Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.


Introduction
Conformal algebras are algebraic structure relevant to physical problems for both relativistic and non-relativistic settings. In the non-relativistic setting, the algebra is called conformal Galilei algebra (CGA) [1] (see also [2]). This is a family of Lie algebras consisting of infinitely many members. Each member is not semisimple, not isomorphic to each other and is labelled by two parameters d and ℓ, where d is a positive integer and ℓ takes a spin value (= 1/2, 1, 3/2, . . . ). The simplest member with ℓ = 1/2 is the Schrödinger algebra which gives the symmetry algebra of free Schrödinger equations in d dimensional space [3,4]. Recently the Schrödinger algebra and ℓ = 1 member of CGA were discussed in the context of non-relativistic AdS/CFT correspondence [5,6,7,8,9]. This caused a renewed interest on CGA. Indeed, CGA with various pairs of (d, ℓ) appears in wide range of physical problems [10,11,12,13,14,15,16,17,18,19,20,21,22,23] (see [24] for more references on ℓ = 1/2 and ℓ = 1 CGA). This observation motivates us to study representation theory of CGA. The present work is a continuation of our previous works on the lowest (or highest) weight representations of CGA done in Refs. [24,25] in which d = 1 CGA with half-integer ℓ are studied. Our aim in this paper is to study the lowest weight Verma modules over d = 1 CGA with integer ℓ. By studying the reducibility of the Verma modules we shall give the irreducible lowest weight modules. We remark that a classification of all finite weight modules over the d = 1 CGA with any ℓ has been done very recently [26]. This is a pure mathematical work containing a classification of all irreducible lowest weight modules (same problem as we discuss in this paper). However, we would like to be more explicit: Namely, we start with the Verma modules and show explicitly how to arrive at the irreducible modules. We give the structure of the irreducible modules by concrete formula. As an application of this scheme we derive hierarchies of partial differential equations defined on the group manifold generated by the CGA. These differential equations are symmetric under the transformation generated by d = 1 CGA.
This paper is organized as follows. In the next section the definition of d = 1 CGA for integer ℓ is given. We also give triangular type decomposition and algebraic anti-involution for later use. In §3 the Verma modules over the CGA is introduced and calculation of the Kac determinant is presented. We observe that the Kac determinant vanishes for many cases. This suggests the existence of singular vectors in the Verma modules. It is shown in §4 that there exist various singular vectors. The formulae of singular vectors are used to construct the partial differential equations in §5. The differential equations are symmetric under the transformation generated by CGA. In §6 we give the irreducible lowest weight modules obtained from the Verma module. Throughout this article we denote the d = 1 CGA with spin ℓ by g ℓ .
2 d = 1 Conformal Galilei algebras g ℓ The complex Lie algebra g ℓ for a fixed integer ℓ has the elements [1]: D, H, C, P n (n = 0, 1, . . . , 2ℓ). (2.1) Their nonvanishing commutators are given by One may see from this that P 0 , P 1 , . . . , P 2ℓ is an abelian ideal of g ℓ , so that g ℓ is not semisimple. It is known that this Lie algebra has no central extensions [9]. The subalgebra spanned by H, D, C is isomorphic to so(2, 1) ≃ sl(2, R) ≃ su(1, 1). The abelian subalgebra spanned by P n n=0,1,...,2ℓ carries the spin ℓ representation of the sl(2, R) subalgebra. This algebra may be realized as generators of transformation of (1+ 1) dimensional spacetime In this realization, H, D and C generates time translation, dilatation and the special conformal representation, respectively. Meanwhile the generator of space translation is represented by P 0 , the Galilei transformation generator by P 1 , the transformation to a reference frame with constant acceleration is given by P 2 and so on. One may introduce the algebraic anti-involution ω : g ℓ → g ℓ by It is not difficult to verify that ω satisfies the required relations: Let us define the degree of the generators based on their commutator with respect to D: With respect to the sign of the degree one may define the triangular decomposition of g ℓ : where, . . , P 2ℓ . This is a decomposition of g ℓ as a direct sum of the vector spaces.

