ALL REAL FORMS OF UQ(SL(4 C)) AND D = 4 CONFORMAL QUANTUM ALGEBRAS

The star operations and reality conditions for the complex quantum alg ebra U(q)(sl(4;C)) providing real quantum algebras U(q)(o(6 - k,k))k = 0, 1,2,3 and U(q)(su(3, 1)) are classified. Standard and non-standard star operations are considered. It appears that only four choices of real forms (one with absolute value of q = 1, three with q real) provi de real Hopf algebra U(q)(su(2, 2)) congruent-to U(q)(o(4, 2)) describ ing D = 4 conformal quantum algebras. We show that only the antipode-e xtended Cartan-Weyl basis of U(q)(sl(4; C)) permits to define real q-d eformed D = 4 conformal algebra generators. In order to obtain the rea l D = 4 Weyl algebra as Hopf subalgebra of U(q)(su(2, 2)) only the non -standard real forms can be employed.