Representations of the q-deformed algebra U-q '(SO4)

We study the nonstandard q-deformation U-q'(so(4)) of the universal envelop ing algebra U(so(4)) obtained by deforming the defining relations for skew- symmetric generators of U(so(4)). This algebra is used in quantum gravity a nd algebraic topology. We construct a homomorphism phi of U-q'(so(4)) to th e certain nontrivial extension of the Drinfeld-Jimbo quantum algebra U-q(sl (2))(x2) and show that this homomorphism is an isomorphism. By using this h omomorphism we construct irreducible finite-dimensional representations of the classical type and of the nonclassical type for the algebra U-q'(so(4)) . It is proved that for q not a root of unity each irreducible finite-dimen sional representation of U-q'(so(4)) is equivalent to one of these represen tations. We prove that every finite-dimensional representation of U-q'(so(4 )) for q not a root of unity is completely reducible. It is shown how to co nstruct (by using the homomorphism phi) tensor products of irreducible repr esentations of U-q'(so(4)). [Note that no Hopf algebra structure is known f or U-q'(so(4)).] These tensor products are decomposed into irreducible cons tituents. (C) 2001 American Institute of Physics.