A Poisson bracket on the space of Poisson structures
aa r X i v : . [ m a t h . DG ] A ug A POISSON BRACKET ON THE SPACE OF POISSONSTRUCTURES
THOMAS MACHON
Abstract.
Let M be a smooth closed orientable manifold and P( M ) thespace of Poisson structures on M . We construct a Poisson bracket on P( M ) depending on a choice of volume form. The Hamiltonian flow of the bracketacts on P( M ) by volume-preserving diffeomorphism of M . We then define aninvariant of a Poisson structure that describes fixed points of the flow equationand compute it for regular Poisson 3-manifolds, where it detects unimodularity.For unimodular Poisson structures we define a further, related Poisson bracketand show that for symplectic structures the associated invariant counting fixedpoints of the flow equation is given in terms of the dd Λ and d + d Λ symplecticcohomology groups defined by Tseng and Yau [16]. Contents
1. Introduction and summary of the construction 22. The differential δ µ
43. The space of Poisson structures 84. The Poisson bracket {⋅ , ⋅} µ
95. Casimirs 136. The isochoric bracket on µ -unimodular structures 147. The invariants 158. Rank-2 Poisson structures 179. The isochoric bracket on symplectic manifolds 20References 22 Introduction and summary of the construction
This paper is concerned with the collection of all distinct Poisson brackets thatcan be defined on a given smooth closed orientable manifold M . We show that thespace of all Poisson brackets (or Poisson structures) on M has, itself, a family ofPoisson brackets, denoted {⋅ , ⋅} µ , depending on a choice of volume form µ for M . Inother words, the space of Poisson brackets on a (smooth closed orientable) manifoldis an infinite-dimensional Poisson manifold. The bracket {⋅ , ⋅} µ is non-linear anddepends cubically on the Poisson tensor.Along with a choice of functional, {⋅ , ⋅} µ induces a Hamiltonian flow on the spaceof Poisson structures, which acts to deform any given Poisson bracket on M . ThisHamiltonian flow is formulated as a PDE on M , and acts by volume-preservingdiffeomorphism, reminiscent of the Hamiltonian formulation of ideal fluid flow andother physical systems [1, 14]. This connnection, along with observations regardingthe Godbillon-Vey invariant for three-dimensional ideal fluids [11], served as themotivation for this work.The vector fields generating the flow equation of the bracket {⋅ , ⋅} µ lie tangentto the symplectic foliation F π defined by the Poisson structure π . Consequentlythe deformation of π induced by the bracket acts only on the symplectic form onthe leaves of F π ; the foliation itself is invariant. Despite this, the global propertiesof F π play a key role. The additional structure carried by a Poisson manifold witha volume form µ is the modular vector field [6, 7, 18], denoted φ µ . The propertiesof φ µ are related to qualitative aspects of F π . As observed by Weinstein [18], φ µ is a Hamiltonian vector field if and only if F π admits a transverse measure, whichis reflected in the properties of the flow equation for the bracket {⋅ , ⋅} µ . A regularPoisson structure on a 3-manifold appears as a non-trivial steady state in the flowequation if and only if it admits a transverse measure. This relationship betweenmeasured foliations and steady solutions is a further property with parallels in thetheory of ideal fluids [12].More generally, a non-trivial steady solution of the flow equation is given by aPoisson vector field, and the set of all such steady solutions forms a subalgebraof Poisson vector fields with respect to the Lie bracket. This subalgebra formsan invariant of the Poisson structure – the set of Poisson vector fields that arisefrom Hamiltonian flow of the bracket {⋅ , ⋅} µ . For unimodular Poisson structures,those admitting a volume form invariant under all Hamiltonian flows, we are ableto define a further invariant, ¯ R ( π ) . We compute this for symplectic structures,where it is given in terms of the d + d Λ and dd Λ symplectic cohomology groupsdefined by Tseng and Yau [16]. In particular, if the symplectic structure satisfiesthe strong Lefschetz property ( e.g . K¨ahler manifolds), then ¯ R ( π ) is given in termsof the second de Rham cohomology group. We note that for a symplectic manifold,Poisson cohomology and homology are always isomorphic to de Rham cohomologygroups, the invariants we define are capable of detecting more information aboutthe symplectic structure.From an algebraic perspective, a key part of our construction is the differential δ µ . This is defined as the adjoint of the Lichnerowicz differential [10] on multivec-tors with respect to the pairing between p -forms and p -vectors given by integrationof the interior product against the volume form µ . The differential δ µ is, of course,related to many existing constructions in Poisson and symplectic geometry, includ-ing the Koszul-Brylinski differential [2,7], as well as the Koszul-Brylinski differential POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 3 associated to a volume form µ [4,19]. In the unimodular case these can be chosen toall agree (up to sign), and in the symplectic case they are further equivalent to thesymplectic differential [9,16]. They are all different, however, for a non-unimodularPoisson structure. The homology of δ µ plays a role in the definition of Casimirs of {⋅ , ⋅} µ , for which we give two examples: the symplectic volume and the Godbillon-Vey class for regular Poisson 3-manifolds. The second of these appears as a classin the homology of δ µ .We have restricted our study to the case of smooth closed orientable manifolds.Compact manifolds with boundary should be easy to accommodate, provided thespace of Poisson structures is restricted to those for which the leaves of the sym-plectic foliation lie tangent to the boundary. The construction should also work foropen manifolds ( e.g . phase space) provided the behaviour at infinity is controlled.1.1. Summary of the construction.
