Real Forms of Complex Quantum Anti de Sitter Algebra $U_q (Sp(4,C))$ and their Contraction Schemes

TL;DR

This paper classifies real forms of the complex quantum anti-de Sitter algebra $U_q(Sp(4,C))$, explores their involutions, and outlines contraction schemes leading to quantum Poincaré algebras with undeformed spatial rotations.

Contribution

It introduces new involutions and contraction schemes for quantum anti-de Sitter algebra, expanding understanding of its real forms and their algebraic limits.

Findings

Identified four types of involutions leading to different real quantum Lie algebras.

Outlined twelve contraction schemes for quantum D=4 anti-de Sitter algebra.

Found that only two contraction schemes yield quantum Poincaré algebra with undeformed O(3) rotations.

Abstract

We describe four types of inner involutions of the Cartan-Weyl basis providing (for $ |q|=1$ and $q$ real) three types of real quantum Lie algebras: $U_{q}(O(3,2))$ (quantum D=4 anti-de-Sitter), $U_{q}(O(4,1))$ (quantum D=4 de-Sitter) and $U_{q}(O(5))$. We give also two types of inner involutions of the Cartan-Chevalley basis of $U_{q}(Sp(4;C))$ which can not be extended to inner involutions of the Cartan-Weyl basis. We outline twelve contraction schemes for quantum D=4 anti-de-Sitter algebra. All these contractions provide four commuting translation generators, but only two (one for $ |q|=1$, second for $q$ real) lead to the quantum \po algebra with an undeformed space rotations O(3) subalgebra.