confirming a conjecture about their structure.

Contribution

It provides a complete classification of these modules, including bounded modules and modules of tensor fields, generalizing known results for the classical Virasoro algebra.

Findings

Classification of irreducible Harish-Chandra modules over toroidal Lie algebras.

Complete description of modules as tensor fields or highest weight types.

Confirmation of Eswara Rao's conjecture from 2004.

Abstract

In this paper, we classify the irreducible Harish-Chandra modules over the full toroidal Lie algebra, which is a natural higher-dimensional analogue of the affine-Virasoro algebra. In particular, we complete the classification of irreducible bounded modules, which were studied by Billig for non-zero level modules [Int. Math. Res. Not. 2006]. As a by-product, we also obtain the classification of irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, which was introduced by Rao-Moody [Comm. Math. Phys. 1994], thereby generalizing the well-known result of O. Mathieu [Invent. Math. 1992] for the classical Virasoro algebra. More precisely, we show that any irreducible Harish-Chandra module over the higher-dimensional Virasoro algebra turns out to be either a quotient of a module of tensor fields on a torus or a highest weight type module up to a twist of an automorphism, as conjectured by Eswara Rao in 2004.