R-matrix realization of two-parameter quantum group U_{r,s}(gl_n)

TL;DR

This paper develops an RTT approach to two-parameter quantum groups, establishing their algebraic structure, properties of quantum determinants, and constructing higher Casimir elements, thereby deepening understanding of their algebraic and representation-theoretic features.

Contribution

It introduces an RTT realization of U_{r,s}(gl_n), analyzes the quantum determinant, and constructs higher Casimir elements for two-parameter quantum groups.

Findings

Quantum determinant is quasi-central in Fun(GL_{r,s}(n)).

U_{r,s}(gl_n) is dual to Fun(GL_{r,s}(n)).

Constructed higher Casimir elements in U_{r,s}(gl_n).

Abstract

We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group Fun(GL_{r,s}(n)) and the quantum enveloping algebra U_{r,s}(gl_n) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det_{r,s}T is a quasi-central element in Fun(GL_{r,s}(n)) generalizing earlier results of Dipper-Donkin and Du-Parshall-Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra U_{r,s}(R) is identified to U_{r,s}(gl_n) as the dual algebra. We then construct n-1 quasi-central elements in U_{r,s}(R) which are analogues of higher Casimir elements in U_q(gl_n).