Symmetries of the Space of Linear Symplectic Connections

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.


Introduction
A (linear) symplectic connection on a symplectic manifold (M, Ω) is a torsion-free affine connection preserving the symplectic form Ω. The space S(M, Ω) of symplectic connections is a symplectic affine space; the difference of two symplectic connections is identified with a section of the bundle S 3 (T * M ) of completely symmetric covariant three tensors, and the symplectic form Ω on S(M, Ω) is given by integrating over M the pairing induced on such sections by Ω. The geometry of the symplectic affine space (S(M, Ω), Ω) is the focus of this note.
Efforts have been made to identify geometrically interesting classes of symplectic connections analogous to classes of connections studied in the Riemannian setting, such as the Levi-Civita connections of Einstein or constant scalar curvature metrics. Such classes are defined in terms of the zeros and critical points of functionals on (S(M, Ω), Ω) equivariant or invariant with respect to some action of some group of symplectomorphisms or Hamiltonian diffeomorphisms.
The geometry of any symplectic space is reflected in its Poisson algebra of functions. Interesting functionals on (S(M, Ω), Ω) are constructed from the curvature of ∇ ∈ S(M, Ω). The simplest interesting classes of symplectic connections that have been studied are those of Ricci type, the preferred symplectic connections (these include the Ricci flat symplectic connections), and the critical symplectic connections. The focus here is on preferred and critical symplectic connections. The critical points of the integral of the trace of the square of the Ricci endomorphism are the preferred symplectic connections first studied by F. Bourgeois and M. Cahen [5,6]. A symplectic connection is preferred if and only if there vanishes the complete symmetrization of the covariant derivative of its Ricci tensor Ric ij = R ij , that is, it solves (δ * Ric) ijk = −∇ (i R jk) = 0. The functionals given by integrating the trace of higher even powers of the Ricci endomorphism play a role in the averaging procedure used by Fedosov in [9], and, as is explained briefly in Section 4.3, their critical points generalize preferred symplectic connections. However, because the first Pontryagin class of a symplectic manifold equals the integral of an expression quadratic in the curvature of the connection, the class of preferred symplectic connections is the only interesting class of symplectic connections obtained as the critical points of a functional given by integrating expressions quadratic in the curvature tensor (this observation is due to Proposition 2.4 of [6]; see also the introduction of [10]).
In [7], M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on S(M, Ω) is Hamiltonian, with a moment map denoted here by K(∇) (see (4.3) for its definition). In [10], the author began the study of the critical symplectic connections, that are defined to be the critical points of the integral E(∇) =´M K(∇) 2 Ω n (where Ω n = 1 n! Ω n ) with respect to arbitrary variations of ∇. A symplectic connection ∇ is critical if and only if the Hamiltonian vector field H K(∇) generated by K(∇) is an infinitesimal automorphism of ∇. Explicitly this means that H(K(∇)) = 0 where H(f ) ijk = (L H f ∇) ij p Ω pk , H f is the Hamiltonian vector field generated by f ∈ C ∞ (M ), and L H f is the Lie derivative along H f . The moment map K(∇) is the sum of a multiple of the twofold divergence of the Ricci tensor of ∇ and a multiple of the complete contraction of the first Pontryagin form of ∇ with Ω∧Ω. By the main theorem of D. Tamarkin [19], a functional on S(M, Ω) given by integrating against Ω n polynomials in the curvature tensor and its covariant derivatives contracted with the symplectic form is independent of the choice of connection if and only if its integrand is, modulo divergences, a polynomial in Pontryagin forms contracted with the symplectic form. It follows that K is in some sense, not made precise here, the simplest symplectomorphism equivariant map from S(M, Ω) to C ∞ (M ).
