Some Researches On Quantum Enveloping Algebras And Conformal Algebras

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,t-Ur,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,t-module 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 quantum3-space. 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 al-gebras of generalized Kac-Moody Lie algebras by some fixed Hopf algebra H. This con-struction generalizes some well-known extensions of quantized enveloping algebras by a Hopf algebra and provides a large of new non-commutative and non-co-commutative Hopf algebras.Fourthly, we introduce left-symmetric conformal algebra to study vertex algebra. A vertex algebra is an algebraic counterpart of a two-dimensional conformal field the-ory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [8], in studying vertex algebra, we have to deal with such a question:whether there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie alge-bras. In Chapter6, we study this question. The definitions of left-symmetric 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 left-symmetric conformal algebras. It provides a large of new non-commutative finite vertex algebras. Finally, we study a conformal analog of left-symmetric bialgebras. The notion of left-symmetric 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 G-invariant parakahler struc-ture. In Chapter7, the notions of left-symmetric conformal co-algebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a C [(?)]-module is equivalent to a parakahler Lie con-formal algebra (see Definition7.18). We also obtain a conformal analog of the S-equation (see [5]), and give a construction of the conformal symplectic double.