New York Journal of Mathematics

New York J. Math. 4 (1998) 177–183.

Heisenberg Lie Bialgebras as Central Extensions

Miloud Benayed and El Mamoun Souidi

Abstract. We determine and study all Lie bialgebra central extensions of ℝ^{2n} by ℝ admitting the Heisenberg algebra H_{2n+1} as the underlying Lie algebra structure. The present work answers the question about the realizability of Lie bialgebra structures on H_{2n+1} as central extensions of ℝ^{2n} endowed with the adapted Lie bialgebra structure, by ℝ.

Contents

1. Introduction 177 2. Central Structures on H_{2n+1} 178 3. Exact Central Structures on H_{2n+1} 180 4. Equivalence of Central Structures on H_{2n+1} 181 5. Conclusion 182 References 182

1. Introduction

A Lie bialgebra is a Lie algebra (𝔤, [ , ]) equipped with a 1-cocycle ε : 𝔤 → ∧^{2}𝔤 (δε = 0) for the extended adjoint action of 𝔤 on ∧^{2}𝔤, whose transpose ε^{∗} : ∧^{2}𝔤^{∗} → 𝔤^{∗} defines a Lie algebra structure on the dual vector space 𝔤^{∗} of 𝔤. A 1-coboundary ε = δr with r ∈ ∧^{2}𝔤 defines a Lie bialgebra structure on a Lie algebra (𝔤, [ , ]) if and only if the Schouten bracket [[r, r]] ∈ ∧^{3}𝔤 is invariant under the extended adjoint action of 𝔤 on ∧^{3}𝔤. Such a Lie bialgebra is called coboundary or exact.

Let 𝔤_{1} and 𝔤_{2} be two Lie bialgebras. A linear map ρ : 𝔤_{1} → 𝔤_{2} is a Lie bialgebra morphism if ρ is a Lie algebra morphism and its transpose ρ^{∗} : 𝔤_{2}^{∗} → 𝔤_{1}^{∗} is also a Lie algebra morphism. A bijective morphism is called an isomorphism.

Two Lie bialgebra structures ε_{1} and ε_{2} on a Lie algebra (𝔤, [ , ]) are called equivalent if there exists a Lie bialgebra isomorphism ρ : (𝔤, [ , ], ε_{1}) → (𝔤, [ , ], ε_{2}).

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received May 2, 1998.

Mathematics Subject Classification. 16W30, 17B56.

Key words and phrases. Heisenberg algebra, Lie bialgebras, central extensions.

The first author would like to thank G. Tuynman for discussions. The second author would like to thank G. Tuynman for the hospitality at Université de Lille I, and also C. Brezinski and J. Mikram, principal investigators of the grant Action Intégrée A/I 1036.

For further details on Lie bialgebras we refer the reader to [3]. In all the sequel the ground field is ℝ.

A Lie bialgebra 𝔤̂ is called a central extension of a Lie bialgebra 𝔤 by ℝ if there exists an exact sequence

0 ⟶ ℝ ⟶^{i} 𝔤̂ ⟶^{π} 𝔤 ⟶ 0

in which i and π are Lie bialgebra morphisms and i(ℝ) is contained in the center of the Lie algebra 𝔤̂. Two central extensions 𝔤̂_{1} and 𝔤̂_{2} of 𝔤 by ℝ will be called equivalent if there exists a Lie bialgebra morphism ρ : 𝔤̂_{1} → 𝔤̂_{2} inducing the identity on ℝ and 𝔤, in the exact sequences defining 𝔤̂_{1} and 𝔤̂_{2}.

The Heisenberg algebra H_{2n+1} is a Lie algebra central extension of the abelian Lie algebra ℝ^{2n} by ℝ, associated with the 2-cocycle γ which is the canonical symplectic 2-form of ℝ^{2n}. A Lie bialgebra structure on the abelian Lie algebra ℝ^{2n} is equivalent to the data of a Lie algebra structure on (ℝ^{2n})^{∗}.

This paper is organized as follows. In Section 2, we describe the set Ext_{big}(ℝ^{2n}, ℝ) of all inequivalent Lie bialgebra central extensions of a given Lie bialgebra structure on the abelian Lie algebra ℝ^{2n}, by ℝ. Let Ext_{big}^{γ}(ℝ^{2n}, ℝ) be the subset consisting of elements in Ext_{big}(ℝ^{2n}, ℝ) which have the Heisenberg algebra as the underlying Lie algebra structure. Such Lie bialgebra structures on H_{2n+1} will be called central. We prove that Ext_{big}^{γ}(ℝ^{2n}, ℝ) is non empty if and only if (ℝ^{2n})^{∗} is an abelian Lie algebra. If the last condition is fulfilled then the set Ext_{big}^{γ}(ℝ^{2n}, ℝ) is parametrized by the endomorphisms of ℝ^{2n}. Section 3 is devoted to give a characterization of exact central structures on H_{2n+1} by means of the associated endomorphims of ℝ^{2n}. In Section 4 we give the orbits of central structures on H_{2n+1} under its automorphisms group action. In the last section we give a motivation of this work as a partial answer of an open question.

