On the Harish-Chandra Homomorphism for Quantum Superalgebras

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.