Journal of Lie Theory Volume 13 (2003) 65–76 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Poisson Algebras of Spinor Functions

Aroldo Kaplan, Linda Saal, and Alejandro Tiraboschi ^{∗}

Communicated by M. Cowling

Abstract. Poisson algebras of spinor-valued functions arise as we extend the classical Hamiltonian formalism to vector-valued symplectic forms.

1. Introduction

The classical Hamiltonian formalism involves a non-degenerate skew-symmetric bilinear form ψ on a finite dimensional real vector space 𝔳; a space of functions, C^{∞}(𝔳), or just the polynomial functions S(𝔳); a Lie algebra of first order differential operators, Vect(𝔳), or Vect_{pol}(𝔳), acting on the function space; and a map from the former to the latter, f ↦ H_{f}. These objects satisfy a number of relations coded into the fact that the Poisson bracket

{f, g} = ψ(H_{f}, H_{g})

defines a Lie algebra structure on the function space.

Furthermore, as H. Weyl observed, both the Heisenberg Lie algebra 𝔫_{ψ} = 𝔳 × ℝ with bracket

[(v, t), (v′, t′)] = (0, ψ(v, v′))

and the symplectic Lie algebra 𝔰𝔭(ψ) are naturally subalgebras of the Poisson Lie algebra — namely those constituted by the polynomials of degree ≤ 1 and of degree 2, respectively. These identifications are compatible with the inclusion 𝔰𝔭(ψ) ↪ Der(𝔫_{ψ}) and, moreover, Der(𝔫_{ψ}) = 𝔰𝔭(ψ) ⊕ 𝔳 ⊕ ℝδ with 𝔳 acting by inner derivations and δ · (v, t) = (v, 2t).

In this article we generalize this formalism to vector-valued skew-symmetric forms

Φ : 𝔳 × 𝔳 → 𝔷

which are symplectic, in the sense that there exist inner products in 𝔳 and in 𝔷 such that the transformations J_{z} ∈ End(𝔳) defined by

(1.1)  ⟨J_{z}u, v⟩_{𝔳} = ⟨z, Φ(u, v)⟩_{𝔷}

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag satisfy J_{z}^{2} = −|z|^{2}I or, polarizing the latter,

∗This work was partially supported by CONICET, CONICOR, SECyT (UNC) and FAMAF (Argentina)

J_{z}J_{w} + J_{w}J_{z} = −2⟨z, w⟩I.

The datum of Φ and the two compatible inner products is therefore equivalent to a structure of C(𝔷)-unitary module on 𝔳.

The inner product in 𝔷 is determined by Φ up to a positive multiple, as we explain below. We will fix one and use it as freely as Φ itself. On the other hand, the existence of one compatible inner product on 𝔳 implies that of infinitely many. Of course, the “Weyl Calculus” should be purely symplectic and not depend on any particular choice of metric in 𝔳, as will indeed be the case.

The Poisson Lie algebra attached to a symplectic Φ will be modeled on the vector space

F̃ = C^{∞}(𝔳 × 𝔷) ⊕ Λ^{2}𝔷^{∗}.

The Hamiltonian vector fields will be ordinary vector fields on 𝔳 × 𝔷 acting on F̃ as linear differential operators. The role of the Heisenberg Lie algebra will be played by 𝔫 = 𝔳 × 𝔷 ≅ 𝔳 ⊕ 𝔷 endowed with the bracket

[(v, z), (v′, z′)] = (0, Φ(v, v′));

𝔫 is a two-step nilpotent Lie algebra with center 𝔷, often called of Heisenberg type [4]. One has

Der(𝔫) ≅ 𝔰𝔭(Φ) ⊕ Hom(𝔳, 𝔷) ⊕ ℝδ,

where

𝔰𝔭(Φ) = {(A, B) ∈ 𝔰𝔩(𝔳) × 𝔰𝔩(𝔷) : Φ(Au, v) + Φ(u, Av) = BΦ(u, v)}.

Moreover, letting

𝔰𝔭_{o}(Φ) = {A ∈ 𝔤𝔩(𝔳) : Φ(Au, v) + Φ(u, Av) = 0}

one has

𝔰𝔭(Φ) ≅ 𝔰𝔭_{o}(Φ) ⊕ 𝔰𝔬(𝔷),

with 𝔰𝔬(𝔷) acting on 𝔳 by a direct sum of spin representations [9]. All these Lie algebras will be realized as subalgebras of the Poisson algebra, defined by algebraic and differential conditions along 𝔷.

A similar calculus is obtained if we replace functions and vector fields on 𝔳 × 𝔷 by objects defined on 𝔳 × S(𝔷), with S(𝔷) the unit sphere in 𝔷 and we replace the covariant derivative D by the induced one on the sphere. While that alternative setup is more natural in some ways, the present one simplifies the calculations considerably and induces the alternative one upon restriction.