2. Central Structures on H_{2n+1}

We endow ℝ^{2n} with a Lie bialgebra structure by taking the abelian Lie algebra structure on ℝ^{2n} and giving a Lie algebra structure [ , ] on its dual vector space (ℝ^{2n})^{∗}. Let “ad” be the adjoint action of (ℝ^{2n})^{∗} on itself and “coad” represents the coadjoint action of (ℝ^{2n})^{∗} on its dual vector space ℝ^{2n}. We denote by Der(ℝ^{2n})^{∗} the vector space of derivations of the Lie algebra (ℝ^{2n})^{∗} and Dex(ℝ^{2n})^{∗} stands for the outer derivations of (ℝ^{2n})^{∗}.

Definition 2.1. An element ω of ∧^{2}(ℝ^{2n})^{∗} is called compatible with the Lie algebra structure on (ℝ^{2n})^{∗} if the condition: coad_{ω̃(x)}(y) = coad_{ω̃(y)}(x) holds for all x, y in ℝ^{2n}, where ω̃ : ℝ^{2n} → (ℝ^{2n})^{∗} is the linear map induced by ω. We let ∧_{c}^{2}(ℝ^{2n})^{∗} to denote the subspace of all such compatible elements of ∧^{2}(ℝ^{2n})^{∗}.

Theorem 2.2. There is a one-to-one correspondence between Ext_{big}(ℝ^{2n}, ℝ) and ∧_{c}^{2}(ℝ^{2n})^{∗} × Dex(ℝ^{2n})^{∗}.

Proof. To any Lie bialgebra central extension ℝ^{2n}⊕ℝ of ℝ^{2n} by ℝ one can associate a couple (ω, f) ∈ ∧_{c}^{2}(ℝ^{2n})^{∗} × Der(ℝ^{2n})^{∗} as follows (see [1]):

[(x, a); (y, b)]_{ℝ^{2n}⊕ℝ} = (0; ω(x, y))
[(ξ, α); (η, β)]_{(ℝ^{2n})^{∗}⊕ℝ} = ([ξ, η]_{(ℝ^{2n})^{∗}} + αf(η) − βf(ξ); 0)

Two elements (ω, f) and (ω′, f′) of ∧_{c}^{2}(ℝ^{2n})^{∗} × Der(ℝ^{2n})^{∗} define equivalent Lie bialgebras if and only if ω′ = ω and f′ = f + ad_{φ} where φ lies in (ℝ^{2n})^{∗}. Reversing the arguments we also get the converse. □

For every ω ∈ ∧_{c}^{2}(ℝ^{2n})^{∗} we let Ext_{big}^{ω}(ℝ^{2n}, ℝ) to be the subspace of Ext_{big}(ℝ^{2n}, ℝ) corresponding to {ω} × Dex(ℝ^{2n})^{∗}. We are interested in the case where ω is the canonical symplectic 2-form γ of ℝ^{2n}:

γ : ℝ^{2n} × ℝ^{2n} → ℝ
((x, y); (x′, y′)) ↦ x · y′ − y · x′

The dots stand here for the inner product in ℝ^{2n}. In order to restrict ourselves to Ext_{big}^{γ}(ℝ^{2n}, ℝ), we must verify the compatibility of γ with the given Lie algebra structure on (ℝ^{2n})^{∗}.

Lemma 2.3. The following conditions are equivalent:

(i) γ ∈ ∧_{c}^{2}(ℝ^{2n})^{∗}. (ii) (ℝ^{2n})^{∗} is an abelian Lie algebra.

Proof. It is enough to prove the implication (i) ⇒ (ii). The fact that γ lies in ∧_{c}^{2}(ℝ^{2n})^{∗} implies the following condition:

∀x ∈ ℝ^{2n}, ∀ξ ∈ (ℝ^{2n})^{∗}, γ̃(coad_{ξ} x) = [ξ, γ̃(x)].

