EXTREMAL VECTORS FOR VERMA TYPE REPRESENTATION OF B 2

Čestmír Burdíka,∗, Ondřej Navrátilb a Department of Mathematics, Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, 120 00 Prague 2, Czech Republic b Department of Mathematics, Czech Technical University in Prague, Faculty of Transportation Sciences, Na Florenci 25, 110 00 Prague, Czech Republic ∗ corresponding author: burdices@kmlinux.fjfi.cvut.cz


Introduction
Representations of Lie algebras are important in many physical models.It is therefore useful to study various methods for constructing them.
The general method of construction of the highestweight representation for the semisimple Lie algebra was developed in [1,2].The irreducibility of such representations (now called Verma modules) was studied by Gelfand in [3].The theory of these representations is included in Dixmier's book [4].
In the 1970's prof.Havlíček with his coworkers dealt with the construction of realizations of the classical Lie algebras, see [5].Our aim in this paper is to show how one can use realizations of the Lie algebra to construct so called extremal vectors of the Verma modules.To work with a specific Lie algebra, we choose Lie algebra so (3,2), which plays an important role in physics, e.g. in AdS/CFT theory, see [6,7].
In the construction of the Verma modules for B 2 , the representations depend on parameters (λ 1 , λ 2 ).For connection with irreducible unitary representations of SO (3,2) we take λ 2 ∈ N 0 , and in section 3 we explicitly construct the factor-Verma representation.Further, we construct a full set of extremal vectors.These vectors are called subsingular vectors in [8].
In this paper, we use an almost elementary partial differential equation approach to determine the extremal vectors in any factor-Verma module of B 2 .It should be noted that our approach differs from a similar one used in [9].First, we identify the factor-Verma modules with a space of polynomials, and the action of B 2 on the Verma module is identified with differential operators on the polynomials.Any extremal vector in the factor-Verma module becomes a polynomial solution of a system of variable-coefficient second-order linear partial differential equations.

The root system for Lie algebra B 2
In the Lie algebra g = B 2 we will take a basis composed by elements We can take as h the Cartan subalgebra with the bases H 1 and H 2 .We will denote λ = (λ 1 , λ 2 ) ∈ h * , for which we have The root systems g = B 2 with respect to these bases H 1 and H 2 are R = ±α k ; k = 1, 2, 3, 4 , where If we choose positive roots

The extremal vectors for
Verma type representation The element |0 will be called the lowest-weight vector.
Let further be where b + -module C|0 is defined by τ λ .
It is clear that W (λ) ∼ U (n − )|0 and it is the U (g)module for the left regular representation, which will be called the Verma module. 1  It is a well-known fact that every U (g)-submodule of the module W (λ) is isomorphic to module W (µ), where for n 1 , n 2 ∈ N 0 = {0, 1, 2, . . .}.For the lowest-weight vector of the representation W (µ) ⊂ W (λ), |0 µ , is fulfilled Such vectors |0 µ will be called extremal vectors W (λ).
From the well-known result for the Verma modules we know that the Verma module W (λ) is irreducible iff If λ 1 ∈ N 0 , resp.λ 2 ∈ N 0 , then the extremal vectors are 1 In Dixmier's book the Verma module M (λ) is defined with where If W (µ) is a submodule W (λ), we will define the U (g)-factor-module Now we can study the reducibility of a representation like that.
Again, the extremal vector is called any nonzero vector v ∈ W (λ|µ) for which there exists ν ∈ h * such that It is clear that In this paper, we find all such extremal vectors in the space W (λ|µ 2 ), where λ 2 ∈ N 0 and µ 2 is given by (1).