Verma modules and Kac determinant
We assume the existence of the lowest weight vector |δ, p defined by The Verma module over g ℓ is defined, as usual (see e.g. [32]), as a module induced from |δ, p : In order to specify the basis of V δ,p ℓ we introduce the ℓ-component vector where m i (i = 1, 2, . . . , ℓ) are non-negative integers. With this the basis of V δ,p ℓ is given by We also introduce the ℓ-component vectors ǫ j = (0, . . . , 0, 1, 0, . . . , 0) ∈ R ℓ , j = 1, 2, . . . , ℓ. (3.4) where the jth entry ofǫ j is 1, and all other entries are 0. It may not be difficult to prove the following relation by induction on k : It follows that the action of g 0 ℓ and g + ℓ on |k,m : From this one see that V δ,p ℓ has a grading structure according to the eigenvalue of D: We refer to N as the level as usual. Now we define an inner product for the vectors in V δ,p ℓ . Let |x and |y be any two vectors in Define the inner product by Next we define the Kac determinant at level N (see e.g. [33]). Let |v 1 , |v 2 , . . . , |v r be a set of basis of the subspace (V δ,p ℓ ) N . We consider the matrix ( v i | v j ) whose entries are the inner products of the basis of (V δ,p ℓ ) N . The determinant of this matrix is called the Kac determinant at level N : ∆ . We give an explicit formulae of the Kac determinants of g ℓ , since ∆ N at level N of g ℓ are given as follows (up to overall sign): Proof . The case with ℓ = 1 shows a deviation from other values of ℓ. We treat the case with ℓ = 1 separately. The basis of V δ,p 1 is specified by two nonnegative integers: The level is N = k + m so that the basis of (V δ,p 1 ) N is given by The product of two vectors in (V δ,p 1 ) N is It is not difficult to verify that P k 2 |m, 0 = 0 if k > m. Thus we have proved the following lemma.

By definition ∆
(1) N is given by (up to sign) N is the determinant of upper triangular matrix. Thus Each factor is calculated by (3.12) as follows: This completes the proof of (3.10). Now let us turn to the case with ℓ ≥ 2. In this case ∆ (ℓ) N = 0 only if N = 1. As we shall see, this fact stems from that at least two rows of ∆ i m i . First we prove the following lemma: Because of (3.5) the right hand side of (3.14) will be a linear combination of the terms However, such terms give nonvanishing contributions only if It follows from Lemma 2 that two rows in ∆ On the other hand the N = 1 subspace is two dimensional with the basis |1,0 , |0,ǫ 1 where0 denotes the zero vector in R ℓ . It is an easy exercise to verify that ∆ (ℓ) 1 = (ℓ + 1) 2 p 2 up to a sign factor. This completes the proof of Proposition 1.
Proof . (i) is the corollary of Proposition 1 (i). Proposition 1 also suggests the existence of singular vectors in V δ,0 ℓ for any ℓ and in V δ,p ℓ with ℓ ≥ 2 and p = 0. In the next section we shall show that this is indeed the case. Thus we establish (ii) and (iii).

Singular vectors in V δ,p ℓ
In this section we give explicit formulae of the singular vectors in V δ,p ℓ . We do not give a complete list of singular vectors, but the list given below is enough to show the reducibility of V δ,p ℓ (Proposition 2) and derive differential equations having g ℓ as a symmetry in the next section. Before giving the list let us recall the definition of singular vectors. A singular vector |v s ∈ V δ,p ℓ is yet another lowest weight vector which is not proportional to |δ, p . Namely, |v s satisfies the conditions: (4.1) According to Proposition 1, singular vectors in V δ,p ℓ may be different for p = 0 and p = 0. We treat these cases separately.

p = 0
In this case the Verma modules over ℓ = 1 algebra have no singular vectors. For the algebra g ℓ with ℓ ≥ 2, singular vectors may exist only in the subspace (V δ,p ℓ ) N with N ≥ 2.

