Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension

TL;DR

This paper classifies irreducible highest weight modules for a class of conformal Galilei algebras with central extension, constructs singular vectors explicitly, and confirms a conjecture about the Kac determinant for the Schrödinger algebra.

Contribution

It provides an explicit construction of singular vectors, complete classification of irreducible modules, and proves a conjecture on the Kac determinant for these algebras.

Findings

Complete classification of infinite-dimensional irreducible modules.

Explicit formulas for singular vectors and Kac determinants.

Verification of a conjecture for the Schrödinger algebra.

Abstract

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrodinger algebra.