This paper benefited substantially from the corrections of the reviewer, to whom we are thankful. Related references to Lie algebras of Heisenberg type, their automorphism groups and associated metaplectic representations are discussed in [1], [2], [3], [4], [5], [8] and [9].

2. Preliminaries

Fix a symplectic Φ : 𝔳 × 𝔳 → 𝔷 and a corresponding inner product ⟨z, w⟩ in 𝔷. This determines the linear family of ordinary skew-forms on 𝔳

ϕ_{z}(u, v) = ⟨z, Φ(u, v)⟩.

which are non-degenerate for all z ≠ 0. It should be emphasized that this nondegeneracy is strictly weaker than the condition for Φ to be symplectic [7], but it does not seem to lead to an analogous generalization of the Weyl Calculus.

For z, w ∈ 𝔷, z ≠ 0, define K_{z,w}, A_{z,w} ∈ End(𝔳) by

(2.1)  ϕ_{z}(K_{z,w}u, v) = −|z|^{2}ϕ_{w}(u, v)

A_{z,w} = \frac{1}{2}(K_{z,w} + ⟨z, w⟩I).

Both operators depend linearly on z and w. Since A_{z,w} = −A_{w,z}, one has a linear map

A : Λ^{2}𝔷 → End(𝔳).

Explicitly,

A : ∑_{i<j} c_{ij}z_{i} ∧ z_{j} ↦ ∑_{i<j} c_{ij}A_{z_{i},z_{j}}.

Identifying 𝔷 with 𝔷^{∗} via the given inner product, ones gets a linear map α ↦ A_{α} from Λ^{2}𝔷^{∗} into End(𝔳).

Note that K_{z,w} and A_{z,w} are defined by the linear family of forms ϕ_{z}, independent of any metric in 𝔳. If, however, a compatible metric is chosen so that the J_{z} are defined by (1.1), then

K_{z,w} = J_{z}J_{w}.

Identify Λ^{2}𝔷^{∗} with the orthogonal Lie algebra 𝔰𝔬(𝔷) in the usual way: B ∈ 𝔰𝔬(𝔷) is identified with the 2-form α_{B}(z, w) = ⟨Bz, w⟩, or, equivalently, α ∈ Λ^{2}𝔷^{∗} is identified with the element B_{α} ∈ 𝔰𝔬(𝔷) such that ⟨B_{α}z, w⟩ = α(z, w).

Recall that the map z ↦ J_{z} extends to a representation of the Clifford algebra C(𝔷) by endomorphisms of 𝔳. The multiplicative subgroup of C(𝔷) generated by the double products zz′, with |z| = |z′| = 1, is Spin(𝔷) and the corresponding representation on 𝔳 is a direct sum of spin representations.

Proposition 2.1. With the identification Λ^{2}𝔷^{∗} ≅ 𝔰𝔬(𝔷),

(a) α ↦ A_{α} is the spin representation of 𝔰𝔬(𝔷),

(b) (A_{α}, B_{α}) ∈ 𝔰𝔭(Φ).

Proof. For (a), just note that the spin representation satisfies

4J_{z∧w}v = J_{z}J_{w}v − J_{w}J_{z}v

(see, e.g., Corollary I.6.3 in [6]) and that, in C(𝔷), zw + wz = −2⟨z, w⟩ for all z, w ∈ 𝔷. For (b), take s, z, w ∈ 𝔷 and u, v ∈ 𝔳 and compute

⟨s, Φ(J_{z}J_{w}u, v)⟩ = ⟨J_{s}J_{z}J_{w}u, v⟩
= −⟨J_{z}J_{s}J_{w}u, v⟩ − 2⟨z, s⟩⟨J_{w}u, v⟩
= ⟨J_{z}J_{w}J_{s}u, v⟩ + 2⟨s, w⟩⟨J_{z}u, v⟩ − 2⟨z, s⟩⟨J_{w}u, v⟩
= ⟨J_{s}u, J_{w}J_{z}v⟩ + 2⟨s, w⟩⟨J_{z}u, v⟩ − 2⟨z, s⟩⟨J_{w}u, v⟩.

Since J_{w}J_{z} = −J_{z}J_{w} − 2⟨z, w⟩I,

Φ(J_{z}J_{w}u, v) + Φ(u, J_{z}J_{w}v) = −2⟨z, w⟩Φ(u, v) + 2⟨z, Φ(u, v)⟩w − 2⟨w, Φ(u, v)⟩z.

Let B_{z,w} be the infinitesimal rotation in 𝔷 defined by

B_{z,w}(z′) = ⟨z, z′⟩w − ⟨w, z′⟩z

and recall that A_{z,w} = \frac{1}{2}(K_{z,w} + ⟨z, w⟩I). One obtains

Φ(A_{z,w}u, v) + Φ(u, A_{z,w}v) = B_{z,w}Φ(u, v).