Proposition 3. Following are the singular vectors in
where n takes the value of a positive integer. The maximal value of n is determined by ℓ and N in such a way that S (2n) and S (2n+1) given below are well-defined.
Proof . We show that the vectors in (4.2) satisfy the conditions in (4.1). It is obvious that the vectors in (4.2) are annihilated by P a ∈ g − ℓ and are eigenvectors of P ℓ with the eigenvalue p. It is also easy to verify that It follows that the vectors in (4.2) are the eigenvectors of D : Finally, from the commutation relations (T (2n+1) ) k |δ, p , k = 1, 2, . . . where n takes a value of positive integer. The maximal value of n is determined by ℓ and N in such a way that T (2n+1) given below is well-defined.
Other coefficients c 0 , d j (j ≤ 1 ≤ n) are determined by the relations Proof . One can prove this in a manner similar to Proposition 3. So, we skip the details of the calculation and present only the final formulae below: where F (x, y, z) is a function of three variables and we do not need the explicit form of it.

p = 0
In this case singular vectors in V δ,0 ℓ may exist for all values of ℓ and N. We have the singular vectors inherited from the case of p = 0.
where n takes the value of a positive integer. The maximal value of n is determined by ℓ and N in such a way thatS (2n) given below is well-defined.
Proof . It is easy to verify the relations: It follows immediately that P k ℓ−1 |δ, 0 is a singular vector with δ + k as the eigenvalue of D. Next we set p = 0 at (4.3), (4.4) and (4.6). Then we find the following reduction (up to overall constant): This shows that the vectors in (4.10) are singular vectors. It is also an easy task to verify the vectors (S (2n) ) k |δ, 0 , (P ℓ−1S (2n) ) k |δ, 0 satisfy the definition of singular vector (4.1).
5 Differential equations symmetric under the kinematical transformations generated by g ℓ

General formalism
The singular vectors obtained in previous section can be used to derive partial differential equations having particular symmetries. The symmetries are generated by g ℓ , i.e., the symmetry group is the exponentiation of g ℓ , and the partial differential equations are invariant under the change of independent variables, i.e., the kinematical symmetries, caused by the group. This can be done by applying the method developed for real semisimple Lie groups in [34]. In this subsection we give a brief review of the method with suitable modification for the present case (see also [25,30,31]). The basic idea is to realize the Verma modules in a space of C ∞ -class functions. Let G be a complex semisimple Lie group and g its Lie algebra. The Lie algebra g has the triangular decomposition g = g + ⊕ g 0 ⊕ g − . The corresponding decomposition of G is denoted by G = G + G 0 G − . Consider the space of C ∞ -class functions on G having the property called right covariance: where Λ ∈ g * (algebra dual to g ℓ ), g ∈ G, H ∈ g 0 , x = e H ∈ G 0 , g − ∈ G − . Because of the right covariance, the functions of C Λ are actually function on G/B with B = G 0 G − , or on G + . We keep using the same notations for the restricted representation space of functions on G + . Then one may define a representation T Λ of G by a left regular action on C Λ : The infinitesimal generator of this action, which is the standard left action of g on C Λ , gives a vector field representation of g on C Λ : We introduce the right action of g on C Λ by the standard formula: One may show by the right covariance that the function f ∈ C Λ has the properties of lowest weight vector: This allows us to realize the Verma module V Λ ≃ U (g + )v 0 with the lowest weight vector v 0 in terms of the function in C Λ and differential operators π R (X), X ∈ g + . Now suppose that the Verma module V Λ has a singular vector. The general structure of a singular vector is v s = P(X 1 , X 2 , . . . , X s )v 0 , X k ∈ g + . (5.6) where P denotes a homogeneous polynomial in its variables. The singular vector v s induces the Verma module V Λ ′ ≃ U (g + )v s with the lowest weight Λ ′ . Thus the differential operator π R (P) is an intertwining operator between the two representation spaces C Λ and C Λ ′ , i.e., π R (P) T Λ (g) = T Λ ′ (g) π R (P). (5.7) Suppose that the operator π R (P) has a nontrivial kernel π R (P)ψ = 0, π R (P) T Λ (g) ψ = T Λ ′ (g) π R (P) ψ = 0.

