Generalized Gauss decomposition of trigonometric R-matrices

Artikel i vetenskaplig tidskrift, 1995

The general formula for the universal R-matrix for quantified nontwisted affine algebras, obtained by the first and third authors, is applied to zero central charge, highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonometric R-matrix in tenser product of these modules. In particular, A(1)((1)) current realization of the universal R-matrix is presented. It gives a new universal presentation for the trigonometric R-matrix with a parameter in tenser product of U-q(sl(2))-Verma modules. Detailed analysis of a scalar factor arising in finite-dimensional representations of the universal R-matrix for any U-q(g) is given. We interpret this scalar factor as a multiplicative bilinear form on highest weight polynomials of irreducible representations and express this form in terms of infinite q-shifted factorials.