Generalized Verma Modules On A Class Of Simple Lie Algebras

As a kind of important non-associative algebras,Lie algebras are one of the important subjects of modern mathematics,which are helpful for us to understand some physical phenomena from algebraic angle.They play an important role in topology,group theory,theoretical physics,quantum mechanics and some other fields.From 1890 to 1894,German mathematician W.Killing and French mathematician E.Cartan completely classified the simple finite-dimensional Lie algebras over the complex field.From then on,more and more mathematicians studied Lie algebras.The representation of Lie algebras is one of the basic research tasks of the Lie theory.As one of the most important representations,Verma modules are widely used in theoretical physics.The classification of Verma modules is a popular topic of domestic and foreign scholars.Generalized Verma modules are the natural generalization of Verma modules.So the study of their structure and irreducibility plays an important role in studying the structure and properties of classical Verma modules.Mazorchuk studied the structure of generalized Verma modules and gave the properties of the generalized Verma module of the Lie algebra of type G2 induced from generalized Witt type Lie algebras and generalized Verma modules induced from complex semi-simple finite-dimensional Lie algebras in turn.At the same time,Khomenko,Futorny,et al.gave several methods to induce generalized Verma modules from simple finite-dimensional modules,Whittaker modules,GelfandZetlin modules and so on.Since 2016,many scholars have studied the simplicity of generalized Verma modules induced from different modules.H.He gave the sufficient criteria of the reducibility of scalar generalized Verma modules on abelian parabolic subalgebras.Y.Cai,G.Liu,Nilsson and K.Zhao determined the simplicities of generalized Verma modules over the special linear Lie algebra sl+2(C)induced from U(h)-free sll+1(C)-modules.Z.Bai and W.Xiao characterized the irreducibility of generalized Verma modules induced from finite-dimensional modules in the Hermite symmetric case.L.Xia,J.Zhu and X.Guo constructed the singular Whittaker modules of the Lie algebra of type B2 and presented the sufficient condition of the irreducibility of generalized Verma modules.In this paper,we focus on two classes of generalized Verma modules of the Lie algebra of type B2 induced from the Whittaker module of sl(2,C)and study its irreducibility.The background and research status of generalized Verma modules and our main research contents are introduced in chapter 1.Then basic knowledge and theorems of Lie algebra are given in chapter 2.In chapter 3 and chapter 4,we use the Whittaker module of sl(2,C)to induce the generalized Verma module of the Lie algebra of type B2 in two cases.The irreducibility of the generalized Verma module is determined by discussing the generating vectors.At last,we make a summary and prospect in chapter 5.