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Some Researches on Quantum Enveloping Algebras and Conformal Algebras
Author: HongYanYong
Tutor: LiFang;WuZhiXiang
School: Zhejiang University
Course: Basic mathematics
Keywords: quantum plane quantum nspace module algebra Verma module weak Hopf algebra HarishChandra homomorphism locally finite subalgebra adjointaction Lie conformal algebra leftsymmetric conformal algebra leftsymmetric conformal bialgebra NovikovPoisson algebra leftsymmetricPoisson algebra Gel’fandDorfman bialgebra vertex algebra matched pair
CLC: O153.3
Type: PhD thesis
Year: 2013
Downloads: 2
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Abstract
The main results of this paper are divided into five parts.Firstly, we study the module algebra structures of quantum enveloping algebras on the quantum space and the adjoint action of quantum algebra Ur,tUr,t is a new quantum enveloping algebra introduced by Wu in [83]. Using the method similar to that in [35], a complete classification of Ur,tmodule algebra structures on the quantum plane is given and we describe these representations when q is not the root of unity and the ground field is C. Moreover, we present a complete classification of module algebra structures of Uq(sl(3)) on the quantum3space. In addition, we investigate the adjoint action of Ur,t when k is a fixed algebraically closed field with characteristic zero and q∈k not a root of unity. The structure of its locally finite subalgebra is given. And, we characterize all its ideals and some primitive ideals of Ur,t.Secondly, we investigate the weak Hopf algebras introduced by Li corresponding to quantum algebras Ug(f(K, H))(see [81]). In Chapter4, we define a new class of algebras denoted by (?)Uqd. When d=((1,1)(1,1)), denote (?)Uqd by (?)1Uq; When d=((0,0)(0,0)), denote (?)Uqd by (?)2Uq. And we study (?)1Uq and (?)2Uq in detail. In some cases, necessary and sufficient conditions for (?)1Uq and (?)2Uq to be weak Hopf algebras are given. The PBW bases of (?)1Uq and (?)2Uq are presented. Finally, representations and the center of (?)1Uq are characterized over C with q∈C not a root of unity.Thirdly, we present a class of extended quantum enveloping algebras Uq(f(K, J)) and some new Hopf algebras, which are certain extensions of quantized enveloping algebras of generalized KacMoody Lie algebras by some fixed Hopf algebra H. This construction generalizes some wellknown extensions of quantized enveloping algebras by a Hopf algebra and provides a large of new noncommutative and noncocommutative Hopf algebras.Fourthly, we introduce leftsymmetric conformal algebra to study vertex algebra. A vertex algebra is an algebraic counterpart of a twodimensional conformal field theory. By an equivalent characterization of vertex algebra using Lie conformal algebra and leftsymmetric algebra given by Bakalov and Kac in [8], in studying vertex algebra, we have to deal with such a question:whether there exist compatible leftsymmetric algebra structures on a class of special Lie algebras named formal distribution Lie algebras. In Chapter6, we study this question. The definitions of leftsymmetric conformal algebra and Novikov conformal algebra are introduced in Chapter2. We show many examples of these algebras in Chapter6. As an application, we present a construction of vertex algebra using leftsymmetric conformal algebras. It provides a large of new noncommutative finite vertex algebras. Finally, we study a conformal analog of leftsymmetric bialgebras. The notion of leftsymmetric bialgebra was introduced by Bai in [5] which is equivalent to a parakahler Lie algebra which is the Lie algebra of a Lie group G with a Ginvariant parakahler structure. In Chapter7, the notions of leftsymmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and leftsymmetric conformal algebras are presented. We show that a finite leftsymmetric conformal bialgebra which is free as a C [(?)]module is equivalent to a parakahler Lie conformal algebra (see Definition7.18). We also obtain a conformal analog of the Sequation (see [5]), and give a construction of the conformal symplectic double.

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