The Use of Quantum Algebras in Quantum Gravity

TL;DR

This paper explores how quantum algebras, specifically nonstandard q-deformed Lie algebras, are used in 2+1 anti-de Sitter quantum gravity, revealing their isomorphism to certain deformed orthogonal group algebras.

Contribution

It demonstrates the isomorphism between deformed algebras in quantum gravity and nonstandard q-deformed orthogonal Lie algebras, clarifying their algebraic structure.

Findings

Deformed algebras in quantum gravity are isomorphic to U'_q(so_n)

The algebra n is related to genus g as n=2g+2

Provides insight into algebraic structures underlying quantum observables

Abstract

After a brief survey of the appearance of quantum algebras in diverse contexts of quantum gravity, we demonstrate that the particular deformed algebras, which arise within the approach of J.Nelson and T.Regge to 2+1 anti-de Sitter quantum gravity (for space surface of genus g) and which are basic for generating the algebras of independent quantum observables, are in fact isomorphic to the nonstandard q-deformed analogues U'_q(so_n) (introduced in 1991) of Lie algebras of the orthogonal groups SO(n), with n linked to g as n=2g+2.