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

2025/05/24 by Chen, Kaixiang, Hu, Naihong, Wang, Hengyi
#17B37 (Primary) 17B35 #81R50 (Secondary) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.18599

Abstract

This paper studies the centre of the multi-parameter quantum group U_q,G(\mathfrakg) introduced by Okado and Yamane, where \mathfrakg is a complex simple Lie algebra, and all parameters are 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)≅ (U⁰_\flat)^W is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the (U⁰_\flat)^W of the two-parameter quantum group U_r,s(\mathfrakg).

Citations

• Explicit generators of the centre of the quantum group
• Natural Elements of Center of Generalized Quantum Groups
• On the center of the quantized enveloping algebra of a simple Lie algebra
• Central elements for quantum affine algebras and affine Macdonald's operators

Related

• On the center of two-parameter (v,t)-quantum groups
• Explicit generators of the centre of the quantum group
• Explicit generators and relations for the centre of the quantum group
• Centers of Generalized Quantum Groups
• Complex semisimple quantum groups and representation theory