Suppose (M, Ω) is compact, and fix a reference symplectic connection ∇ 0 . Here there are constructed, for each (s, t) ∈ R × R × , a Lie bracket (( · , · )) s,t on and a Hamiltonian action of the Lie algebra ( h, (( · , · )) s,t ) on (S(M, Ω), Ω). The associated moment map is essentially a linear combination of K and the Ricci tensor (see (6.16) for the precise expression when (s, t) = (1, 1); it requires some preliminary discussion to make sense of it). The choice of ∇ 0 is unimportant in the sense that the Lie algebras associated with different choices of ∇ 0 are isomorphic. In fact, the brackets (( · , · )) s,t (and corresponding Hamiltonian actions) for which neither s nor t is 0 are all pairwise isomorphic, so isomorphic to (( · , · )) = (( · , · )) 1,1 ; the reason for considering the full family is that it interpolates between the degenerate cases (s, t) = (1, 0) and (s, t) = (0, 1) (this is discussed further below). In (1.1), the factor C ∞ (M ) should be seen as the central extension of the Lie algebra ham(M, Ω) of Hamiltonian vector fields, while Γ(S 2 (T * M )) should be seen as the Lie algebra of infinitesimal gauge transformations of the symplectic frame bundle. The Lie bracket (( · , · )) combines and extends these actions. The somewhat complicated definition of the bracket (( · , · )) is summarized now. The precise definition is (6.14) of Section 6.1; see also Theorem 6.4. The algebra S(M ) whose elements are formal linear sums of completely symmetric tensors on (M, Ω) (see Section 3 for the precise meaning) carries two Lie brackets, a differential Lie bracket · , · induced from the Schouten bracket on symmetric contravariant tensors on M and an algebraic Lie bracket ( · , · ) induced fiberwise by regarding the symmetric tensor algebra as the associated graded algebra of the Weyl algebra. The definition of (( · , · )) requires Lemma 3.5, showing that these Lie brackets are compatible in the sense of bi-Hamiltonian systems; precisely, the differential, in the Lie algebra cohomology of ( · , · ), of the symmetrized covariant derivative along ∇ equals the Schouten bracket · , · (see (2) of Lemma 3.5). (Although it seems that this claim about the Schouten bracket must be known in some form to experts, I do not know a reference.) This means that any linear combination [ · , · ] s,t = s( · , · ) + t · , · is again a Lie bracket.
The space S(M ) is too big because its 1-graded part comprises all one-forms on M , rather than the closed one-forms, that are dual to symplectic vector fields. Because the [ · , · ] = [ · , · ] 1,1 bracket of closed one forms in S(M ) need not be closed, it is inadequate to restrict to the subspace of S(M ) the 1-graded part of which comprises closed one-forms (although something can be done in this directions; see Lemma 3.11). However, the subspace H(M ), comprising formal sums α 0 + α 1 + · · · such that α k ∈ Γ(S k (T * M )) and α 1 = −dα 0 is a subalgebra of ( S(M ), [ · , · ]), and this subalgebra plays a prominent role in the discussion. Lemmas 3.7 and 3.11 describe its structure.
Although ( S(M ), [ · , · ]) cannot be made to act on (S(M, Ω), Ω) in any obvious way, the subalgebra H(M ) admits a symplectic affine action on (S(M, Ω), Ω) such that the ideal H 4 (M ) = k≥4 Γ(S k (T * M )) acts trivially. This action is weakly Hamiltonian, but the cocycle obstructing equivariance can be computed explicitly. This yields a central extension that acts in a Hamiltonian manner on (S(M, Ω), Ω), with moment map M. Since the Lie ideal H 4 (M ) acts trivially, quotienting by it yields an action of the Lie algebra ( h, (( · , · ))).
The action of the group of gauge transformations of the symplectic frame bundle of M on the space of affine connections, possibly having torsion, preserving Ω, is Hamiltonian with a moment map constructed in the manner of Atiyah and Bott in [1]. See Theorem 5.1 for the precise statement. Using the symplectic form the Lie algebra of infinitesimal gauge transformations can be identified with the space Γ(S 2 (T * M )) equipped with a constant multiple of the algebraic Lie bracket mentioned above. The obstruction to this Lie algebra acting on S(M, Ω) is given by the Schouten bracket. However, an extended bracket on Γ(S 2 (T * M )) ⊕ Γ(S 3 (T * M )) built from the Poisson bracket and the algebraic bracket does act on S(M, Ω), by symplectic affine transformations. This extended action can be combined in a coherent way with the action of ham(M, Ω) on S(M, Ω). While the resulting action is only weakly Hamiltonian, the cocycle measuring nonequivariance of the moment map can be computed explicitly and from it there is constructed the Hamiltonian action of ( h, (( · , · )) s,t ). This action combines, at the infinitesimal level, the actions of Ham(M, Ω) and the group of gauge transformations to yield a Hamiltonian action with a moment map M that simultaneously extends the moment map K for the action of the Lie algebra ham(M, Ω) of Hamiltonian vector fields and the Atiyah-Bott moment map.