Throughout, M will be a smooth closedorientable manifold of dimension n , and we will use ‘manifold’ as such. Let P( M ) denote the space of Poisson structures on M , A ( M ) be the space of multi-vectorfields, and Ω ( M ) the space of differential forms. A 2-vector π ∈ A ( M ) defines aPoisson structure [8, 17] iff(1) [ π, π ] = , where [ ⋅ , ⋅ ] is the Schouten-Nijenhuis bracket. The Poisson bracket defined by π is denoted { ⋅ , ⋅ } π , and is given by { f, g } π = π ( df ∧ dg ) . Our construction beginswith the differential δ µ , which is defined as the adjoint of the Lichnerowicz differen-tial [10], X ↦ [ π, X ] , on multivectors with respect to the pairing between a p -form β and p -vector X given by(2) ( β, X ) µ = ∫ M ( ι X β ) µ, so that δ µ is defined by the equation ( β, [ π, Y ]) µ = ( δ µ β, Y ) µ , where Y ∈ A p − ( M ) . We will call a functional F ∶ P ( M ) → R admissible if it canbe written as the integral of some smooth kernel depending on π and its derivatives(up to arbitrary order). The functional derivative of F at π takes values in thesmooth part of the dual to the tangent space T π P ( M ) . We write this dual spaceas D π P ( M ) and it is isomorphic to the space of 2-forms modulo δ µ -exact 3-forms,Ω ( M )/ δ µ Ω ( M ) . Then for two admissible functionals F and G we define thePoisson bracket { F, G } µ = ( δ µ β F ∧ δ µ β G , π ) µ , where β F is a 2-form representing the functional derivative of F at π . As δ µ =
0, thebracket does not depend on the choice of representative forms β F and β G . Moreover, { F, G } µ is once again an admissible functional. For a Hamiltonian functional F ,the flow equation on P ( M ) of the bracket is given by ∂ t π = [ V F , π ] , where the vector field V F = π ( δ µ β F , ⋅ ) preserves the volume form µ and is tangent tothe symplectic foliation defined by π . Of particular interest are those vector fields V F which are Poisson, i.e . [ V F , π ] =
0. These form a subalgebra, for two functionals F and G giving Poisson vector fields, V { F,G } µ is also Poisson, and we have(3) [ V F , V G ] = V { F,G } µ , THOMAS MACHON which can be compared with the relation [ H f , H g ] = H { f,g } π for Hamiltonian vectorfields of functions f and g .Studying these fixed points leads to the definition an invariant of the pair ( π, µ ) ,which we denote R ( π, µ ) , which is defined as a short exact sequence,0 ÐÐÐÐ→ ker ˜ T ÐÐÐÐ→ ker T ÐÐÐÐ→ P ( π, µ ) ÐÐÐÐ→ . ker T comprises elements of the smooth part of the dual tangent space D π P ( M ) which give Poisson vector fields, and ker ˜ T is the subset which gives the zero vectorfield. The quotient, P ( π, µ ) , therefore comprises the distinct Poisson vector fieldsarising this way, and which forms a subalgebra of Poisson vector fields using (3). Wecompute the invariant R ( π, µ ) for regular Poisson structures on 2- and 3-manifolds,showing that it is related to the existence of a transverse measure for the symplecticfoliation of π . For example a regular Poisson structure on a 3-manifold has P ( π, µ ) ≠ µ , the space of which we writeas P ( M ) µ . This is a Poisson submanifold of the bracket { ⋅ , ⋅ } µ , and on the space P ( M ) µ we call { ⋅ , ⋅ } µ the isochoric bracket. We then define the correspondingisochoric invariant ¯ R ( π ) . For symplectic manifolds the groups comprising this in-variant are finite-dimensional, in particularker T µ = H dd Λ ( M ) , ¯ P ( π ) ⊂ H d + d Λ ( M ) , where H kdd Λ ( M ) and H kd + d Λ ( M ) are the symplectic cohomology groups defined byTseng and Yau [16]. Using the dd Λ lemma, one can show that if the symplecticstructure satisfies the strong Lefschetz property, then ¯ R ( π ) is described completelyby H DR ( M, R ) , the second de Rham cohomology group (in particular ¯ P ( π ) = The differential δ µ Preliminaries.
