Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harish-Chandra Theorem for the Multi-Parameter Quantum Groups of Okado-Yamane Type

Kaixiang Chen, Naihong Hu, Hengyi Wang

Abstract

This paper is devoted to studying the centre of the multi-parameter quantum group $U_{q,G}(\mathfrak{g})$ introduced by Okado and Yamane, where $\mathfrak{g}$ is a complex simple Lie algebra, and all parameters lie in general position. We mainly establish the Harish-Chandra theorem, proving that the Harish-Chandra homomorphism is an isomorphism; in particular, we determine the centre $Z(U_{q,G})\cong (U^0_\flat)^W$ is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the $(U^0_\flat)^W$ of the two-parameter quantum group $U_{r,s}(\mathfrak{g})$.

Harish-Chandra Theorem for the Multi-Parameter Quantum Groups of Okado-Yamane Type

Abstract

This paper is devoted to studying the centre of the multi-parameter quantum group introduced by Okado and Yamane, where is a complex simple Lie algebra, and all parameters lie in general position. We mainly establish the Harish-Chandra theorem, proving that the Harish-Chandra homomorphism is an isomorphism; in particular, we determine the centre is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the of the two-parameter quantum group .

Paper Structure

This paper contains 13 sections, 25 theorems, 75 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Specialized multi-parameter quantum groups and the Rosso form
  4. Weight modules of $U_{q,G}$
  5. Injectivity and the image of Harish-Chandra homomorphism $\xi$
  6. Injective properties
  7. The image $\operatorname{Im}(\xi)$
  8. Central elements and the Harish-Chandra theorem
  9. Rosso form realization and central elements
  10. The Harish-Chandra theorem
  11. The Centre of $U_{q,G}(\mathfrak{g})$
  12. Minimal generating system $\Psi_{\min}$
  13. Presentation of $Z(U_{q,G})$

Key Result

Proposition 1

HW25X25 Let $n= \operatorname{rank} (\mathfrak{g})$, parameters $r$ and $s$ be in general position, $U=U_{r,s}(\mathfrak{g})$, and $\breve{U}=\breve{U}_{r,s}(\mathfrak{g})$ be the weight lattice type of $U$, then we have the following Harish-Chandra theorem:

Theorems & Definitions (49)

  • Proposition 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Lemma 1
  • proof
  • Definition 2
  • Proposition 4
  • Theorem 5
  • proof
  • ...and 39 more