Universal T-matrix for Twisted Quantum gl(N)

Abstract

The Universal T-matrix is the capstone of the structure that consists of a quantum group and its dual, and the central object from which spring the T-matrices (monodromies) of all the associated integrable models. A closed expression is obtained for the case of multiparameter (twisted) quantum $gl(N)$. The factorized nature of standard quantum groups, that allows the explicit expression for $U\hskip-1mm T $ to be obtained with relative ease, extends to some nonstandard quantum groups, such as those based on $A_n^{(2)}$, and perhaps to all. The paper is mostly concerned with parameters in general position, but the extension to roots of unity is also explored, in the case of $g\ell(N)$. The structure of the dual is now radically different, and an interesting generalization of the $q$-exponential appears in the formulas for the Universal T- and R-matrices. The projection to quantum $sl(N)$ is simple and direct; this allows, in particular, to apply recent results concerning deformations of twisted $gl(N)$ to the semisimple quotient.