The Schouten-Nijenhuis bracket [ ⋅ , ⋅ ] ∶ A p ( M ) × A q ( M ) → A p + q − ( M ) is an extension of the Lie bracket to multivector fields. For a vectorfield X ∈ A ( M ) and p-vector Y ∈ A p ( M ) , [ X, Y ] = L X Y , the Lie derivative. Inparticular, for Y ∈ A ( M ) we obtain the Lie bracket. π defines a map π ∶ Ω ( M ) → A ( M ) , α ↦ π ( α, ⋅ ) . In particular we have ι π ( α ∧ β ) = ι π ( α ) β, and if α , β are exact we obtain the Poisson bracket defined by π { f, g } π = ι π ( df ∧ dg ) = ι π ( df ) dg, where f, g ∈ C ∞ ( M ) and π ( df ) = H f is the Hamiltonian vector field of f . Theintegrability condition [ π, π ] = π forms an integrable sub-bundle of T M , this is the symplectic foliation defined by π , denoted F π .The interior product ι X is not a derivation on Ω ( M ) for X ∈ A q ( M ) , q ≠ e.g . [8]). Lemma 1.
For α ∈ Ω ( M ) , β ∈ Ω p ( M ) , X ∈ A ( M ) , ι X ( α ∧ β ) = ι X ( α ) β + α ∧ ι X β. POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 5
Lemma 2.
For α ∈ Ω p ( M ) , β ∈ Ω n − p + q ( M ) , X ∈ A q ( M ) , q ≤ p , ι X α ∧ β = ( − ) ( p + ) q α ∧ ι X β. The modular vector field.
In the presence of a volume form, µ , one may definethe modular vector field [6, 7, 18] φ µ of π by the equationdiv ( H f ) = L φ µ f, where the divergence of a vector field div V is given by L V µ = div V µ.
Alternatively φ µ may be defined by ι φ µ µ = d ( ι π µ ) . The modular vector field φ µ preserves both the volume form and the Poisson struc-ture, i.e. L φ µ π = [ φ µ , π ] = φ µ =
0. If µ is replaced by a different volumeform, φ µ changes by a Hamiltonian vector field. φ µ is hence a well-defined objectin Poisson cohomology and its flow defines the 1-parameter modular automorphismgroup of the Poisson manifold. A Poisson structure is unimodular if there is avolume form making the modular vector field vanish. i.e . dι π µ =
0. If a Poissonstructure is unimodular with respect to the volume form µ we say it is µ -unimodular.2.1.2. Poisson cohomology.
For P ∈ A p ( M ) , Q ∈ A q ( M ) , R ∈ A r ( M ) , the Schouten-Nijenhuis bracket satisfies the graded Jacobi identity ( − ) p ( r − ) [ P, [ Q, R ]] + ( − ) q ( p − ) [ Q, [ R, P ]] + ( − ) r ( q − ) [ R, [ P, Q ]] = . In particular this implies that the Lichnerowicz differential [10] δ L ∶ A p ( M ) → A p + ( M ) , X ↦ [ π, X ] , squares to zero. The associated cohomology is the Poisson cohomology, H ( M, π ) .The lower-order cohomology groups are well-known. H ( M, π ) corresponds toCasimir functions of π . H ( M, π ) corresponds to Poisson vector fields moduloHamiltonian vector fields ( i.e . non-Hamiltonian automorphisms of π ). H ( M, π ) corresponds to deformations of π modulo trivial deformations generated by diffeo-morphism of M .2.2. Defining δ µ . Consider the interior produce between forms and multivectors ⟨ ⋅ , ⋅ ⟩ ∶ Ω p ( M ) × A p ( M ) → C ∞ ( M ) . Integration with respect to µ allows us to define a bilinear pairing ( ⋅ , ⋅ ) µ ∶ A p ( M ) × Ω p ( M ) → R given by(4) ( β, X ) µ = ∫ M ⟨ β, X ⟩ µ. Definition 1.
