Harish-Chandra theorem for two-parameter quantum groups

This paper is devoted to investigating the center of two-parameter quantum groups Ur,s⁢(g) via establishing the Harish-Chandra homomorphism. Based on the Rosso form and the representation theory of weight modules, we prove that when the rank g is even, the Harish-Chandra homomorphism is an isomorphism, and in particular, the center of the quantum group U˘r,s⁢(g) of the weight lattice type is a polynomial algebra KK[z_ϖ1,…,z_ϖn], where canonical central elements zλ (λ∈Λ⁺) are turned out to be uniformly expressed. For the rank g to be odd, we figure out a new invertible extra central generator z*, which does not survive in Uq⁢(g), then the center of U˘r,s⁢(g) contains (Formula presented), where ℓ=2, except ℓ=4 for D_2⁢k+1.