Writing this condition for ξ = γ̃(y) with y ∈ ℝ^{2n} and using the fact that γ ∈ ∧_{c}^{2}(ℝ^{2n})^{∗}, we obtain

∀x, y ∈ ℝ^{2n}, [γ̃(x), γ̃(y)] = [γ̃(y), γ̃(x)]

which implies that [γ̃(x), γ̃(y)] = 0 for all x, y in ℝ^{2n}. The non-degeneracy of γ means that γ̃ : ℝ^{2n} → (ℝ^{2n})^{∗} is a vector space isomorphism. So we conclude that (ℝ^{2n})^{∗} is an abelian Lie algebra. □

Remark 2.4. The condition ω ∈ ∧_{c}^{2}(ℝ^{2n})^{∗} implies that the range of ω̃ is an abelian Lie subalgebra of (ℝ^{2n})^{∗}. The converse is false in general, as one can see in the following example. We endow (ℝ^{4})^{∗} with the Lie algebra structure defined by the only non vanishing bracket [X_{2}^{∗}, X_{3}^{∗}] = X_{4}^{∗}, where (X_{k}^{∗})_{k=1}^{4} is the dual basis of the canonical ordered basis (X_{k})_{k=1}^{4} of ℝ^{4}. Let ω be the element of ∧^{2}(ℝ^{4})^{∗} given by ω(X_{1}, X_{2}) = 1 and vanishing elsewhere. The range of ω̃ is abelian but ω ∉ ∧_{c}^{2}(ℝ^{4})^{∗}, for example coad_{ω̃(X_{1})}(X_{4}) = −X_{3} ≠ 0 = coad_{ω̃(X_{4})}(X_{1}).

As a consequence of the previous lemma and theorem we get the following.

Proposition 2.5. Ext_{big}^{γ}(ℝ^{2n}, ℝ) is non empty if and only if (ℝ^{2n})^{∗} is an abelian Lie algebra. If the last condition is fulfilled then Ext_{big}^{γ}(ℝ^{2n}, ℝ) is parametrized by the space End(ℝ^{2n}) of all endomorphisms of ℝ^{2n}.

Henceforth we assume that ℝ^{2n} is an abelian Lie bialgebra, i.e., ℝ^{2n} and (ℝ^{2n})^{∗} are endowed with the zero Lie brackets. Otherwise, as we have seen, Ext_{big}^{γ}(ℝ^{2n}, ℝ) will be empty.

The set Ext_{big}^{γ}(ℝ^{2n}, ℝ) is viewed as the Heisenberg algebra H_{2n+1} endowed with Lie bialgebra structures, parametrized by End(ℝ^{2n}), such that each structure makes H_{2n+1} as a central extension of the abelian Lie bialgebra ℝ^{2n} by ℝ. Such a structure on H_{2n+1} will be called central. Let us specify the central structures on H_{2n+1} in terms of the corresponding 1-cocycles ε : H_{2n+1} → ∧^{2}H_{2n+1} (the transposes of Lie brackets in H_{2n+1}^{∗}: See the proof of Theorem 2.2).

The center Z(H_{2n+1}) of the Lie algebra H_{2n+1} is one dimensional, let Z be a non zero element of Z(H_{2n+1}). Let (X_{k})_{k=1}^{2n} be the canonical ordered basis of ℝ^{2n} then ((X_{k})_{k=1}^{2n}, Z) is (the canonical) basis of H_{2n+1}. The only non vanishing Lie brackets on H_{2n+1} are given by:

For all 1 ≤ i, j ≤ n , [X_{i}, X_{n+j}] = δ_{ij}Z

where δ_{ij} is the Kronecker’s symbol.

Definition 2.6. A central structure on H_{2n+1} is the data of a 1-cocycle ε_{f} : H_{2n+1} → ∧^{2}H_{2n+1} where f ∈ End(ℝ^{2n}) satisfying:

(i) ε_{f}(Z) = 0. (ii) For all X ∈ ℝ^{2n}, ε_{f}(X) = Z ∧ f(X).

3. Exact Central Structures on H_{2n+1}

An arbitrary element r of ∧^{2}H_{2n+1} can be written as

r = ∑_{1≤i<j≤2n} α_{ij}X_{i} ∧ X_{j} + ∑_{i=1}^{2n} β_{i}X_{i} ∧ Z

where α = (α_{ij})_{1≤i,j≤2n} is an antisymmetric matrix and the β_{i} are in ℝ.

Lemma 3.1. The couboundary δr′ of r′ = ∑_{i=1}^{2n} β_{i}X_{i} ∧ Z vanishes.

Proof. For all H ∈ H_{2n+1} we have:

(δr′)(H) = ∑_{i=1}^{2n} β_{i}([H, X_{i}] ∧ Z + X_{i} ∧ [H, Z]).

