Representations of the q-deformed algebra Uq′(so4)

Abstract

We study the nonstandard q-deformation U q Ј(so 4 ) of the universal enveloping algebra U͑so 4 ͒ obtained by deforming the defining relations for skew-symmetric generators of U͑so 4 ͒. This algebra is used in quantum gravity and algebraic topology.We construct a homomorphism of U q Ј(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 homomorphism we construct irreducible finitedimensional representations of the classical type and of the nonclassical type for the algebra U q Ј(so 4 ). It is proved that for q not a root of unity each irreducible finitedimensional representation of U q Ј(so 4 ) is equivalent to one of these representations. We prove that every finite-dimensional representation of U q Ј(so 4 ) for q not a root of unity is completely reducible. It is shown how to construct ͑by using the homomorphism ͒ tensor products of irreducible representations of U q Ј(so 4 ). ͓Note that no Hopf algebra structure is known for U q Ј(so 4 ).͔ These tensor products are decomposed into irreducible constituents.

On representations of $U'_qso_n$

“…Up to a renormalization of the basis vectors, this is essentially just an extension of the results in e.g. [7], Section VI from the finite dimensional modules to the full Verma module. We give a brief outline here how this could be proved similar to our approach for U ′ q so 3 modules, see Prop.…”

“…Lemma 5.2. (see [7], Section VI) Let q not be a root of unity, let m = [λ] and let (v µ ) be the basis of weight vectors of V m as in Lemma 5.1. The action of the generators on v µ , with…”