When M is compact, there is constructed (see (6.17) for the precise definition) from M a functional N : S(M, Ω) → R. There is a graded symmetric bilinear pairing on H(M ) that descends to h. Pairing the moment map M with itself with respect to this pairing yields N. The critical points of N are the solutions of The equations (1.2) interpolate between the equations for preferred and critical symplectic connections. The case (s, t) = (1, 0) yields the equations for the preferred symplectic connections introduced by Bourgeois and Cahen [6], and the case (s, t) = (0, 1) recovers the equations for the critical symplectic connections introduced by the author in [10].
Remark 1.1. In [10] it is shown that on a 2-manifold a preferred symplectic connection is critical. Consequently, on a 2-dimensional symplectic manifold a preferred symplectic connection solves (1.2) for any (s, t) ∈ R 2 .
Section 3 gives the background needed to formulate and prove Lemma 3.5. Section 4 recalls the definitions and basic characterizations of critical and preferred symplectic connections. Section 5 discusses the failure of the symplectomorphism group to act in a Hamiltonian manner on S(M, Ω). Finally, the construction of (( · , · )) s,t and its action on S(M, Ω) is given in Section 6.

2.1.
Smooth means C ∞ , and all manifolds, bundles, sections, etc. are smooth, unless otherwise indicated. Throughout, (M, Ω) is a connected 2n-dimensional symplectic manifold oriented by the Liouville volume form Ω n = 1 n! Ω n . For simplicity, M is often assumed compact, although most claims generalize straightforwardly to the noncompact setting.
For a finite-dimensional real vector space V, S k (V) and Λ k (V) denote the kth symmetric and antisymmetric powers of V. Purely algebraic constructions on vector spaces extend straightforwardly to vector bundles and their spaces of sections, and the same notations will be used in both contexts. For a smooth vector bundle E → M , Γ(E) denotes the vector space of smooth sections of E, and Λ k (E) and S k (E) denote the kth antisymmetric and symmetric powers of E.
Tensors are usually indicated using the abstract index conventions (see [17] or [20]). Indices are labels and do not indicate a choice of frame. An index is in either up or down position. Up (down) indices label contravariant (covariant) tensors. For example, ∇ i ∇ j X k (which could also be written (∇ 2 X) ij k ) indicates the second covariant derivative of the vector field X k . Were a frame E i chosen, The up index k on X k is simply a label that indicates that X is a contravariant vector field. The summation convention is used in the following form: repetition of a label in up and down position indicates the trace pairing. Enclosure of indices in square brackets (parentheses) indicates complete antisymmetrization (symmetrization) over the enclosed indices, so that, for example, a ij = a (ij) + a [ij] indicates the decomposition of a contravariant two-tensor into its symmetric and skew-symmetric parts. An index included between vertical bars | | is omitted from an indicated (anti)symmetrization; for example 2a The conventions are illustrated by the following definitions. The curvature R ijk l and torsion τ ij k of an affine connection ∇ are defined by (Sometimes, for readability, Ric is written instead of R ij .) The antisymmetric bivector Ω ij inverse to Ω ij is defined by Ω ip Ω pj = −δ j i . Indices are raised and lowered using Ω ij and Ω ij by contracting with these tensors consistently with the conventions X i = X p Ω pi and X i = Ω ip X p . When it is raised or lowered, an index's horizontal position is maintained. The symplectic sharp and flat operators on a vector field X i and a oneform α i are defined by X i = X p Ω pi and α i = Ω ip α p . These are inverses, so that (X ) = X. For a vector field X, X i and X are synonyms and both notations will be used.

