REAL FORMS OF COMPLEX QUANTUM ANTI-DE-SITTER ALGEBRA Uq(Sp(4;C)) AND THEIR CONTRACTION SCHEMES

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: Uq(O(3, 2)) (quantum D = 4 anti-de-Sitter), Uq(O(4, 1)) (quantum D = 4 de-Sitter) and Uq(O(5)). We give also two types of inner involutions of the Cartan-Chevalley basis of Uq(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 Poincaré algebra with an undeformed space rotations O(3) subalgebra. Partially supported by the Swiss National Science Foundation On leave of absence from the Institute for Theoretical Physics, University of Wroc law, ul. Cybulskiego 36, 50205 Wroc law, Poland On leave of absence from the Institute of Physics, Pedagogical University, Plac S lowiański 6, 65020 Zielona Góra, Poland