Gauss decomposition of trigonometric R-matrices

TL;DR

This paper demonstrates how the universal R-matrix for quantized affine algebras yields the Gauss decomposition of trigonometric R-matrices, providing explicit formulas and calculations for specific cases.

Contribution

It applies the universal R-matrix formula to derive the Gauss decomposition of trigonometric R-matrices for quantized affine algebras, including explicit results for $A_1^{(1)}$.

Findings

Derived explicit formulas for the Gauss decomposition of trigonometric R-matrices.

Presented calculations for the $A_1^{(1)}$ case.

Obtained new formulas for the R-matrix with a parameter in tensor products of $U_q(sl_2)$-Verma modules.

Abstract

The general formula for the universal R-matrix for quantized nontwisted affine algebras by Khoroshkin and Tolstoy is applied for zero central charge highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonomitric R-matrix in tensor product of these modules. Explicit calculations for the simplest case of $A_1^{(1)}$ are presented. As a consequence new formulas for the trigonometric R-matrix with a parameter in tensor product of $U_q(sl_2)$-Verma modules are obtained.