Recall that 𝔫 = 𝔳 × 𝔷 ≅ 𝔳 ⊕ 𝔷 endowed with the bracket

[(v, z), (v′, z′)] = (0, Φ(v, v′))

is a two-step nilpotent Lie algebra, with center 𝔷. Its algebra of derivations has the following structure [9]. If (A, B) ∈ 𝔰𝔭(Φ), then (v, z) ↦ (Av, Bz) is a derivation of 𝔫, yielding an inclusion 𝔰𝔭(Φ) ↪ Der(𝔫). Hom(𝔳, 𝔷) is also contained in Der(𝔫), as the abelian subalgebra consisting of the maps (v, z) ↦ (0, T (v)), T ∈ Hom(𝔳, 𝔷). Furthermore, one has the semidirect sum decomposition

(2.2)  Der(𝔫) = 𝔰𝔭(Φ) ⊕ Hom(𝔳, 𝔷) ⊕ ℝδ

and the direct sum decomposition

(2.3).  𝔰𝔭(Φ) = 𝔰𝔭_{o}(Φ) ⊕ 𝔰𝔬(𝔷)

Both 𝔰𝔭(Φ) and 𝔰𝔭_{o}(Φ) are real reductive Lie algebras, the latter acts trivially on the center, 𝔰𝔬(𝔷) acts by rotations on 𝔷 and by the spin representation on 𝔳 and δ(v, z) = (v, 2z). In matrix form, if 𝔷 ≅ ℝ^{m} and 𝔳 ≅ ℝ^{n}, then

Der(𝔫) ≅ \{(\begin{matrix}A & 0 \\ C & B \end{matrix}) : Φ(Au, v) + Φ(u, Av) = BΦ(u, v)\}

where A, B, C, are n × n, m × m and m × n real matrices, respectively,

𝔰𝔭(Φ) ≅ \{(\begin{matrix}A & 0 \\ 0 & B \end{matrix}) : Φ(Au, v) + Φ(u, Av) = BΦ(u, v), tr(A) = tr(B) = 0\}

𝔰𝔭_{o}(Φ) ≅ \{(\begin{matrix}A & 0 \\ 0 & 0 \end{matrix}) : Φ(Au, v) + Φ(u, Av) = 0, \},

Hom(𝔳, 𝔷) ≅ \{(\begin{matrix}0 & 0 \\ C & 0 \end{matrix})\}, δ ≅ \{(\begin{matrix}2I_{n} & 0 \\ 0 & I_{m} \end{matrix})\}.

The presence of the summand 𝔰𝔬(𝔷) shows that the inner product in 𝔷 is determined by Φ up to a positive scalar. Indeed, 𝔰𝔭(Φ) and 𝔰𝔭_{o}(Φ) are defined by Φ and are both real reductive, hence the sum of an abelian and a semisimple subalgebra. Therefore 𝔰𝔬(𝔷) is the semisimple part of the centralizer of 𝔰𝔭_{o}(Φ) in 𝔰𝔭(Φ) and, as such, it is canonically attached to Φ. Since it acts in the standard irreducible manner on 𝔷 and preserves the inner product, the latter is unique up to homotheties. Furthermore,

𝔰𝔬(𝔷) ≅ \{(\begin{matrix}A_{B} & 0 \\ 0 & B \end{matrix}) : B ∈ 𝔰𝔬(m)\}.

where B ↦ A_{B} is a direct sum of spin representations of 𝔰𝔬(m).

Let

F = C^{∞}(𝔫) = C^{∞}(𝔳 × 𝔷).

Those functions f(v, z) which are polynomial in both v and z constitute a bigraded subspace of F

F^{(·,·)} = ⨁_{p,q≥0} F^{(p,q)},

where F^{(p,q)} are the polynomials which are homogeneous of degree p in v and of degree q in z.

For any subspace P ⊂ F we let

P̃ = P ⊕ Λ^{2}𝔷^{∗}.

An element of F̃ can be viewed as a function F : 𝔳 × 𝔷^{3} → ℝ of the form

F (v, z_{1}, z_{2}, z_{3}) = f(v, z_{1}) + α(z_{2}, z_{3}),

with f smooth and α bilinear and skew-symmetric.

