On the Harish-Chandra Homomorphism for Quantum Superalgebras

TL;DR

This paper develops a Harish-Chandra homomorphism for quantum superalgebras, describing its image explicitly, and proves an isomorphism between the center of the algebra and a ring of exponential super-invariants, extending classical results.

Contribution

It introduces the Harish-Chandra homomorphism for quantum superalgebras and establishes a Harish-Chandra type theorem relating the center to super-invariants, with an explicit description and basis construction.

Findings

Explicit description of the Harish-Chandra homomorphism image.

Isomorphism between the center of $ ext{U}_q( ext{g})$ and super-invariant ring.

Basis of the center constructed using quasi-R-matrix.

Abstract

In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra $\mathrm{U}_q(\mathfrak{g})$ associated with a simple basic Lie superalgebra $\mathfrak{g}$ and give an explicit description of its image. We use it to prove that the center of $\mathrm{U}_q(\mathfrak{g})$ is isomorphic to a subring of the ring $J(\mathfrak{g})$ of exponential super-invariants in the sense of Sergeev and Veselov, establishing a Harish-Chandra type theorem for $\mathrm{U}_q(\mathfrak{g})$. As a byproduct, we obtain a basis of the center of $\mathrm{U}_q(\mathfrak{g})$ with the aid of quasi-R-matrix.