$R$-Matrix Realization of Two-Parameter Quantum Group $U_{r,s}(\mathfrak {gl}_n)$

Abstract

We provide a Faddeev–Reshetikhin–Takhtajan’s RTT approach to the quantum group $\mathrm{Fun}(\mathrm{GL}_{r,s}(n))$ and the quantum enveloping algebra $U_{r,s}(\mathfrak {gl}_n)$ corresponding to the two-parameter $R$-matrix. We prove that the quantum determinant ${\det }_{r,s}T$ is a quasi-central element in $\mathrm{Fun}(\mathrm{GL}_{r,s}(n))$ 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 $U_{r,s}(\mathfrak {gl}_n)$ as the dual algebra. We then construct $n-1$ quasi-central elements in $U_{r,s}(R)$ which are analogs of higher Casimir elements in $U_q(\mathfrak {gl}_n)$.

Keywords

Quantum groups / Determinants / Casimir elements / Yang–Baxter equations