The Bialgebras Of Extended Affine Lie Algebras And The Representations Related To The W(a, B) Algebras

In this thesis, we mainly study the bialgebra structures, quantizations of extended affine Lie algebras and the representations related to the W(a, b) algebras.During the investigation of quantum groups, V.Drinfel’d introduced the notion of Lie bialgebras in1983. Quantization of Lie algebras and bialgebras is an important way to produce new quantum groups. In order to study quantum groups associated with extended affine Lie algebras, it is rather necessary to first know quantizations and Lie bialgebra structures on extended affine Lie algebras. Therefore, investigations of Lie bialgebra structures and quantizations on extended affine Lie algebras is definitely an important problem. However, it seems to us that there is not much known on this aspect.In Chapter2, we shall investigate the Lie bialgebra structures on the Lie algebra sl2(Cq[x,y]) and sl2(Cq[x±1, y±1]). Due to the fact that, compared with some classical infinite dimensional graded Lie algebras, there are inner derivations (from itself to its tensor space) which are hidden more deeply in itself’s interior algebraic structure, some new techniques need to be employed in order to find them out. It is proved that all Lie bialgebra structures on sl2(Cq[x, y]) and sl2(Cq[x±1, y±1]) are triangular coboundary.In Chapter3, the extended affine Lie algebra sl2(Cq[x±1, y±1]) is quantized from three different points of view, which produces three noncommutative and noncocom-mutative Hopf algebra structures, and yield other three quantizations by an isomor phism of sl2(Cq[x±1, y±1]) correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra sl2(Cq[x,y]).Due to extreme importance in mathematics and physics, representations of the Vi-rasoro algebra have been widely studied in mathematical and physical literatures. We study a a class of Lie algebra W(a, b), which is closely related to the Virasoro algebra and its modules. W(a,b) embraces many meaningful algebras, such as Heisenberg-Virasoro algebra, W(2,2) algebra, etc and it is very natural and desirable to consider representations of W(a,b). Note that some special cases of W(a,b) naturally appear as subalgebras of many interesting infinite-dimensional graded Lie (super)algebras, e.g., the W1+∞algebra, some Block type Lie algebras, N=2super-Virasoro alge-bras, Schrodinger-Virasoro algebras etc. One of our motivations is that just as results on representations of the Virasoro algebra are widely used in classifications of repre-sentations of Lie (super)algebras which contain the Virasoro algebra as a subalgebra, one can expect that results on representations of W(a, b) may be used in that of Lie (super)algebras which contain some W(a, b) as a subalgebra.In Chapter4, indecomposable modules of the intermediate series over W(a,b) are classified. It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module. Furthermore, if a(?) Q, an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s. We would like to emphasis here that, due to the crucial fact that the parameter a is not necessarily an integer, modules of the intermediate series over W(a,b) may have a rather complicated structure, which is very different from that of the Virasoro algebra.Lie conformal algebras resemble Lie algebras in many ways. On the one hand Lie conformal algebras turn out to be an adequate tool for the study of infinite dimensional Lie algebras satisfying the locality property. On the other hand Lie conformal algebras are closely related to vertex operator algebras. It is noteworthy that Virasoro algebra is one of the main examples of Lie conformal algebras. The W-algebra W(2,2), which is a special case of W(a, b), was first introduced in order to investigate a classification of vertex operator algebras generated by weight2vectors. Consequently, the structure and properties on W(a, b) conformal algebra are significant.In Chapter5, after recalling the concepts of formal distribution Lie algebras and conformal algebra, we construct the W(a, b) conformal algebra and then consider the conformal derivations of W(a, b) conformal algebra. We also determine the conformal module of rank one with respect to the W(a,b) conformal algebra.