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: 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 cannot 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, the second for q real) lead to the quantum Poincare algebra with an undeformed space rotation O(3) subalgebra.