Hierarchies of differential equations
Now let us apply the scheme in §5.1 to the group generated by g ℓ . We parametrize an element of G + as g = exp(tH) exp ℓ−1 n=0 x n P n . Then the right action of g + ℓ yields From (5.8), Proposition 3 and Proposition 4 we obtain the following hierarchies of partial differential equations.
Proposition 6. If p = 0 then the following equations are invariant (in the sense of §5.1 ) under the group generated by g ℓ .
We have obtained highly nontrivial differential equations. To have a close look at the equations, we give examples of the hierarchies of equations for n = 1, 2. For n = 1, the equations (5.10)-(5.12) are as follows: The corresponding equations for n = 2 are given by By a similar method we obtain, from (5.8) and Proposition 5, the invariant equations for p = 0. Proposition 7. If p = 0, then the following equations are invariant (in the sense of §5.1 ) under the group generated by g ℓ .
6 Irreducible lowest weight modules of g ℓ We have shown that the Verma modules over g ℓ are reducible in many cases (Proposition 2). It is known that the Verma module is, in a sense, the largest lowest weight module. That is, one can derive all irreducible lowest weight modules starting from the Verma module V δ,p ℓ . The purpose of this section is to obtain some types of irreducible lowest weight modules explicitly. Our results are summarized in the next theorem.  (ii) if 2δ + k = 0 for any nonnegative integer k then the module isomorphic to the infinite dimensional module of sl(2, R).
Theorem 1 coincides with the results in [26].

Proof for p = 0
The Verma modules V δ,p ℓ are reducible for ℓ ≥ 2 so we restrict ourselves to ℓ ≥ 2. We consider the quotient module V δ,p ℓ /I (2) where I (2) is the largest g ℓ -submodule of V δ,p ℓ . Since there is no singular vectors in the N = 1 subspace (V δ,p ℓ ) 1 , I (2) will be induced by the singular vector in the N = 2 subspace. Lemma 3. There exists precisely one (up to an overall constant) singular vector in the N = 2 subspace (V δ,p ℓ ) 2 . This singular vector is given by Proof . The basis of (V δ,p ℓ ) 2 is |2,0 , |1,ǫ 1 , |0, 2ǫ 1 , |0,ǫ 2 . The singular vector v (2) s is a linear combination of the basis: It must satisfy the condition Thus c 1 = c 2 = 0. Furthermore, the condition C v Define be the lowest weight vector in V δ,p ℓ /I (2) . Then D u It follows that the basis of V δ,p ℓ /I (2) is given by we define the level N (2) in the quotient space V δ,p ℓ /I (2) by { |1,0 , |0,ǫ 1 } and { |2,0 , |1,ǫ 1 , |0, 2ǫ 1 } form a basis of N (2) = 1 and N (2) = 2 subspaces of V δ,p ℓ /I (2) , respectively. It follows that the Kac determinants of N (2) = 1, 2 subspaces are given by (ℓ + 1) 2 p 2 and 4(ℓ + 1) 6 p 6 , respectively. Therefore there exist no singular vectors in N (2) = 1, 2 subspaces. On the other hand, one finds a singular vector in the N (2) = 3 subspace. Lemma 4. There exists precisely one (up to overall constant) singular vector in the N (2) = 3 subspace of V δ,p ℓ /I (2) . The singular vector is given by Proof . The lemma can be proved in a way exactly similiar to Lemma 3. We here give the basis of the level N (2) = 3 subspace and omit the detailed proof.

