[Colored solutions of Yang-Baxter equation from representations of U_{q}gl(2)]

TL;DR

This paper explores the structure and representations of a multiparameter quantum group U_q(gl(2)), leading to new colored solutions of the Yang-Baxter equation through explicit construction of the universal R-matrix.

Contribution

It provides an explicit construction of the multiparameter universal R-matrix as a quantum double intertwiner without Reshetikhin's transformation, and classifies representation types for generic and root of unity q.

Findings

Derived standard and nonstandard colored R-matrix solutions

Classified representation theory into two types based on q

Explicitly described Hopf algebra structure and maps

Abstract

We study the Hopf algebra structure and the highest weight representation of a multiparameter version of $U_{q}gl(2)$. The commutation relations as well as other Hopf algebra maps are explicitly given. We show that the multiparameter universal ${\cal R}$ matrix can be constructed directly as a quantum double intertwiner, without using Reshetikhin's transformation. An interesting feature automatically appears in the representation theory: it can be divided into two types, one for generic $q$, the other for $q$ being a root of unity. When applying the representation theory to the multiparameter universal ${\cal R}$ matrix, the so called standard and nonstandard colored solutions $R(\mu,\nu; {\mu}', {\nu}')$ of the Yang-Baxter equation is obtained.