Differential equations for extremal vectors
Let λ 2 ∈ N 0 and µ 2 be given by equation (1).It is easy to see that the basis in the space W (λ|µ 2 ) is given by the vectors Now by direct calculation we obtain It is possible to rewrite the action by the second order differential operators (see [10,11]) on the polynomial functions z 1 , z 2 , z 3 a z 4 , which are in variable z 2 up to the level λ 2 .If we put , we obtain from equations (3) for the action on poly- no. / Extremal Vectors for Verma Type Representation of B 2 where The conditions for extremal vectors (2) are now where ν 1 and ν 2 are complex numbers.The condition on the degree of the polynomial f (z 1 , z 2 , z 3 , z 4 ) in variable z 2 can be rewritten in the following way

The extremal vectors
The extremal vectors are in one-to-one correspondence to polynomial solutions of the systems of equations ( 5), which are in variable z 2 of maximal degree λ 2 .
You can find all such solutions in the appendix.
For any λ 1 and λ 2 there exists a constant solution f (z 1 , z 2 , z 3 , z 4 ) = 1.But such a solution gives v = |0 , which is not interesting.
A further solution exists only in the cases For , and we obtain the extremal vector For λ 1 + λ 2 + 1 ∈ N 0 and 2λ 1 + λ 2 + 4 ≤ 0 we find the solution where Then we can rewrite the solution from the appendix in the following way: For λ 1 being a half integer, i.e. λ 1 = 1 − 1 2 , where 1 ∈ Z, we have where For these solutions we obtain the extremal vectors If λ 1 is an integer we have λ 1 ≤ −2.The solution of the differential equations in this case is where and the extremal vectors are

Appendix: Polynomial solutions of differential equations
To obtain extremal vectors we need to find the polynomial solutions of the system of equations ( 5), which are of less degree than (λ 2 + 1) in the variable z 2 .
To simplify the solution of the first equations, we put where The first order equations are equivalent to the conditions and so g(t, x 1 , x 2 , x 3 ) = g(t).
The equations of the second order give the system of three equations As we want to obtain polynomial solutions f (z 1 , z 2 , z 3 , z 4 ), which are in variable z 2 of less or equal degree λ 2 ∈ N 0 , there must be solution g(t) of the system (6), which is the polynomial in √ t of less or equal degree λ 2 .

Case 2 (ρ
. The function that corresponds to the extremal vector is in this case where g(t) is the solution of system (7).As in event 1 we find that the non-constant polynomial solutions ).The function for the extremal vectors is where g(t) is the solution of the system As we assume that λ 2 ∈ N 0 ,for each λ 1 , λ 2 this system has the solution This solution corresponds to the function which is a non-constant polynomial for This function is a polynomial in the variable z 2 of degree 2λ 1 + 2λ 2 + 2. It gives sought solutions for 2λ 1 + λ 2 + 4 ≤ 0.
Thus, function (9) provides a permissible solution for the ).In this case, the function that can match the extremal vector is where g(t) is the solution of system (8).So To give a polynomial solution, which we have found, to this function there must be λ 1 ∈ N 0 and λ 1 + λ 2 + 1 ∈ N 0 .But in this case, the degree of polynomial f in the variable z 2 is greater than λ 2 and, therefore, is not a permissible solution. vol.
no. / Extremal Vectors for Verma Type Representation of B 2 Case 5 (ρ 1 = 2λ 1 + λ 2 + 3, ρ 2 = 0.).The function corresponding to the possible extremal vectors is where function g(t) meets the equation This equation has two linearly independent solutions where F (α, β; γ; t) is the hypergeometric function These solutions correspond to the functions For at least one of these functions to be a nonconstant polynomial, must be 2λ 1 +λ 2 +3 ∈ N, i.e. 2λ 1 +λ 2 +2 ∈ N 0 .If 2λ 1 + λ 2 + 3 is even, we get the solution and for 2λ 1 + λ 2 + 3 odd, we have the solution and, therefore, f is in the variable z 2 of a polynomial of degree not exceeding λ 2 .If 2λ 1 + λ 2 + 3 is even and λ 2 is odd, i.e. λ 1 is an integer, the function in the variable z 2 is a polynomial of degree 2λ 1 +λ 2 +3.Thus admissible solutions get only λ 1 ≤ −2.