All real forms of Uq(sl(4;C)) and D = 4 conformal quantum algebras

Abstract

The star operations and reality conditions for the complex quantum algebra Uq(sl(4;C)) providing real quantum algebras Uq(o(6-k,k)) k = 0, 1, 2, 3 and Uq(su(3,1)) are classified. Standard and non-standard star operations are considered. It appears that only four choices of real forms (one with mod q mod =1, three with q real) provide real Hopf algebra Uq(su(2,2)) approximately=Uq(o(4,2)) describing D = 4 conformal quantum algebras. The authors show that only the antipode-extended Cartan-Weyl basis of Uq(sl(4;C)) permits one to define real q-deformed D = 4 conformal algebra generators. In order to obtain the real D = 4 Weyl algebra as the Hopf subalgebra of Uq (su(2,2)) only the non-standard real forms can be employed.