2.2.
Let Symp(M, Ω) be the group of compactly supported symplectomorphisms of (M, Ω). Its Lie algebra symp(M, Ω) comprises compactly supported vector fields X that are locally Hamiltonian, meaning L X Ω = 0; equivalently X = Ω(X, · ) is closed. Define the Hamiltonian vector field

2.
3. An action of a Lie group G on a symplectic manifold (M, Ω) is symplectic if G acts by symplectic diffeomorphisms. The associated Lie algebra homomorphism from the Lie algebra g of G to symp(M, Ω), defined by x → X x p = d dt t=0 exp(−tx) · p, is weakly Hamiltonian if there is a map µ : M → g * such that for each x ∈ g, the Hamiltonian vector field H µ(x) equals the vector field X x determined by the action. If µ is equivariant with respect to the given action of G on M and the coadjoint action of G on g * , then the action is Hamiltonian and µ is a moment map. When G is simply connected, this equivariance is equivalent to the requirement that when viewed as a map from g to C ∞ (M ), the map µ be a Lie algebra homomorphism, {µ(x), µ(y)} = µ([x, y]), and the induced action of g is said to be Hamiltonian if it satisfies this last condition. For a weakly Hamiltonian action that is not necessarily Hamiltonian, the associated µ is called a nonequivariant moment map. Since some authors do not require equivariance in the definition of a moment map, the redundant expression equivariant moment map is sometimes used for clarity. These definitions are applied in the infinite-dimensional context, in which case they are to be understood formally.

2.4.
This subsection fixes terminology related to affine actions of a Lie algebra. This is needed mainly in Section 6. An affine representation of a Lie algebra (g, [ · , · ]) on the affine space A is a Lie algebra homomorphism ρ : g → aff(A), that is, a linear map satisfying ρ([a, b]) = L(ρ(a))ρ(b) − L(ρ(b))ρ(a) for all a, b ∈ g. An affine action of a Lie algebra (g, [ · , · ]) on the affine space A is a biaffine map π : g × A → A such that π(0, p) = p for all p ∈ A and a · (b · p) − b · (a · p) = [a, b] · p − p for all p ∈ A and all a, b ∈ g. That this identity is an equality of vectors in V explains the need for the p on its right-hand side. That π be biaffine means that the maps p → a · p = π(a, p) and a → a · p = π(a, p) are affine maps from A to A for all a ∈ g and from g to A for all p ∈ A. (Note that a biaffine map need not be affine with respect to the product affine structure on g × A.) Lemma 2.1. For an affine space A and a Lie algebra g, a map ρ : g → aff(A) is an affine representation if and only if π : g × A → A defined by π(a, p) = p + ρ(a)p is an affine action.
First one shows that ρ is linear if and only if π is biaffine and π(0, p) = 0 for all p ∈ A. Then one checks directly that, in this case, The representation ρ is said to be associated with the affine action π. The stabilizer of p 0 ∈ A under the affine action of g means the subalgebra {a ∈ g : a · p 0 = p 0 } = {a ∈ g : ρ(a)p 0 = 0}.
A typical example is the affine action X ·∇ = ∇+L X ∇ of the Lie algebra of vector fields on M on the space of affine connections on M . The associated affine representation is ρ(X) = L X ∇.
A symplectic affine space means an affine space A equipped with a symplectic form Ω invariant under the action of the group of translations of A. An affine representation ρ : g → aff(A) on a symplectic affine space (A, Ω) is symplectic affine if each ρ(a) is symplectic, meaning the linear part L(ρ(a)) satisfies Ω(L(ρ(a))u, v) + Ω(u, L(ρ(a))v) = 0 for all a ∈ g and u, v ∈ V .
Similarly, an affine action is symplectic affine if the corresponding affine representation is symplectic.

2.5.
An affine connection is symplectic if ∇ i Ω jk = 0. For background about symplectic connections see [2] or [8]  , and it is mainly in Section 5.1 that reference will be made to T(M, Ω). Throughout this paper symplectic connection means a torsion-free symplectic connection, while symplectic affine connection means a connection preserving Ω but possibly having torsion.