We will consistently identify a vector space 𝔲 with its tangent space at each point and denote by

D_{u}f = uf = u · f

(u ∈ 𝔲) the derivative of the function f in the direction u. If X is a vector field on 𝔲, D_{u}X will denote the canonical covariant derivative of X in the direction u. Constant vector fields will be identified with the corresponding elements of 𝔲. If X is a vector field on 𝔫 = 𝔳 × 𝔷, we will sometimes write X = X′ + X′′ with X′, X′′, tangent to 𝔳 and 𝔷, respectively. If X, Y, are vector fields on 𝔫 which are tangential to 𝔳, we will denote by ϕ(X, Y) the function on 𝔫

ϕ(X, Y)(v, z) = ϕ_{z}(X_{(v,z)}, Y_{(v,z)}) = ⟨z, Φ(X_{(v,z)}, Y_{(v,z)})⟩

3. The main result

The differential equation

(3.1)  D_{z}D_{x}f(v, s) + |s|^{−2}D_{K_{s,z}(x)}f(v, s) = 0

where s, z ∈ 𝔷 and v, x ∈ 𝔳, is linear and homogeneous of degree -1 in each of the variables v and s. Therefore, the set E ⊂ F of its solutions is a linear subspace, containing

E^{(·,·)} = ⨁_{p,q≥0} E^{(p,q)}, E^{(p,q)} := E ∩ F^{(p,q)},

as a dense subspace. For emphasis: E^{(p,q)} consists of the polynomial functions on 𝔳 × 𝔷 of bidegree (p, q) which satisfy the equation (3.1).

Theorem 3.1. There exist an extension of the natural action of Vect(𝔫) on F = C^{∞}(𝔫) to a bilinear map Vect(𝔫) × F̃ → F,

(X, F) ↦ X · F,

a linear map F̃ → Vect(𝔫)

F ↦ H_{F}

and a bilinear operation F̃ × F̃ → F̃

(F, G) ↦ {F, G},

satisfying the following properties. Let X, Z, be vector fields on 𝔫 tangential to 𝔳 and 𝔷 respectively, and let F, G ∈ F̃, with F = f + α, f ∈ F, α ∈ Λ^{2}𝔷^{∗}. Then:

(a) ϕ(H′_{F} , X) = X · F and ⟨H′′_{F}, Z⟩ = Z · α

(b) f ∈ E ⇔ D_{Z}(H_{f}) = 0

(c) H_{H_{α}f} = D_{H_{α}}(H_{f}) + A_{α}(H_{f})

(d) [H_{F} , H_{G}] = H_{{F,G}}

(e) F̃ is a Lie algebra under {·, ·} and E, Ẽ, Ẽ^{(·,1)} are subalgebras.

(f) E^{(1,1)}⊕E^{(0,1)} is a subalgebra, isomorphic to 𝔫 = 𝔳⊕𝔷 as a graded Lie algebra.

(g) E^{(2,1)} and Ẽ^{(2,1)} are subalgebras, with 𝔰𝔭_{o}(Φ) ≅ E^{(2,1)} and 𝔰𝔭(Φ) ≅ Ẽ^{(2,1)}

(h) Der(𝔫) ≅ Ẽ^{(2,1)} ⊕ F^{(1,1)} ⊕ ℝδ and, with the identification in (f), the first two terms act on 𝔫 by inner derivations of the Poisson algebra.

Proof. Define the bilinear map Vect(𝔫) × F̃ → F by

Y · (f + α) = D_{Y} f + Y · α

where f ∈ F, α ∈ Λ^{2}𝔷^{∗} and the last term is the function on 𝔫 defined by

(3.2)  (Y · α)(v, s) = −ϕ_{s}(A_{α}v, Y′_{(v,s)}) − α(s, Y′′_{(v,s)})

which is clearly smooth. By definition, Y · F̃ ⊂ F and Y · f = D_{Y} f = Y f for f ∈ F.

Define the Hamiltonian vector field associated to a F = f + α ∈ F̃ as the element in Vect(𝔫) given by H_{f+α} = H_{f} + H_{α}, where H_{f} is determined by the conditions

(3.3)  ϕ_{s}((H′_{f})_{(v,s)}, X) = D_{X}f(v, s),  H′′_{f} = 0

while

(3.4)  (H_{α})_{(v,s)} = −A_{α}v − B_{α}s.

If we set f_{z}(v) = f(v, z), then (3.3) says that (H_{f})_{(·,z)} is the usual Hamiltonian vector field of f_{z} with respect to the symplectic form ϕ_{z}. The Hamiltonian H_{α} of a 2-form, on the other hand, is an infinitesimal spin in 𝔳 and an infinitesimal rotation in 𝔷, namely, those determined by −α.

Finally, define the Poisson bracket in F̃ by

(3.5)  {f + α, g + β} = {f, g} + H_{α} · g − H_{β} · f + [α, β],

where [α, β] denotes the Lie bracket in Λ^{2}𝔷^{∗} ≅ 𝔰𝔬(𝔷) and {f, g} = H_{g} · f. We will now prove that the objects just defined satisfy (a) to (h).

