Harish-Chandra Theorem for Two-parameter Quantum Groups

Abstract

This paper is devoted to investigating the centre of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ via establishing the Harish-Chandra homomorphism. Based on the Rosso form and the representation theory of weight modules, we prove that when rank $\mathfrak{g}$ is even, the Harish-Chandra homomorphism is an isomorphism, and in particular, the centre of the quantum group $\breve{U}_{r,s}(\mathfrak{g})$ of the weight lattice type is a polynomial algebra $\mathbb{K}[z_{\varpi_1},\cdots,z_{\varpi_n}]$, where canonical central elements $z_\lambda \; (\lambda \in \Lambda^+)$ are turned out to be uniformly expressed. For rank $\mathfrak{g}$ to be odd, we figure out a new invertible extra central generator $z_*$, which doesn't survive in $U_q(\mathfrak g)$, then the centre of $\breve{U}_{r,s}(\mathfrak{g})$ contains $\mathbb{K}[z_{\varpi_1},\cdots,z_{\varpi_n}]\otimes_\mathbb K\mathbb K[z_*^{\frac{1}{\ell}}, z_*^{-\frac{1}{\ell}}]$, where $\ell=2$, except $\ell=4$ for $D_{2k+1}$.