Since [H_{2n+1}, H_{2n+1}] = Z(H_{2n+1}) and Z is central in the Lie algebra H_{2n+1} then (δr′)(H) = 0, for all H ∈ H_{2n+1}. □

Henceforth we assume that r = ∑_{1≤i<j≤2n} α_{ij}X_{i}∧X_{j}. The matrix α determines r completely and vice-versa.

Lemma 3.2. [2] Every element r of ∧^{2}H_{2n+1} defines an exact Lie bialgebra structure δr on H_{2n+1}, i.e., the Schouten bracket [[r, r]] is ad-invariant, for all r in ∧^{2}H_{2n+1}.

Let us remark that an exact Lie bialgebra δr on H_{2n+1} is necessarily central. Our aim is to give a characterization of central structures on H_{2n+1} that are exact.

Proposition 3.3. An endomorphism f of ℝ^{2n} defines an exact central structure on

H_{2n+1} if and only if the matrix of f (in the basis (X_{k})_{k=1}^{2n}) has the form (\begin{matrix}A & B \\ C & ^{t}A \end{matrix})

where A, B, C are n × n matrices with B and C antisymmetric. The corresponding

r-matrix has the form α = (\begin{matrix}−B & A \\ −^{t}A & C \end{matrix}).

Proof. Let r = ∑_{1≤i<j≤2n} α_{ij}X_{i} ∧ X_{j} be an element of ∧^{2}H_{2n+1}. Set α = (α_{ij}). We verify there exists an f ∈ End(ℝ^{2n}) with matrix (f_{ij}) (in the canonical basis) such that δr = ε_{f} if and only if f_{ik} = α_{i,n+k}, f_{n+k,k} = 0 for all 1 ≤ k ≤ n, and f_{ik} = α_{k−n,i}, f_{k−n,k} = 0 for all n + 1 ≤ k ≤ 2n. Using these relations and the antisymmetry of the matrix α, we obtain the following conditions on f_{ij}: for all 1 ≤ p, q ≤ n,

f_{p,n+q} = −f_{q,n+p}, f_{n+p,n+q} = f_{q,p}, f_{n+p,q} = −f_{n+q,p}.

In other words, the matrix of f is necessarily of the form (f_{ij}) = (\begin{matrix}A & B \\ C & ^{t}A \end{matrix}) where

B and C are n × n antisymmetric matrices. Hence the corresponding r-matrix α can be written as

α = (\begin{matrix}−B & A \\ −^{t}A & C \end{matrix}) .

Reversing the arguments, we also get the converse. □

Remark 3.4. If there exists 1 ≤ k ≤ n such that f_{n+k,k} ≠ 0 (or f_{k−n,k} ≠ 0 for n + 1 ≤ k ≤ 2n) then the central structure defined by f on H_{2n+1} is not exact.

We denote by I_{p} the identity matrix p × p.

Example 3.5. The exact central structures on H_{3} are all of type a I_{2}, where a ∈ ℝ. The corresponding r-matrix is given by r = aX_{1} ∧ X_{2}.

4. Equivalence of Central Structures on H_{2n+1}

Two central structures ε_{f} and ε_{g} on H_{2n+1}, defined by f and g in End(ℝ^{2n}) respectively, will be called equivalent if they define equivalent Lie bialgebra structures, i.e., if there exists a Lie algebra automorphism φ of H_{2n+1}, say φ ∈ Aut(H_{2n+1}), such that the following diagram commutes:

H_{2n+1} ⟶^{φ} H_{2n+1}
ε_{f} ↓  ↓^{ε_{g}}
∧^{2}H_{2n+1} ⟶^{φ⊗φ} ∧^{2}H_{2n+1}

We define the “extended” symplectic group of ℝ^{2n} by

S(2n, ℝ) := \{A ∈ GL(2n, ℝ) | ∃s ∈ ℝ^{∗} : ^{t}AJA = sJ\} where J = (\begin{matrix}0 & I_{n} \\ − I_{n} & 0 \end{matrix}) .

Proposition 4.1. Let f and g be two endomorphisms of ℝ^{2n} with the matrixes M_{f} and M_{g} in the canonical basis of ℝ^{2n}, respectively. The central structures ε_{f} and ε_{g} on H_{2n+1} are equivalent if and only if there exists A ∈ S(2n, ℝ) such that M_{g} = sAM_{f}A^{−1}, where s is defined by ^{t}AJA = sJ.

The proof is an immediate consequence of the following realization of Aut(H_{2n+1}).