(a) By (3.2), (3.3) and (3.4), one has

\begin{aligned}ϕ_{s}((H_{f})_{(v,s)}, X) & = D_{X}f(v, s), \\ ϕ_{s}((H′_{α})_{(v,s)}, X) & = X · α(v, s), \\ ⟨(H′′_{α})_{(v,s)}, Z⟩ & = −⟨B_{α}s, Z⟩ = −α(s, Z) = Z · α(v, s). \end{aligned}

(b) Differentiate the equation ϕ_{s}((H_{f})_{(v,s)}, X) = D_{X}f(v, s) with respect to s in the direction z. Since s ↦ ϕ_{s} is linear and X can be assumed to be a constant vector field, one gets

(3.6)  ϕ_{z}((H_{f})_{(v,s)}, X) + ϕ_{s}((D_{z}H_{f})_{(v,s)}, X) = D_{X}D_{z}f(v, s).

By (2.1) with u = X and v = H_{f},

ϕ_{s}(H_{f}, K_{s,z}X) = −|s|^{2}ϕ_{z}(H_{f}, X)

and therefore

(3.7)  K_{s,z}(X) · f = ϕ_{s}(H_{f}, K_{s,z}X) = −|s|^{2}ϕ_{z}(H_{f}, X).

We see from (3.6), (3.7) and the non-degeneracy of ϕ_{z}, that

f ∈ E  ⇔  K_{s,z}(X)f + |s|^{2}D_{X}D_{z}f = 0  ⇔  D_{z}H_{f} = 0.

(c) Differentiating the function X · f(v, s) = ϕ_{s}((H_{f})_{(v,s)}, X) with respect to v in the direction (H_{α})_{(v,s)} = −A_{α}v − B_{α}s while taking X ∈ 𝔳 constant, we get

−ϕ_{B_{α}(s)}((H_{f})_{(v,s)}, X) + ϕ_{s}((D_{H_{α}}H_{f})_{(v,s)}, X)) = [H_{α}, X] · f(v, s) + XH_{α} · f(v, s)

On one hand,

−ϕ_{B_{α}(s)}(u, v) = −⟨B_{α}(s), Φ(u, v)⟩ = ⟨s, B_{α}(Φ(u, v))⟩.

Since (A_{α}, B_{α}) ∈ 𝔰𝔭(Φ),

⟨s, B_{α}(Φ(u, v))⟩ = ⟨s, Φ(A_{α}u, v)⟩ + ⟨s, Φ(u, A_{α}v)⟩ = ϕ_{s}(A_{α}u, v) + ϕ_{s}(u, A_{α}v),

so that

−ϕ_{B_{α}(s)}((H_{f})_{(v,s)}, X) = ϕ_{s}(A_{α}(H_{f})_{(v,s)}, X) + ϕ_{s}((H_{f})_{(v,s)}, A_{α}(X)).

On the other hand,

[H_{α}, X] = D_{H_{α}}(X) − D_{X}(H_{α}) = 0 + A_{α}(X).

Therefore

ϕ_{s}(A_{α}(H_{f})_{(v,s)}, X) + ϕ_{s}((H_{f})_{(v,s)}, A_{α}(X)) + ϕ_{s}((D_{H_{α}}H_{f})_{(v,s)}, X)
= (A_{α}(X) + XH_{α}) · f(v, s).

The terms ϕ_{s}((H_{f})_{(v,s)}, A_{α}(X)) = A_{α}(X)f(v, s) cancel out and XH_{α} · f(v, s) = ϕ_{s}((H_{H_{α}f})_{(v,s)}, X), so the equation becomes

ϕ_{s}((A_{α}H_{f} + D_{H_{α}}H_{f})_{(v,s)}, X) = ϕ_{s}((H_{H_{α}f})_{(v,s)}, X),

proving the assertion.

(d) If F = f and G = g are in F, the assertion reduces to the standard identity for ordinary symplectic forms, because of the remark after (3.4). If F = α and G = β are in Λ^{2}𝔷^{∗}, (3.5) reduces to {α, β} = [α, β]. Since

(H_{α})_{(v,s)} = −A_{α}v − B_{α}s

and α ↦ A_{α} and α ↦ B_{α} are Lie algebra morphisms,

\begin{aligned}[H_{α}, H_{β}]_{(v,s)} & = −[A_{α}, A_{β}](v) − [B_{α}, B_{β}](s) \\ & = −A_{[α,β]}v − B_{[α,β]}s = (H_{[α,β]})_{(v,s)} = (H_{\{α,β\}})_{(v,s)}. \end{aligned}

Finally, if F = α ∈ Λ^{2}𝔷^{∗} and G = g ∈ F,

\begin{aligned}[H_{α}, H_{g}] & = D_{H_{α}}(H_{g}) − D_{H_{g}}(H_{α}) \\ & = H_{H_{α}g} − A_{α}(H_{g}) + A_{α}(H_{g}) = H_{H_{α}g} = H_{\{α,g\}}. \end{aligned}

The first equality is just the definition of the commutator of two vector fields. The second follows from

D_{H_{α}}(H_{g}) = H_{H_{α}g} − A_{α}(H_{g}),

which is (c), and from

(D_{H_{g}}H_{α})_{(v,s)} = D_{H_{g}}(−A_{α}v − B_{α}s) = −A_{α}(H_{g}),

while the third equality follows from the definition of {α, g}.

