Towards $\ell$-conformal Galilei algebra via contraction of the conformal group

TL;DR

This paper demonstrates how the Inönü-Wigner contraction of certain conformal groups can produce a family of $ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$-conformal Galilei algebras, revealing new algebraic structures in theoretical physics.

Contribution

It introduces a method to derive $ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$-conformal Galilei algebras from conformal groups via contraction, expanding the understanding of algebraic symmetries.

Findings

Derived a family of $ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$-conformal Galilei algebras

Showed these algebras include various conformal extensions of the Galilei algebra

Connected conformal group contractions to new algebraic structures in physics

Abstract

We show that the In\"{o}n\"{u}-Wigner contraction of $so(\ell+1,\ell+d)$ with the integer $\ell>1$ may lead to algebra which contains a variety of conformal extensions of the Galilei algebra as subalgebras. These extensions involve the $\ell$-conformal Galilei algebra in $d$ spatial dimensions as well as $l$-conformal Galilei algebras in one spatial dimension with $l=3$, $5$, ..., $(2\ell-1)$.