QUANTUM ALGEBRA DEFORMING MAPS, CLEBSCH-GORDAN-COEFFICIENTS, COPRODUCTS, R AND U MATRICES

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch-Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang-Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2 x j of SU (2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.