Axiomatic Inductive Construction Of Higher Rank Quantum Groups And Quantization Of Jacobson-Witt Algebras

The main objects of this thesis are the inductive construction of higher rank quantum groups, the quantization of Jacobson-Witt algebra and the quantum generalized projective representation for quantum group of type A. We have four parts:The first part gives the inductive construction of higher rank quantum groups via quantum quasisymmetric algebra. In the second part, we obtain the matching realization of Uq(sln+1) of higher rank in the quan-tum Weyl algebra Wq(2n) using quantum differential operators. The third part quantizing the entire Jacobson-Witt algebra W(n). In the last part, we construct the so called quantum generalized projective representation of type A and obtain a sufficient and necessary condition for the irreducibility of this representation. The logical correlation among these four parts are as follows: The first two parts focus on the study of the inductive construction of higher rank quantum groups. The former uses quasisymmetric algebra and the latter uses quantum differential operators. Actually, the matching realization in the second part quantizes a Lie subalgebra of W(n) generated by W-1, W0and an abelian subalgebra of W1, which is the first step at quantizing the entire Lie algebra W(n) in the third part. Since quantum generalized projective represen-tation in the fourth part is a kind of generalization of the matching realization in the second part, the results in the second part can provide some useful infor-mation for the construction of quantum generalized projective representation in the fourth part.In the first chapter, we introduce the notion of quantum symmetric alge-bra and quantum quasisymmetric algebra. In1998, Rosso gave a axiomatic realization of the upper triangular part of the quantum group associated with a symetrizable Cartan matrix using quantum symmetric algebra. In2013, Fang and Rosso realized the entire one-parameter quantum group with Cartan data, which is also an axiomatic construction. We begin with a quantum group Uq(gn-1) of rank n-1, a suitable irreducible representation of it and its dual representation, using quantum quasisymmetric algebra, and finally get a quan-tum group Uq(gn) of rank n, where the Cartan matrix of Uq(gn) is obtained from that of Ug(gn-1) by adding a suitable line and a suitable column.Chapter two realizes the higher rank quantized universal enveloping alge-bra Uq(sln+1) as certain quantum differential operators in Wq(2n) defined over the quantum divided power algebra Aq(n) of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq(sln+1). The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.In the third chapter, we further quantize the entire Jacobson-Witt algebra W(n) on the base of chapter two. Firstly, we quantize each W(n)l according to the module structure, and then get a module Vq of Uq(sln). Secondly, we made it a Hopf bimodule of Uq(sln) by some standard procedure, which enable us to built a quantum symmetric algebra over it and finally obtain the quantization of the enveloping algebra of W(n). This question is still an open hard question proposed by Naihong Hu when he was a Humbolt research fellow.The last chapter generalizes the matching realization of Uq(sln+1) to the quantum generalized projective representation Aq(n)(?) V of Uq(sln+1). In par-ticular, if we take V to be the trivial representation, then the quantum general-ized projective representation of Uq(sln+1) is just the matching realization. we get a sufficient and necessary condition for the irreducibility of the quantum generalized projective representation after giving its detailed construction. As an application, for any given finite dimensional irreducible Uq(sln)-module, we obtain a new family of infinite-dimensional irreducible Uq(sln+1)-modules, which are in general not highest-weight type.