General Construction of Nonstandard R h - matrices as Contraction Limits of R q - matrices

A class of transformations of Rq-matrices is introduced such that the q → 1 limit gives explicit nonstandard Rh-matrices. The transformation matrix is singular itself at q → 1 limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for sl(2) but also for algebras of higher dimensions. Explicit constructions are presented starting with Uq(sl(2)) and Uq(sl(3)), while choosing Rq for (fund. rep.)⊗(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the h-Borel subalgebra on the corresponding classical Borel subalgebra. For Uh(sl(2)) this map is extended to the whole algebra and compared with another one proposed by us previously. [email protected] [email protected] Permanent address: Department of Theoretical Physics, University of Madras, Guindy Campus, Madras600025, India Laboratoire Propre du CNRS UPR A.0014 The R-matrices for the fundamental representations of the nonstandard h-deformations of sl(2) and so(4)(≃ sl(2)⊗ sl(2)) were obtained [1,2] through a specific contraction of the corresponding q-deformed R-matrices. A similarity transformation of the 4 × 4 Rq-matrix for the fundamental representation of Uq(sl(2)) was performed using a transforming matrix singular itself at the q → 1 limit, but in such way that all singularities cancel out for the transformed R-matrix giving the finite nonstandard Rh-matrix. Following the previous practice [1,2], we refer to this combined process of similarity transformation and subsequent cancellation of singularities at the q → 1 limit as contraction procedure. This technique was generalized to higher dimensional algebras [3] considering again the N × N dimensional R-matrices for the fundamental representations of q-deformed sl(N), for example. Other relevant references can be found in [1-3]. We present here an operatorial generalization of this approach directly applicable to R-matrices of arbitrary dimensions. For brevity and simplicity we start with ( 2 ⊗ j) representation i.e. 2(2j+1)×2(2j+1) dimensional Rq-matrix for Uq(sl(2)). Then we will indicate possible generalizations in different directions, using Uq(sl(3)) as a particular example. The universal R-matrix for Uh(sl(2)) has been given a particularly convenient form [4,5]. For ( 2 ⊗ j) representation this reduces to Rh = 