We define the differential δ µ ∶ Ω p ( M ) → Ω p − ( M ) as the adjoint ofthe Lichnerowicz differential with respect to the pairing (4). For X ∈ A p − ( M ) , β ∈ Ω p ( M ) it is given by ([ π, X ] , β ) µ = ( X, δ µ β ) µ . THOMAS MACHON
Since the Lichnerowicz differential squares to zero, it follows immediately that δ µ =
0. Before exploring more properties of δ µ we state Cartan’s identity for theSchouten-Nijenhuis bracket ι [ P,Q ] = [[ ι P , d ] , ι Q ] , where [ ⋅ , ⋅ ] is the graded commutator of linear endomorphisms on Ω ( M ) , satisfying [ Θ , Φ ] = Θ ○ Φ − ( − ) θφ Φ ○ Θ , with θ , φ the gradings. Whether [ ⋅ , ⋅ ] denotes the Schouten-Nijenhuis bracket orthe commutator should be inferred from context. We can then give an explicit formof δ µ . Lemma 3. (5) δ µ = ( − ) p ([ ι π , d ] − ι φ µ ) . Proof. ( β, [ π, X ]) µ = ∫ M ι [ π,X ] βµ = ∫ M β ∧ ι [ π,X ] µ, where we have used Lemma 2. Using Cartan’s formula this is given by ∫ M β ∧ ι π dι X µ − β ∧ dι π ι X µ + ( − ) p − β ∧ ι X ι φ µ µ. Using Stokes’ theorem and Lemma 2 on each of the three terms yields ∫ M ( − ) p + ι X dι π β ∧ µ − ( − ) p + ι X ι π dβµ + ( − ) p − ι X ι φ µ βµ. (cid:3) The differential is hence related to the Koszul-Brylinski differential [2, 7] δ KB = [ ι π , d ] . There is another differential one may define. The volume form µ inducesan isomorphism ∗ ∶ A q ( M ) → Ω n − q ( M ) given by ∗ ∶ X ↦ ι X µ. There is, therefore, a further differential operator [4,19] Ω p ( M ) → Ω p − ( M ) definedas ∗ δ L ∗ − . A short calculation shows that this operator is given by(6) [ ι π , d ] + ι φ µ . Note that, ignoring the factor of ( − ) p , the sign of ι φ µ differs between (5) and(6), and they are not the same differential. Up to sign, all these differentials agreewhen the Poisson structure is unimodular and φ µ =
0, and in the symplectic casethey further agree with the operator d Λ , which may also be constructed using thesymplectic star [9, 16] ∗ s ∶ Ω k ( M ) → Ω m − k ( M ) .The modular vector field φ µ satisfies div H f = L φ µ f where H f is the Hamiltonianvector field π ( df, ⋅ ) . We can extend this result to the case of vector fields π ( α, ⋅ ) foran arbitrary 1-form α . Lemma 4.
For α ∈ Ω ( M ) , (7) div π ( α, ⋅ ) = δ µ α. Proof.
Using Lemma 1, ι π ( α, ⋅ ) µ = − α ∧ ι π µ . Thendiv π ( α, ⋅ ) = − dα ∧ ι π µ + α ∧ dι π µ = ( δ µ α ) µ (cid:3) POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 7
Hamiltonian vector fields obey the identity(8) [ H f , H g ] = H { f,g } . All vector fields of the form π ( α, ⋅ ) are tangent to the symplectic foliation definedby π . Since this is integrable, Frobenius’ integrability theorem implies that theirLie bracket must also be tangent to the symplectic foliation. The exact nature ofthis is given by the following which extends the identity (8). Lemma 5.
For α , β ∈ Ω ( M ) , let W α = π ( α, ⋅ ) , similarly for β , then [ W α , W β ] = π ( dπ ( α ∧ β ) , ⋅ ) + π ( ι W α dβ − ι W β dα, ⋅ ) . This can also be written as [ W α , W β ] = − π ( δ µ ( α ∧ β ) + ( δ µ α ) β − ( δ µ β ) α, ⋅ ) . Proof.