(e) Our Poisson bracket is clearly bilinear and skew-symmetric, so we must prove that it satisfies Jacobi’s identity. This identity holds in F because of the corresponding classical statement for scalar-valued forms, while Λ^{2}𝔷^{∗} is already a Lie algebra. Therefore we need to verify it in the cases {α, {f, g}} and {f, {α, β}} (α, β ∈ Λ^{2}𝔷^{∗}, f, g ∈ F).

In the first case, we have

{α, {f, g}}(v, s) = H_{α} · {f, g}(v, s), {f, g}(v, s) = ϕ_{s}((H_{f})_{(v,s)}, (H_{g})_{(v,s)})

so that

{α, {f, g}}(v, s) = H_{α} · {f, g}(v, s)
= ϕ_{−B_{α}(s)}((H_{f})_{(v,s)}, (H_{g})_{(v,s)}) + ϕ_{s}((D_{H_{α}}H_{f})_{(v,s)}, (H_{g})_{(v,s)})
+ ϕ_{s}((H_{f})_{(v,s)}, (D_{H_{α}}H_{g})_{(v,s)}).

But

\begin{aligned}ϕ_{−B_{α}(s)}((H_{f})_{(v,s)}, (H_{g})_{(v,s)}) & = −⟨B_{a}(s), Φ((H_{f})_{(v,s)}, (H_{g})_{(v,s)})⟩ \\ & = ⟨s, B_{α}Φ((H_{f})_{(v,s)}, (H_{g})_{(v,s)})⟩ \end{aligned}

which can be written as

⟨s, Φ((A_{α}H_{f})_{(v,s)}, (H_{g})_{(v,s)}) + Φ((H_{f})_{(v,s)}, (A_{α}H_{g})_{(v,s)})⟩
= ϕ_{s}((A_{α}H_{f})_{(v,s)}, (H_{g})_{(v,s)}) + ϕ_{s}((H_{f})_{(v,s)}, (A_{α}H_{g})_{(v,s)}).

Therefore

{α, {f, g}} = ϕ(A_{α}H_{f}, H_{g}) + ϕ(H_{f}, A_{α}H_{g}) + ϕ(D_{H_{α}}H_{f}, H_{g}) + ϕ(H_{f}, D_{H_{α}}H_{g})
= ϕ(A_{α}H_{f}, H_{g}) + ϕ(H_{f}, A_{α}H_{g}) + ϕ(H_{H_{α}f} − A_{α}H_{f}, H_{g}) +
ϕ(H_{f}, H_{H_{α}g} − A_{α}H_{g})
= ϕ(H_{H_{α}f}, H_{g}) + ϕ(H_{f}, H_{H_{α}g})
= ϕ(H_{{α,f}}, H_{g}) + ϕ(H_{f}, H_{{α,g}})
= {{α, f}, g} + {f, {α, g}}.

In the other case,

{f, {α, β}} = {f, [α, β]} = −H_{[α,β]} · f = −[H_{α}, H_{β}] · f
= −H_{α}H_{β} · f + H_{β}H_{α} · f = −H_{α} · {β, f} + H_{β} · {α, f}
= −{α, {β, f}} + {β, {α, f}} = {{f, α}, β} + {α, {f, β}}.

We now prove that E, Ẽ ⊂ F̃ are subalgebras. From (b), we can deduce that E is a subalgebra if and only if D_{z}H_{{f,g}} = 0 for z ∈ 𝔷 and f, g ∈ E. The last equation follows from (d) and the fact that D_{z} is a derivation on vector fields. Since H_{{α,f}} = H_{H_{α}f}, it follows from (c) and (b) that D_{z}(H_{{α,f}}) = 0 for all z ∈ 𝔷. Because of (b), {α, f} ∈ E. So, Ẽ is a subalgebra as well.

For f ∈ E^{(·,1)} define f_{o} ∈ C^{∞}(𝔳, 𝔷) by f(v, z) = ⟨f_{o}(v), z⟩. Let now f, g ∈ E^{(·,1)} and α ∈ Λ^{2}𝔷^{∗}. Because of (b), H_{f} and H_{g} depend only on v ∈ 𝔳. Therefore, the functions

{f, g}(v, s) = ϕ_{s}((H_{f})_{v}, (H_{g})_{v})

and

\begin{aligned}\{α, f\}(v, s) & = H_{α} · f(v, s) = −D_{A_{α}v}f(v, s) − D_{B_{α}s}f(v, s) \\ & = −⟨D_{A_{α}v}f_{o}(v), s⟩ − ⟨B_{α}s, f_{o}(v)⟩ \end{aligned}

are both smooth in v and linear in s and therefore lie in E^{(·,1)}. Hence both E^{(·,1)} and Ẽ^{(·,1)} are subalgebras.

