Double-bosonization Theory And Majid’s Conjecture

In this thesis, according to double-bosonization theory [44] and FRT-construction theory [12], we explain in detail how to obtain step by step, and starting with Uq(sl2), all quantized enveloping algebras associated with finite dimensional complex simple Lie algebras. There are four parts. In the first part, we give the general inductive construction for the classical quan-tized enveloping algebras by the Corollary 1.6. We describe in detail the inductive construction of high-one rank quantized enveloping algebras for type B, C, D from Uq(sln) in the second chapter. Then we generalize the double-bosonization construction, and give the construction of exceptional quantum groups for type G2, F4 for the first time in the third chapter, We also ex-plore the inductive construction of quantized enveloping algebras for type E in the last chapter, and draw the tree of quantum groups generated by the double-bosonization construction.Starting with a quasitriangular Hopf algebra H and a dual pair of braided groups B, B* in the braided category (?)H(H(?)), then there is a unique Hopf algebra struture on the tensor space B*(?）H(?)B. Majid expected that this new Hopf algebra is the high rank one when H is a classical quantized enveloping algebra. Starting from type A, we choose (co)braided vector algebras V(R’,R) and Vv(R’,R21-1) as braided groups. A key step is to obtain the following rule by the examples of low ranks:in order to describe the structure of new quantum groups, we only need to focus on the diagonal and skew-diagonal elements in the matrix m±, which can be obtained easily. Then we get the general inductive construction for the classical quantized enveloping algebras by this observation.We choose another dually paired braided groups V(P,R), V∨(P,R21-1/) in the second chapter to construct Uq(so2n+1) from the standard R-matrix for type A. Inspired by the theorem 8 in [12], we generalize and build double-bosonization construction for the general R-matrix Rvv, given by Theorem 2.19, which depends on the weakly quasitriangular dual pair(Theorem 2.13) we build for general R-matrix. Moreover, we find an ingenious method to get quantum groups normalization constant A and braided groups we need. Ac-cording to these results, we construct inductively quantized enveloping alge-bras for type C, D from the’symmetric square’sym2 V and the second exterior power ∧2V of the vector representation for Uq(sln). Starting with A1 and its spin3/2 representation, and 8-dimensional spin representation of type B3, we also construct quantized enveloping algebras of type G2,F4 for the first time in the third chapter.In the last chapter, we obtain all quantized enveloping algebras for the E series from Uq(D5)’s 16-dimensional spin representation, the 27-dimensional minimal fundamental representation for Uq(E6) and 56-dimensional minimal fundamental representation for Uq(E7). Finally, we give Theorem 4.8 to sum up all the above inductive constructions, and draw the structure of the tree generated by these inductive constructions, with nodes indexed by the quantized enveloping algebras.