Properties Of Two-Parameter Quantum Groups And Quantizations Of The Generalized Witt Algebra

This dissertation focuses on the study of some properties of two-parameter quan-tum groups and quantizations of the generalized Witt algebra over a field of character-istic zero.The first part of this thesis is devoted to investigating the construction and fac-torization theory of simple modules for restricted two-parameter quantum groups of type B and G. When some conditions on the parameters r and s hold, restricted two-parameter quantum group is a Drinfel’d double of its Borel subalgebra. We analyse and construct the simple modules for restricted two-parameter quantum groups of type B and G by using Radford’s construction of simple modules for D(H) in [1], where H is a Hopf algebra satisfying certain conditions and D(H) is Drinfel’d double of H. Then we give the necessary and sufficient conditions for the two kinds of simple modules to factor as the tensor product of a one-dimensional module with another module, which is also a module for the quotient of the restricted two-parameter quantum group by central group-like elements, respectively.In the second part of the thesis we determine the structure of the derivation algebra of the two-parameter quantum group Ur,s+(B3) and the automorphism group of the Hopf algebra Ur,s≥0(B3). Inspired by the method for one-parameter quantum group given by Launois and Lopes in [2], we characterize the structure of the derivation algebra of the quantum group Ur,s+(B3) and show that the Hochschild cohomology group of degree1of this algebra is a three-dimensional vector space over the base field C. Further, we construct the augmented Hopf algebra Ur,s≥0(B3) and determine the groups of (Hopf) algebra automorphisms of this (Hopf) algebra.Finally, in the third part, we give the basic twists and quantizations of the general-ized Witt algebra over a field of characteristic zero. First, we construct Drinfel’d twists by using two-dimensional non-abelian subalgebra of Lie algebra and obtain some gen-eral formulas. Then we give the quantizations of the generalized Witt algebra over a field of characteristic zero. We also classify the quantizations on the positive part subal- gebra W+into five cases according to the coefficients appearing in the coalgebra struc-tures. Second, we construct various extended twists depending on four-dimensional subalgebras and five-dimensional subalgebras, and then obtain the quantizations of the generalized Witt algebra by these twists. On one hand, these conclusions give a general way of quantizing other algebras. On the other hand, they also provide the academ-ic foundation for studying the quantizations of restricted simple modular Lie algebra W(n;1) and its restricted universal enveloping algebra u(W(n;1)) in characteristic p.