The group of diffeomorphisms acts on the space of affine connections on the right, by pullback.

Symmetric tensors on symplectic manifolds
3.1. The symmetric tensor algebra S(V), which is, by definition, a quotient of the tensor algebra, can be identified, as a graded vector space, with the graded vector space of homogeneous degree k polynomials on the dual vector space V * , and the product corresponds to multiplication of polynomials. Elements of the formally completed symmetric tensor algebraŜ(V) = k≥0 S k (V) are identified with formal infinite sums of symmetric tensors, and correspond with formal power series on V * .
Let An are ideals with respect to the commutative algebra structure determined by . The p = 0 case of (2.2) extends to a graded symmetric pairing on S(V) in such a way that S k (V) and S l (V) are orthogonal if k = l. Likewise, as commutative algebras, S(M ) and S(M ) are filtered by the ideals S k (M ) and S k (M ) comprising sums of tensors of degree at least k, and the pairing (2.3) extends to a graded symmetric pairing on S(M ).

3.2.
If (V, Ω) is a symplectic vector space, then S(V * ) carries a Lie bracket, determining with a Poisson algebra, that can be defined most simply as the unique Poisson bracket ( · , · ) on (S(V * ), ) such that (α, β) = Ω ij α i β j for all α, β ∈ V * . By definition, ( · , · ) has degree −2. The bracket ( · , · ) can be described explicitly as follows. The space of functions on V carries the Poisson structure determined by Ω. Transporting this Poisson structure to S(V * ) via the identification of S(V * ) with the ring of polynomials on V, yields the bracket ( · , · ) on S(V * ). Concretely, for α ∈ S k (V * ), the corresponding polynomialα ∈ Pol k (V) isα(u) = α i 1 ...i k u i 1 · · · u i k , and it generates the Hamiltonian vector field In the case l = 1 and k > 1, (α, β) = kβ p α pi 1 ...i k−1 is simply k times interior multiplication of the vector field β i in α. The Poisson bracket (3.1) extends toŜ(V * ) with the same definition.
The additive filtration of the Weyl algebra at level k is spanned by k-fold products of elements of V. Its associated graded algebra is isomorphic as a vector space to the symmetric algebra of V and carries a Poisson bracket defined by projecting the commutator of elements of level k and l onto level k + l − 2. Applying this construction to (V * , Ω) yields the algebraic bracket (3.1).
Remark 3.2. Another construction of (3.1) can be given in terms of the notion of prolongation. This is recalled following [12]; see also [18,Chapter 7.3] and [14, Chapter I.1].
The first prolongation u for all u, v ∈ V}, and, for k ≥ 2, the kth prolongation g (k) of g is defined inductively by g (k) = (g (k−1) ) (1) . As vector spaces, When W = V, so g ⊂ End(V), define g (0) = g and g (−1) = V. Using the identification (3.2), a graded Lie bracket is defined on the direct sum ⊕ k≥−1 g (k) and its formal completion k≥−1 g (k) as follows. Define an antisymmetric bilinear pairing [ · , · ] : , and it is straightforward to check that the image is contained in g (k+l) and that the resulting bracket satisfies the Jacobi identity, so makes ⊕ k≥−1 g (k) and k≥−1 g (k) into Lie algebras. In the case (V, Ω) is a symplectic vector space and g = ( V, Ω), this can be seen as follows. If the differential of the flow of a vector field on V preserves the symplectic frame bundle V×Sp(V, Ω) then the (k+1)st coefficient of the Taylor expansion of the vector field takes values in g (k) . The Lie bracket is that induced by taking jets of Lie brackets of vector fields. Since W = V, an element of g (0) can be viewed as an endomorphism of V; the definition (3.3) is made so that the bracket [ · , · ] agrees on g (0) with the algebraic commutator of endomorphisms.
