Harish-Chandra Theorem for Two-parameter Quantum Groups
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arXiv
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| Format: | Preprint |
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2024
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| author | Hu, Naihong Wang, Hengyi |
| author_facet | Hu, Naihong Wang, Hengyi |
| contents | 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 $z_λ\; (λ\in Λ^+)$ 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 $D_{2k+1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_12793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harish-Chandra Theorem for Two-parameter Quantum Groups Hu, Naihong Wang, Hengyi Quantum Algebra 17B37 (Primary) 17B35, 81R50 (Secondary) 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 $z_λ\; (λ\in Λ^+)$ 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 $D_{2k+1}$. |
| title | Harish-Chandra Theorem for Two-parameter Quantum Groups |
| topic | Quantum Algebra 17B37 (Primary) 17B35, 81R50 (Secondary) |
| url | https://arxiv.org/abs/2402.12793 |