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

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.

14 pages

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