1 REPRESENTATIONS OF THE q-DEFORMED ALGEBRA U ′ q (so4)

We study the nonstandard q-deformation U ′ q (so 4) of the universal enveloping 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 and algebraic topology. We construct a homomor-phism φ of U ′ q (so 4) to the certain nontrivial extension of the Drinfeld–Jimbo quantum algebra U q (sl 2) ⊗2 and show that this homomorphism is an isomorphism. By using this homomor-phism 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 dimensional representation of U ′ q (so 4) is equivalent to one of these representations. 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 construct (by using the homomorphism φ) tensor products of irreducible representations of U ′ q (so 4). (Note that no Hopf algebra structure is known for U ′ q (so 4).) These tensor products are decomposed into irreducible constituents.