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 $z_{\hat{I}}»(\hat{I}»\in \hat{I}{›}^{+})$ are turned out to be uniformly expressed. For rank $\mathfrak{g}$ to be odd, we figure out a new invertible extra central generator $z_{\ast }$, which doesn't survive in $U_q(\mathfrak{g})$, then the centre of ${\breve{U}}_{r,s}(\mathfrak{g})$ contains $\mathbb{K}[z_{{\varpi }_1},\cdots ,z_{{\varpi }_n}]{\otimes }_{\mathbb{K}}\mathbb{K}[z_{\ast }^{\frac{1}{\ell }},z_{\ast }^{-\frac{1}{\ell }}]$, where $\ell =2$, except $\ell =4$ for $D_{2k+1}$.

20 pages. The final version. Accepted in Forum Mathematicum, 2025, currently ahead of print