On simple modules over conformal Galilei algebras

TL;DR

This paper classifies simple weight modules with finite-dimensional weight spaces over conformal Galilei algebras in one spatial dimension, using oscillator representations and a functor from Heisenberg algebra modules.

Contribution

It introduces a functor linking Heisenberg algebra modules to conformal Galilei algebra modules and classifies simple weight modules with finite-dimensional weight spaces.

Findings

Constructed a functor transforming simple modules with nonzero central charge.

Described the structure of simple highest weight modules.

Classified simple weight modules with finite-dimensional weight spaces.

Abstract

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.