The Harish-Chandra Homomorphisms Of Lie Superalgebras And Quantum Superalgebras

The representations of quantum（super）algebras are closely connected to Rmatrices and knot invariants.This thesis mainly discusses Harish-Chandra homomorphism for quantum superalgebra U_q（g）associated to a basic Lie superalgebra g,and establishes a relationship between its representations and center.Inspired by the cases of Lie（super）algebras and quantum algebras,we define Harish-Chandra map Ξ for U_q（g）to be the composite （?） where π is the projection from U_q（g）to its Cartan subalgebra U0,γ-ρ is an automorphism of U0 induced by the Weyl vector p.1.We show that Ξ is an injective homomorphism of algebras,the proof is based on the key observation that Z（U_q（g））is contained in the homogeneous component U0 of degree 0 of U_q（g）.We mention that this is nontrivial when g=A（n,n）.2.Using the character formulas for Verma modules and typical simple modules,certain automorphisms of U_q（g）and nontrivial homomorphisms between Verma modules,we prove that the image of Ξ is contained in(U_ev⁰)_sup^W.3.We discuss two constructions of the center of U_q（g）via the skew-pairing between its positive part and negative part.This induces a linear map Ψ:k?Z K_ev（U_q（g））→Z（U_q（g））,where K_ev（U_q（g））is the subring generated by finitedimensional highest weight simple U_q（g）-modules M=Lq（λ）with 2λ ∈Zφ.4.We establish an algebra isomorphism from(U_ev⁰)_sup^W to k?Z Jev（g）,where Jev（g）is a subring of the ring of exponential super-invariants J（g）,introduced by Sergeev and Veselov.Applying the isomorphism,we finally proved that when g≠A（1,1）,Ψ andΞ induce algebra isomorphisms k?Z K_ev（U_q（g））≌Z（U_q（g））≌(U_ev⁰)_sup^W.Thus,we prove the Harish-Chandra type theorem of quantum superalgebras,and obtain a basis of Z（U_q（g））by the forementioned central elements.