The preceding can be recast in the following manner. The graded linear map η : is a central extension of Lie algebras, where ι is the inclusion of R as the central ideal S 0 (V * ) ⊂ (S(V * ), ( · , · )). The map η is an algebraic version of the operator associating to the polynomial Remark 3.3. A horizontal distribution on a symplectic fibration such that the corresponding parallel transport preserves the fiberwise symplectic structure is also called a symplectic connection (see for example [16]). The symplectic affine connections considered here are symplectic connections in this more general sense, although the converse is clearly false. For this reason (and also to honor their role in Fedosov's deformation quantization) what are here called symplectic connections are sometimes called Fedosov connections; here the notions are distinguished terminologically as linear and nonlinear symplectic connections.
The two notions encapsulate different points of view. Each is equivalent to the data of a principal connection on what might be called the symplectic frame bundle of M , the point being that symplectic frame bundle has two possible meanings, depending on whether symplectic refers to the symplectic linear group or the symplectomorphism group of the reference symplectic vector space (V, Ω). The linear symplectic frame bundle is the reduction of the usual linear frame bundle determined by restricting to symplectic frames. A much flabbier notion is obtained by instead considering as frames at p ∈ M all symplectomorphisms from (V, Ω) to (T p M, Ω) mapping 0 ∈ V to 0 ∈ T p M . Although to define such a nonlinear symplectic frame bundle rigorously requires restricting the class of symplectomorphisms considered, or working formally, e.g., with infinite jets, morally there results a principal bundle with structure group an infinite-dimensional group of (perhaps formal) symplectomorphisms of (V, Ω), and, by construction, F is a reduction of this bundle. Correspondingly, there are two quite distinct notions of symmetries of S(M, Ω). The usual one is to consider principal bundle automorphisms of the linear symplectic frame bundle F, regarded as a principal bundle for the finite-dimensional group G = Sp(V, Ω) of linear symplectic transformations. On the other hand, the symplectic frame bundle F can be regarded as a subbundle of a principal bundle with structure group some infinite-dimensional Lie group G of symplectomorphisms of (V, Ω) fixing the origin.
The automorphism group of F comprises the G-equivariant bundle maps from F to itself. The group G(F) of gauge transformations of F is its subgroup comprising bundle automorphisms of F covering the identity. A gauge transformation is naturally identified with a map from F to G, equivariant with respect to the action of G on itself by conjugation, or, equivalently, with a section of the bundle F × G G associated with F by this action. Via this identification the group structure corresponds to fiberwise composition. An infinitesimal gauge transformation is a section of the adjoint bundle Ad(F). These form a Lie algebra L(F) under fiberwise Lie bracket.
In the case of a frame bundle, a (infinitesimal) gauge transformation can be identified with a section of the bundle End(T M ) of endomorphisms of the tangent bundle. Precisely, an element Similarly, an infinitesimal gauge transformation of F is identified with a section α i j ∈ Γ(End(T M )) such that α [ij] = 0; in general it is more convenient to identify α with the corresponding element α ij of S 2 (M ). Alternatively, the Lie algebra g is isomorphic as a G-module to S 2 (V * ) via the map A → A defined by A (x, y) = Ω(Ax, y), and via this isomorphism Γ(Ad(F)) is identified with (S 2 (M ), [ · , · ]).
The preceding all makes formal sense with the nonlinear symplectic frame bundle in place of F, and with the corresponding structure group G in place of G. Such notions can be given rigorous sense at the Lie algebra level. The formal analogue of the infinitesimal G-gauge transformations is the Lie algebra ( S 2 (M ), [ · , · ]), corresponding to the fiberwise action of the truncated prolongation ( i≥0 g (i) , [ · , · ]) (as in Remark 3.2), comprising formal sums of covariant symmetric tensors of rank at least two.
The formal adjoint δ * ∇ : S k (M ) → S k+1 (M ) of δ ∇ is given by When it is not necessary to indicate the dependence on ∇ the subscripts are omitted and there are written δ and δ * instead of δ ∇ and δ * ∇ . When δ * is applied to functions the subscript is always omitted, because δ * f = −df = H f . Note that the Poisson bracket {f, g} of f, g, ∈ C ∞ (M ) is expressible as {f, g} = (δ * f, δ * g).