(f) For u ∈ 𝔳 and z ∈ 𝔷 define the real-valued functions on 𝔫:

(3.8)  q_{u}(v, s) = ⟨s, Φ(u, v)⟩,  c_{z}(v, s) = ⟨s, z⟩.

Then the map Θ : u + z ↦ q_{u} + c_{z} determines a Lie isomorphism

N ≅ E^{(1,1)} ⊕ E^{(0,1)}.

Indeed, we easily see that H_{q_{u}} = u and H_{c_{z}} = 0, so, from (b), q_{u} + c_{z} ∈ E^{(1,1)} ⊕ E^{(0,1)}. Also,

{q_{u}, q_{u′}}(v, s) = (H_{q_{u′}} · q_{u})(v, s) = ⟨s, Φ(u, u′)⟩ = c_{Φ(u,u′)}(v, s),

while {q_{u}, c_{z}} = (H_{c_{z}} · q_{u}) = 0 and {c_{z}, c_{z′}} = 0. Since the bracket in 𝔫 is given by [u + z, u′ + z′] = Φ(u, u′), we have

\begin{aligned}Θ([u + z, u′ + z′]) & = c_{Φ(u,u′)} = \{q_{u}, q_{u′}\} = \{q_{u} + c_{z}, q_{u′} + c_{z′}\} \\ & = \{Θ(u + z), Θ(u′ + z′)\}, \end{aligned}

so Θ is a Lie morphism. To see that it is surjective, let g ∈ E^{(1,1)}. Since g is bilinear, there exist T ∈ Hom(𝔳, 𝔷) such that

(3.9)  g(v, s) = g_{T} (v, s) := ⟨s, T v⟩, for v ∈ 𝔳, s ∈ 𝔷.

Because g ∈ E, H_{g} is constant along 𝔷. Therefore

⟨s, Φ((H_{g})_{v}, X)⟩ = ϕ_{s}((H_{g})_{v}, X) = X · g(v, s) = ⟨s, T X⟩.

We conclude that Φ((H_{g})_{v}, X) = T X and, consequently, H_{g} is also independent of v. Letting u = H_{g} ∈ 𝔳,

g(v, s) = ⟨s, T v⟩ = ⟨s, Φ(u, v)⟩,

from which g = q_{u}. On the other hand, if g ∈ E^{(0,1)}, then g(v, s) = ⟨s, z⟩ with z ∈ 𝔷, hence g = c_{z}, showing that the map is onto.

(g) We must prove that the operators G ↦ {F, G} with F ∈ E^{(2,1)} realize, upon restriction to E^{(1,1)}⊕E^{(0,1)} ≅ 𝔫, all of 𝔰𝔭_{o}(Φ), viewed as subalgebra of Der(𝔫).

Let f ∈ E^{(2,1)} = E ∩ F^{(2,1)}, i.e., f(v, z) is a homogeneous polynomial of degree 2 in v and of degree 1 in 𝔷, satisfying the differential equation (3.1). As we have already observed, its Hamiltonian H_{f} is just the classical Hamiltonian relative to ϕ_{s} and it is independent of s ∈ 𝔷. Therefore

ϕ_{s}((H_{f})_{v}, v′) + ϕ_{s}(v, (H_{f})_{v}) = 0,

showing that H_{f} ∈ 𝔰𝔭_{o}(Φ). Since H_{g}f = {f, g}, we conclude that E^{(2,1)} acts on 𝔫 as 𝔰𝔭_{o}(Φ).

To see that E^{(2,1)} is isomorphic to 𝔰𝔭_{o}(Φ) define, for any Q ∈ 𝔰𝔭_{o}(Φ), the function

(3.10)  p_{Q}(v, s) := \frac{1}{2}⟨s, Φ(Qv, v)⟩ v ∈ 𝔳, s ∈ 𝔷

and prove that H_{p_{Q}} = Q. Indeed, ϕ_{s}(Qu, v) + ϕ_{s}(u, Qv) = 0, so

ϕ_{s}((H_{p_{Q}})_{(v,s)}, X) = X · p_{Q}(v, s) = \frac{1}{2}ϕ_{s}(QX, v) + \frac{1}{2}ϕ_{s}(Qv, X) = ϕ_{s}(Qv, X).

Therefore p_{Q} ∈ E^{(2,1)}. Moreover, two functions in E^{(2,1)} with the same Hamiltonian are equal, therefore f ↦ H_{f} is the inverse map of Q ↦ P_{Q}. Because of (d), F ↦ H_{F} is a Lie morphism. From Proposition 2.1 and the fact that Ẽ^{(2,1)} = E^{(2,1)}⊕Λ^{2}𝔷^{∗}, we conclude that 𝔰𝔭(Φ) ≅ Ẽ^{(2,1)} and, therefore, that Ẽ^{(2,1)} acts as 𝔰𝔭(Φ).

