Quantization And Generalized Verma Representation Of Block Type Lie Algebras

Lie algebras were introduced by Norway mathematician S.Lie and German mathematician W.Killing to study the concept of infinitesimal transformations respectively.Because of its broad application in differential equation and differential geometry,this algebra was developed rapidly.As it is well known that,the structure theory and representation theory,especially for infinite dimensional Lie algebras,are two of the most important topics in the theory of Lie algebras.Quantum groups were introduced by V.G.Drinfeld and M.Jimbo respectively to study the Yang-Baxter equations in mathematics and physics in the eighties of the twentieth century.According to quantum groups theory,quantum group is not a group,but a special Hopf algebra.Quantum group is essentially a formal deformation of the universal enveloping algebra of a Lie algebra(?),the semiclassical structure associated with such a deformation is a Lie bialgebra structure on(?).A Lie bialgebra is a Lie algebraαprovided with a Lie cobracket which is related to the Lie bracket by a certain compatibility condition.We know that there is a 1-1 correspondence between triangular Lie bialgebras and solutions of the CYBE, while giving all the solutions of the CYBE is still an open problem.In quantum groups,in order to produce new quantum groups,constructing quantization of Lie bialgebras is an important and efficient approach.Therefore,one of the most important intent of study bialgebras is to quantize these algebras.In recent years it has been great interest to quantum deformations of commutator relations of Lie algebra and related Hopf structures.The quantum deformation of a Lie algebra is obtained by adding one parameter q,which is reduced to the original Lie algebra when taking the limit q→1;some properties of the original Lie algebra remain.In 1958,R.Block introduced a class of infinite dimensional simple Lie algebras (usually referred to as Lie algebras of Block type).Since then,many authors studied generalized Block algebras.Partially due to their close relations to the Virasoro algebra and Cartan type Lie algebras(Generalized Block type Lie algebras include Cartan S or H type Lie algebra,Virasoro-like and its q-analog algebra),these algebras have attracted some attentions in the literature.It is well known that Virasoro algebras and Cartan type simple Lie algebras play an important role in many areas of mathematics and physics.The theory of the representations of Virasoro algebras is well developed,however,the representation theory of Caftan type Lie algebras is far from being well developed.In order to better understand the representation theory of Cartan type Lie algebras,it is very natural to first study representations of special cases of Caftan type Lie algebras.The foundation of the theory of highest weight module over complex simple finite dimensional Lie algebras due to the original paper by D.Verma,where the family of universal highest weight modules(Verma modules)was introduced and studied.Since this,many famous and deep results about Verma modules were obtained.Some results have been generalized to certain infinite dimensional Lie algebras,e.g.(affine)Kac-Moody algebra,Virasoro algebra,quantum algebras, Yangians and forth.So far the theory of Verma modules is not completed and there are many interesting unsolved questions and problems.There have been several attempts to generalize Verma modules,and one of the most natural of them is to generalize Verma modules themselves.Generalized Verma modules can be applied to study the structure of usual Verma modules for different algebras.The present paper includes four parts.In the first part we investigative the Lie bialgebra structures on Block type Lie algebras and q-analog Virasoro-like algebras. We obtain that all such Lie bialgebras are coboundary triangular.The proof of the main result is based on the heavy computation of the set of all derivations and we can consider the Block type Lie algebras as a module of intermediate series of Virasoro algebra.We obtain that the first cohomology group of these Lie algebras with coefficients in its certain modules is trivial.In the second part we use the general method of quantization by Drinfeld's twist to quantize explicitly the Lie bialgebra structures on Block type and q-analog Virasoro-like algebras. In the third part we construct a class of generalized Verma modules over some other Block type Lie algebra and determine its irreducibility.In the four part, the quantum deformations of Heisenberg-Virasoro algebra(?)_q is presented and the central extension of this algebra is investigated.Finally,we give its nontrivial quantum group structure.