Proof . The Ricci identity yields Combining (3.6) with By the definition (3.7), the Schouten bracket of functions is trivial. In (3.7) the signs are chosen so that X , Y = [X, Y ] for X, Y ∈ Γ(T M ). More generally, if X ∈ symp(M, Ω) then X , α = L X α. In particular, L H f α = α, df = δ * f, α for f ∈ C ∞ (M ). Since symp(M, Ω) is a Lie algebra, the Schouten bracket of closed one-forms is a closed one-form. The operator δ * is a homomorphism from (C ∞ (M ), { · , · }) to (ham(M, Ω), · , · ), as Since the pairing (3.7) does not depend on the choice of ∇ ∈ S(M, Ω), it should be regarded as an object of a differential topological character attached to the symplectic manifold. In particular, the Schouten bracket is Symp(M, Ω)-equivariant. The corresponding infinitesimal statement, the infinitesimal symplectomorphism equivariance of the Schouten bracket, is equivalent to the statement that the Lie derivative L X along X ∈ symp(M, Ω) is a derivation of · , · . Alternatively this is a consequence of the identity X , α, β = L X α, β in conjunction with the Jacobi identity.
3.6. Two Lie brackets [ · , · ] and · , · on a vector space g are compatible if any linear combination of them is again a Lie bracket. For background on compatible Lie brackets see, for example, [3,4,11], and the references therein. A straightforward computation shows that compatibility of two Lie brackets is equivalent to the condition that each of the brackets is a cocyle with respect to the other; this means · , · is a cocyle of the cochain complex of (g, [ · , · ]) with coefficients in its adjoint representation, and likewise with [ · , · ] and · , · interchanged. From this second characterization it follows that if one of two Lie brackets is a coboundary in the Lie algebra cohomology of the other, then the brackets are compatible. Lemma 3.5 gives an interpretation of δ * in terms of the Lie algebra cohomology of ( · , · ).

The Cahen-Gutt moment map K : S(M, Ω) → C ∞ (M ) is defined by
The map K was defined by M. Cahen and S. Gutt in [7] (see also [2] or [13]  Indication of proof . For a complete proof see [10]. LetR ijk l be the curvature of∇ = ∇ + tΠ ij k and label with a¯the tensors derived from it; for exampleR ij is the Ricci curvature of∇. Then    for its kth power as a fiberwise endomorphism. Since (Rc •k ) ij = (−1) k+1 (Rc •k ) ji , (Rc •k ) ij is symmetric if k is odd, and skew-symmetric if k is even. In particular, tr Rc •2k+1 = 0 and Proof . For Π ∈ T ∇ S(M, Ω), by (4.6) and (4.11), the last equality because the integral of a divergence vanishes. Here ∇(t) is a path in S(M, Ω) and c ∈ R. By Lemma 4.8, E is constant along the flows (4.14).
where g iq g j q = g i p g j q Ω pq = Ω ij and (g −1 ) ij = −g ji . The covariant derivatives associated with principal connections on F preserve the symplectic structure, but need not be torsionfree, and the action of G(F) does not preserve the torsion, for, by (5.1), if ∇ is torsion-free, then ∇ g is torsion-free if and only if ∇ [i g j] k = 0. This need not be true in general. The action (5.1) is affine (as in Section 2.4). Its linear part L(g) :
Arguing as in the proof of Lemma 5.4 shows directly that (5.17) defines a symplectic affine action if q = p + 1 and r = 3 = s. In this case, (5.17) has the form (5.16), whatever is the value of p.
6 Hamiltonian action of the extended Lie algebra on (S(M, Ω), Ω) In this section M is supposed compact. This is needed only to guarantee convergence of the integrals that appear. With appropriate qualifications, everything extends to noncompact M , but this requires fussing that would be distracting here. The dependence of M p,q,r α on the reference connection ∇ 0 is not indicated so that the notation does not become too cluttered. By In the case p = −1, q = 3, and r = 3, there is written simply M for M −1,3,3 . Lemma 6.2 computes the Poisson brackets { {M α , M β } }. The following preliminary lemma is needed in its proof.