(h) Identifying 𝔫 with E^{(1,1)} ⊕E^{(0,1)}, 𝔰𝔭_{0}(Φ) with E^{(2,1)} and Hom(𝔳, 𝔷) with F^{(1,1)}, we must prove that the action of Der(𝔫) on 𝔫 is by inner derivations of the Poisson bracket. Equivalently, that the functions q_{u}, c_{z}, p_{Q} and g_{T}, as defined in (3.8), (3.9) and (3.10), satisfy the following commutation relations:

(i) {p_{Q}, q_{u}} = q_{Qu}

(ii) {α, q_{u} + c_{z}} = q_{A_{α}u} + c_{B_{α}z}

(iii) {g_{T} , q_{u}} = c_{T u}

(iv) {g_{T} , c_{z}} = 0

for all u ∈ 𝔳, z ∈ 𝔷, α ∈ Λ^{2}𝔷^{∗}, T ∈ Hom(𝔳, 𝔷) and Q ∈ 𝔰𝔭_{o}(Φ).

To prove (i), just compute

q_{u}(v, s) = −H_{p_{Q}}q_{u}(v, s) = −⟨s, Φ(u, Qv)⟩ = ϕ_{s}(Qu, v) = q_{Qu}(s, v).

For (ii),

{α, q_{u}}(v, s) = H_{α} · q_{u}(v, s) = −D_{A_{α}v}q_{u} − D_{B_{α}s}q_{u}
= −⟨s, Φ(u, A_{α}v)⟩ − ⟨B_{α}s, Φ(u, v)⟩
= −⟨s, Φ(u, A_{α}v)⟩ + ⟨s, B_{α}Φ(u, v)⟩

= −⟨s, Φ(u, A_{α}v)⟩ + ⟨s, Φ(A_{α}u, v)⟩ + ⟨s, Φ(u, A_{α}v)⟩ = ⟨s, Φ(A_{α}u, v)⟩

= q_{A_{α}u}(v, s),

and, similarly,

{α, c_{z}}(v, s) = H_{α} · c_{z}(v, s) = −D_{B_{α}s}c_{z} = −⟨B_{α}s, z⟩ = ⟨s, B_{α}z⟩ = c_{B_{α}z}(v, s).

Finally, (iii) follows from

{g_{T} , q_{u}}(v, s) = H_{q_{u}} · g_{T} (v, s) = D_{u}g_{T} (v, s) = ⟨s, T u⟩ = c_{T u}(v, s),

and (iv) from

{q_{T} , c_{z}}(v, s) = H_{c_{z}} · g_{T} (v, s) = 0.

Remark 3.2. F^{(·,1)} is not closed under { , }. Also, in general, {F, G} ≠ H_{F} ·G; instead one has the identity (3.5). For example, let F = f and G = α. Then {f, α} = −H_{α}f = (A_{α}v)f+(B_{α}s)f while (H_{f})_{(v,s)}·α = −ϕ_{s}(A_{α}v, H′_{f})−α(s, H′′_{f}) = ϕ_{s}(H_{f}, A_{α}v) = (A_{α}v)f.

References

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[2] Damek, E., and F. Ricci, A class of non-symmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139–142.

[3] Galina, E., A. Kaplan and F. Levstein, The Oscillator Representation and groups of Heisenberg Type, Comm. Math. Phys. 210 (2000), 309–321.

[4] Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE, Trans. Amer. Math. Soc. 258 (1980), 147–153.

[5] Kaplan, A., and F. Ricci Harmonic Analysis on Groups of Heisenberg type, Lect. Notes in Math 992 (1983), 416–435.

[6] Lawson, H., and M. Michelsohn, “Spin Geometry,” Princeton U. Press, (1989).

[7] Levstein, F., and A. Tiraboschi, Classes of 2-step nilpotent Lie algebras , Comm. in Algebra 27 (1999), 2425–2440.

[8] Riehm, C., The automorphism group of a composition of quadratic forms, Trans. Amer. Math. Soc. 269 (1982), 403–414.

[9] Saal, L., The automorphism group of a Lie algebra of Heisenberg Type, Rend. Sem. Mat. Univ. Pol. Torino 54 (1996), 101–113.

Aroldo Kaplan and Alejandro Tiraboschi CIEM-FAMAF Medina Allende y Haya de la Torre Ciudad Universitaria (5000) Córdoba Córdoba Argentina kaplan@famaf.unc.edu.ar tirabo@famaf.unc.edu.ar Linda Saal FAMAF Medina Allende y Haya de la Torre Ciudad Universitaria (5000) Córdoba Córdoba Argentina saal@famaf.unc.edu.ar

Received June 20, 2001 and in final form Mai 29, 2002