The vector v
(3) s is singular only in the quotient space V δ,p ℓ /I (2) and not in V δ,p ℓ itself. Such vector is called subsingular [35,36].
(iii) If 2 ≤ λ ≤ ℓ − 1, then there exists precisely one (up to overall constant) singular vector in the The singular vector is given by . (6.10) Proof . (i) Can be trivially proved.
This implies the uniqueness of the formula (6.10) of the singular vector. We remark that if λ = ℓ then the vector corresponds to P ℓ−λ−1 does not exist. Thus α λ+1 = 0, so that no singular vectors at level ℓ + 1.
Now we consider the module V (ℓ) ℓ = V δ,p ℓ /I (2) / · · · /I (ℓ) . The basis of this space is H k P m ℓ−1 u is the lowest weight vector defined by (6.7) with λ = ℓ. By Lemma 5, there exist no singular vectors in the subspaces of V (ℓ) ℓ labelled by N (ℓ) = 1, 2, . . . , ℓ. For N (ℓ) ≥ ℓ + 1 the singular vectors may be written as Analogous to the proof of Lemma 5, one can see that the conditions P ℓ+1 v It is easy to see that Thus there are two possibilities: (a) α = 0 and δ is arbitrary (so we take δ = 0). In this case there exists precisely one singular vector at N = 1 and it is given by P ℓ−1 |δ, 0 .
We treat these cases separately.
(iii) A singular vector at level N (λ) = λ + 1 subspace is written as It follows that α = 0, and β is arbitrary. . From (3.5) we see that P n |k = 0 for all k and n. Therefore, only the sl(2, R) subalgebra (spanned by H, D, C) of g ℓ has nontrivial action on V (ℓ) ℓ . Namely, V (ℓ) ℓ is isomorphic to a sl(2, R)-module. V (ℓ) ℓ is also a graded vector space. Each subspace (labelled by the positive integer k) is one dimensional with the basis vector |k . Since C |k = k(2δ + k − 1) |k − 1 , if 2δ + k − 1 = 0 then |k is the unique singular vector in V (ℓ) ℓ . The quotient module V (ℓ) ℓ /U (g + ℓ ) |k is k dimensional irreducible module of sl(2, R).
(b) δ = 0 : Let J = U (g + ℓ )H |0, 0 and consider the quotient V 0,0 ℓ /J . This space is spanned by P m 1 ℓ−1 · · · P m ℓ 0 |v 0 with the lowest weight vector |v 0 . Now we repeat the same process as in the previous cases and see that the space shrinks step by step to the linear span of P m 2 ℓ−2 · · · P m ℓ 0 |v 0 , P m 3 ℓ−3 · · · P m ℓ 0 |v 0 , etc. Finally, we arrive at the irreducible one dimensional space which gives the trivial representation of g ℓ .

Concluding remarks
The main results of this work are the following: Explicit formulae of singular vectors in V δ,p ℓ over g ℓ (Proposition 3, 4 and 5). Hierarchies of partial differential equation symmetric under the transformations generated by g ℓ (Proposition 6 and 7). Irreducible lowest weight modules of g ℓ (Theorem 1). In the present work and in [24,25,26] the irreducible lowest/highest weight modules of the d = 1 CGA and invariant differential equations have been investigated in full detail. However, the structure of irreducible modules of the CGA for d ≥ 2 (especially higher values of ℓ) is still an open problem. Another interesting problem may be a relation of the representation of CGA and orthogonal polynomials. In this regard we would like to cite Ref. [37] wherein a relation between the representation of the Schrödinger group and a discrete matrix orthogonal polynomial has been discussed in detail. However to the best of our knowledge no such relationship between orthogonal polynomials and the representation of CGA is known. A key observation for such polynomials might be the formulae of singular vectors presented in §4. Because they are multivariable polynomials on P n 's and orthogonal with respect to the inner product (3.9). A thorough investigation of this aspect is beyond the scope of the current work and hence will be reported elsewhere.