Real forms of complex quantum anti-de-Sitter algebra U q(Sp(4; [formula omitted])) 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 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 Poincaré algebra with an undeformed space rotation O(3) subalgebra.