Classification of simple weight modules over the 1-spatial ageing algebra

TL;DR

This paper classifies simple weight modules, including Harish-Chandra modules, over the 1-spatial ageing algebra age(1), revealing unique modules with infinite-dimensional weight spaces and extending to modules over the Schrödinger algebra.

Contribution

It provides the first complete classification of simple weight modules over age(1), especially those with infinite-dimensional weight spaces, and constructs new modules over the Schrödinger algebra.

Findings

Classified all simple weight modules over age(1).

Identified modules with infinite-dimensional weight spaces.

Constructed new simple modules over the Schrödinger algebra.

Abstract

In this paper we use Block's classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra age(1). Most of these modules have infinite dimensional weight spaces and so far the algebra age(1) is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrodinger algebra S that have a simple age(1)-submodule thus constructing many new simple weight S-modules.