Quantizations Of Cartan-type Lie Algebras And Restricted Two-Parameter Quantum Groups Of Type B

The paper consists of two parts. In the first part we quantize Cartan-type Lie algebras of W and S-series. In the second part from two-parameter quantum groups of type B, we construct a family of finite-dimensional Hopf algebras, i.e. restricted two-parameter quantum groups of type B.1. Quantizations of Cartan-type Lie algebras of W and S-seriesIn chapter 2, we construct explicit Drinfel'd twists which completely depend on the classical Yang-Baxter r-matrix. We use the general quantization method by a Drinfel'd twist to quantize explicitly the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures. To study quantizations of Cartan-type simple modular Lie algebra of W type in characteristic p, we work over the so-called " positive " part subalgebra W⁺ of the generalized-Witt algebra W and develop the quantization integral forms for the Z-form W_Z⁺ in characteristic 0. Over a field of characteristic p, W⁺ contains a maximal ideal J₁ (see Lemma 2.2.2) and the corresponding quotient is exactly the Jacobson-Witt algebra W(n; 1) ( for the Cartan-type simple modular Lie algebra of W type ). We go on two steps of our reduction on the quantization integral forms for the Z-form W_Z⁺ in characteristic 0: modulo p reduction and modulo " restrictedness " reduction process, we get the expected finite-dimensional quantizations of the restricted universal enveloping algebra for the Jacobson-Witt algebra W(n; 1), i.e. the truncated p-polynomial noncocom-mutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra W(n; 1). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p^1+npn with indeterminate t, or of dimension p^npn with specializing t into a scalar in K in characteristic p. Our results generalize a work of C. Grunspan ( J. Algebra 280 (2004), 145-161, [39]) in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from modulo reduction approach (different from C. Grunspan [39] whose treatment did not work for the modular case ) with some techniques from the modular Lie algebra theory. We find there exist n the so-called basic Drinfel'd twists whose pairwise different products among them afford the possible Drin-fel'd twists and some horizontal twists. Accordingly, we get more quantization integral forms for the Z-form W_Z⁺ in characteristic 0 and the Jacobson-Witt algebra W(n; 1) in characteristic p.In chapter 3, we investigate the generalized Cartan-type S Lie algebras in characteristic 0 and the special algebra S(n; 1) using almost the same methods, but more complicated computations. Furthermore, we prove that the twisted structures given by the Drinfel'd twists with different product lengths are not isomorphic with each other. A remarkable point is that these Hopf algebras contain the well-known Radford algebra [76] as a Hopf subalgebra.2. Restricted two-parameter quantum groups of type BIn chapter 4, we restrict the parameter r and s of two-parameter quantum groups U_r,s(so_2n+1) discovered by Bergeron-Gao-Hu [12] to be roots of unity. We construct a family of finite-dimensional Hopf algebras u_r,s(so_2n+1) of dimension l²ⁿ²⁺²ⁿ as a quotient of U_r,s(so_2n+1) by a Hopf ideal I_n, which is generated by the central elements. We show that these Hopf algebras are pointed and of Drinfel'd doubles of a certain Hopf subalgebra b. Our results on the skew-primitive elements of u_r,s(so_2n+1) enable us to determine when two such restricted two-parameter quantum groups are isomorphic. We determine the left and right integrals of b and use these results in combining with a result of L. H. Kauffmann and D. E. Radford from [51] to give necessary and sufficient conditions for these Hopf algebras to possess a ribbon element. Indeed, the ribbon element of these pointed Hopf algebras provides an important invariant.Since the classification of the finite-dimensional Hopf algebras seems very far off, it is quite useful to have various means of constructing examples of finite-dimensional Hopf algebras. Our results on quantizations of Cartan-type Lie algebras and restricted two-parameter quantum groups of type B provide many new examples of finite-dimensional Hopf algebras.