Representations of Classical Lie Algebras from their Quantum Deformations

TL;DR

This paper explores how quantum deformations of Lie algebras, specifically the Poincaré, de Sitter, and anti-de Sitter groups, can be used to construct classical Lie algebra representations, revealing new links between quantum and classical structures.

Contribution

It introduces a method to derive classical Lie algebra representations from their quantum deformations using irrational functions of deformed generators.

Findings

Constructed Poincaré algebra from de Sitter groups via irrational functions.

Generalized anti-deformation to SO(p+2,q) and SO(p+1,q+1) cases.

Provided explicit representations for the U_q(so(2,1)) case.

Abstract

We make use of a well-know deformation of the Poincar\'e Lie algebra in $p+q+1$ dimensions ($p+q>0$) to construct the Poincar\'e Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincar\'e Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this ``anti-deformation'' for the $SO(p+2,q)$ and $SO(p+1,q+1)$ cases with arbitrary $p$ and $q$. Similar results have been established for $q$ deformations $U_q(so(p,q))$ with small $p$ and $q$ values. Combining known results on representations of $U_q(so(p,q))$ (for $q$ both generic and a root of unity) with our ``anti-deformation'' formulae, we get representations of classical Lie algebras which depend upon the deformation parameter $q$. Explicit results are given for the simplest example (of type $A_1$) i.e. that associated with $U_q(so(2,1))$.