Intertwining operators for l-conformal Galilei algebras and hierarchy of invariant equations

TL;DR

This paper develops a representation theory for the l-conformal Galilei algebra and derives hierarchies of invariant partial differential equations, expanding understanding of nonrelativistic conformal symmetries.

Contribution

It introduces a new representation framework for g_{l}^{d} and constructs hierarchies of invariant equations based on these representations.

Findings

Derived hierarchies of invariant PDEs for d=1,2

Developed Verma modules and singular vectors for g_{l}^{d}

Extended the algebra's representation theory to nonrelativistic conformal symmetries

Abstract

l-Conformal Galilei algebra, denoted by g{l}{d}, is a non-semisimple Lie algebra specified by a pair of parameters (d,l). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g{l}{d} with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g{l}{d} and vector field representations for d = 1, 2.