Lemma 4.2. The automorphisms group of H_{2n+1} is given by

G = \{(\begin{matrix}A & 0 \\ v & s \end{matrix}) ∈ GL(2n + 1, ℝ) | \begin{matrix}^{t}AJA = sJ, with s ∈ ℝ^{∗}, and \\ v = (v_{1}, ⋯, v_{2n}) ∈ ℝ^{2n} arbitrary \end{matrix} \} .

Proof. We distinguish the following classes of automorphisms of H_{2n+1} that we represent by their matrices in the canonical basis ((X_{k})_{k=1}^{2n}, Z).

i) The symplectic automorphisms: (\begin{matrix}A & 0 \\ 0 & 1 \end{matrix}) where A is a 2n × 2n symplectic

matrix, i.e., ^{t}AJA = J.

ii) The inner automorphisms: (\begin{matrix}I_{2n} & 0 \\ v & 1 \end{matrix}) where v ∈ ℝ^{2n}.

iii) The dilatations: (\begin{matrix}r I_{2n} & 0 \\ 0 & r^{2} \end{matrix}) where r > 0.

iv) The inversion: (\begin{matrix}0 & I_{n} & 0 \\ I_{n} & 0 & 0 \\ 0 & 0 & −1 \end{matrix}).

Every automorphism of H_{2n+1} can be written as a product of these automorphisms. It is easy to see that Aut(H_{2n+1}) ⊂ G. Reciprocally, consider an element

σ = (\begin{matrix}A & 0 \\ v & s \end{matrix}) of G. By multiplying this matrix by an inversion, we can assume that

s > 0. By multiplying σ by the dilatation of coefficient \frac{1}{\sqrt{s}} followed by a convenient

inner automorphism, we obtain a matrix of the form (\begin{matrix}B & 0 \\ 0 & 1 \end{matrix}), where B is a 2n×2n

symplectic matrix. Thus σ is an automorphism of H_{2n+1}, i.e., G ⊂ Aut(H_{2n+1}) hence G = Aut(H_{2n+1}). □

5. Conclusion

This work is motivated by the open question about the classification of all Lie bialgebra structures on a given (nilpotent) Lie algebra.

Every nilpotent Lie algebra can be obtained by successives central extensions from an abelian Lie algebra. Our hope was to use the notion of Lie bialgebra central extensions in order to classify all Lie bialgebra structures on a fixed nilpotent Lie algebra.

Our test Lie bilagebra was the Heisenberg algebra where all Lie bialgebra structures are known. Let us recall this result in terms of our previous notations.

Theorem 5.1. [4] Each Lie bialgebra structure ε : H_{2n+1} → ∧^{2}H_{2n+1} on H_{2n+1} is one of the following two forms:

(i) ε(Z) = 0 and for all X ∈ ℝ^{2n}, ε(X) = Z ∧ f(X) with f ∈ End(ℝ^{2n}). (ii) ε(Z) = Z∧A and ∀X ∈ ℝ^{2n}, ε(X) = Z∧N_{A}(X)+ \frac{1}{2}(X∧A+γ(X, A)γ^{−1}) with

A ∈ ℝ^{2n} − {0} and N_{A} is the endomorphism of ℝ^{2n} given by N_{A} = γ(A, U) I_{2n} + U ⊗ γ̃(A) − A ⊗ γ̃(U) + λA ⊗ γ̃(A), U ∈ ℝ^{2n}, λ ∈ ℝ.

Following our study, the first family (i) is the central structures on H_{2n+1} realized by the notion of Lie bialgebra central extensions. The second family (ii) cannot be obtained by this notion: One can see that for central extensions ε(Z) vanishes necessarily. This is not the case for the second family (ii).

References

[1] M. Benayed, Central extensions of Lie bialgebras and Poisson Lie groups, Journal of Geometry and Physics 16 (1995), 301–304, MR 96d:17020.

Heisenberg Lie Bialgebras 183

[2] M. Cahen and C. Ohn, Bialgebra structures on the Heisenberg algebra, Bulletin de la Classe des Sciences de l’Academie Royale de Belgique 75 (1989), 315–321, MR 93d:16048.

[3] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994, MR 95j:17010.

[4] I. Szymczak and S. Zakrzewski, Quantum deformations of the Heisenberg group obtained by geometric quantization, Journal of Geometry and Physics 7 (1990), 553–569, MR 92i:58066.

Université des Sciences et Technologies de Lille UFR de Mathématiques-URA au CNRS D751 59655 Villeneuve d’Ascq Cedex; France.

benayed@gat.univ-lille1.fr

Université Mohammed V. Faculté des Sciences Département de Mathématiques et Informatique B.P 1014. Rabat; Maroc.

souidi@maghrebnet.net.ma

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-12.html.