Differential Operators And Singular Vectors Of The Verma Modulus

Lie algebras are divided into typical lie algebras An,Bn,Cn,Dn(n?1)and exceptional lie algebras E6,E7,E8,F4,G2(where E6 and E7 are the children of E8).The theory of the highest weight representation is very representative in the field of Lie algebra research,and has a wide range of applications in mathematics,physics and other disciplines.The most important module is Verma module.The research of Verma module can help us have a deeper and clearer understanding of lie algebra.In the study of the singular vector with the highest weight of the lie algebra Verma module,Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight ? over sl(n,C).Xiao present an explicit formula of the solution S?(1)for every positive root a and show directly that S?(1)is a polynomial if and only If and only if<?+?,?>is a nonnegative integer(Where p is half of the sum of the positive roots).Xiao obtains the expression of singular vector in sl(n,C)which is essentially consistent with Xu,and then gives the singular vector expression of Verma module of Cn.For an exceptional lie algebra E6,E7,E8,F4 and G2,the basis is finite,so it is relatively easy to calculate the singular vectors of an exceptional lie algebra.In the remaining two groups of Bn and Dn,the structure of Bn is more complex than that of An and Cn,and the differentiation operator of Bn is not very commutative.Therefore,this paper mainly introduces the singular vector of Bn,gives the expression of the differential operator of Bn,introduces a system of differential equations to study the singular vector in Verma module by using the straight roots and differential operator,The expression of Verma modular singular vector of any straight root of type Bn is given and the corresponding proof is given by using the differential operator of type Bn.The expression of Verma modulus singular vector of any straight root of type B2 is calculated and proved.The expression of singular vector with weight Se1+en·? in M(?)is also given and proved.The main work and contributions of this paper are as follows:Firstly,This paper introduces the concept of straight roots,calculates and gives the straight roots of typical lie algebras and exceptional lie algebras,An example is given to illustrate the reason why Verma modulus singular vector is calculated by using the straight root of Bn instead of the positive root.Secondly,a new group of bases in type Bn is given.The lie algebra multiplication table is given for B2 and B3,According to the complexity of differential operator structure and the commutativity of its structure,the general formula of differential operator in Bn is found and proved by calculation and observation.Thirdly,The formula of the singular vector that satisfies<?+?,e1-e2>=?1,<?+?,e2?>=?2,and<?+?,??>=k(where k=1,2)in Bn type weight is Se1·? and Se1+e2·?is calculated.According to the characteristic of B2 singular vector when k=1,2,the expressions of the singular vector of all straight roots in B2 are given,and the proof is given by using differential operator.Fourth,According to the characteristics of B2 type singular vector and a lot of calculation and observation,the expression of singular vector with Bn weight Se1+en·? is given,and the proof is given by using differential operator.Through calculation and observation,we guess the expression that Bn weight is Se1·? singular vector,which is convenient for our subsequent work.Finally,the expression of singular vector with weight Se1+en·? in Dn type is given.