Representations of ℓ-conformai Galilei algebra and hierarchy of invariant equation

Abstract

The ℓ-conformai Galilei algebra, denoted by g_ℓ(d), is a particular non-semisimple Lie algebra specified by a positive integer d and a spin value ℓ. The algebra g_ℓ(d) admits central extensions. We study lowest weight representations, in particular Verma modules, of g_ℓ(d) with the central extensions for d = 1,2. We give a classification of irreducible modules over d &equal; 1 algebras and a condition of the Verma modules over d &equal; 2 algebras being reducible. As an application of the representation theory, hierarchies of differential equations are derived. The Lie group generated by g_ℓ(d) with the central extension is a kinematical symmetry of the differential equations.