Using Cartan’s formula and Lemma 4 we may write ι [ W α ,W β ] µ = − ( δ µ α ) ι W β µ + ( δ µ β ) ι W α µ + d ( ι W α ι W β µ ) . Now consider the last term in the above equation. First note that ι W α ι W β µ = − π ( α, β ) ι π µ + α ∧ β ∧ ι π ∧ π µ. By direct computation we then find d ( ι W α ι W β µ ) = − dπ ( α ∧ β ) ∧ ι π µ − ( ι W α dβ − ι W β dα, ⋅ ) ∧ ι π µ − δ µ βι W α µ + δ µ αι W β µ. This requires using the identity 2 ι π dι π = dι π ι π which follows from the integrabilitycondition (1). Applying Lemma 1 then yields the result. To obtain the secondform, consider the expression δ µ ( α ∧ β ) , this may be evaluated explicitly as δ µ ( α ∧ β ) = − ( δ µ α ) β + ( δ µ β ) α − dι π ( α ∧ β ) + ι W β dα − ι W α dβ. (cid:3) Note that δ µ ( α ∧ β ) + ( δ µ α ) β − ( δ µ β ) α does not depend on µ , and measures thefailure of δ µ to be a derivation. If both α and β are exact, we recover (8). Finallywe establish the following triple identity. Lemma 6.
Let α , β , γ ∈ Ω ( M ) . Then δ µ ( α ∧ β ∧ γ ) = − δ µ ( α ∧ β ) ∧ γ − ( δ µ γ ) α ∧ β + ↻ , where ↻ denotes the sum of cyclic permutations with respect to α , β , γ .Proof. Direct computation yields δ µ ( α ∧ β ∧ γ ) = ( dπ ( α, β ) + ι π ( α, ⋅ ) dβ − ι π ( β, ⋅ ) dα ) ∧ γ + α ∧ β ( − ι π dγ + ι φ µ γ ) + ↻ . Using δ µ α = − ι π dα + ι φ µ α and the expression for δ µ ( α ∧ β ) from the proof of Lemma 5gives the result. (cid:3) Remark . Another way of viewing this result is that the obstruction to δ µ (andsimilar differentials) being a derivation on Ω ( M ) lies in degree 2 (since π ∈ A ( M ) ). THOMAS MACHON
Homology of δ µ . The identity δ µ = [ φ µ , π ] =
0. We write H p ( M, ( π, µ )) for the p th homology group defined by δ µ . If the Poisson structure is unimodular, we maychoose µ such that φ µ =
0, then δ µ = δ KB , and H p ( M, ( π, µ )) ≅ H p ( M, π ) , where H p ( M, π ) is the p th Poisson homology group.
Proposition 1.
There is a natural pairing H q ( M, π ) × H n − q ( M, ( π, µ )) → R givenby (9) ( β, X ) µ , for β , X representatives of classes in H n − q ( M, ( π, µ )) and X respectively.Proof. We need only establish that (9) does not depend on the choice of represen-tative multi-vectors and differential forms ( X and β ). Bilinearity of (4) and thedefinition of δ µ gives the result. (cid:3) The differential ideal.
For a Poisson structure, let F π be the symplectic fo-liation of M defined by π and I ( F π ) ⊂ Ω ( M ) the ideal of forms, defined by thecondition that α ∈ Ω k ( M ) is an element of I k ( F π ) if α ( V , ⋯ , V k ) =
0, for all vectorfields V i tangent to F . Then we have the following result. Proposition 2. I ( F π ) is a differential ideal of Ω ( M ) with respect to δ µ .Proof. This follows as the modular vector field is tangent to the symplectic folia-tion [18] and because I ( F π ) is a differential ideal with respect to exterior differen-tiation. (cid:3) Remark . This holds for the Koszul-Brylinski differential [ ι π , d ] and the differential ∗ δ L ∗ − also. 3. The space of Poisson structures
On a smooth closed orientable manifold M , we denote the space of Poissonstructures on M by P ( M ) , which is given as P ( M ) = { X ∈ A ( M ) ∣ [ X, X ] = } . P ( M ) is connected. At π ∈ P ( M ) the tangent space T π P ( M ) is given by T π P ( M ) = { X ∈ A ( M ) ∣ [ π, X ] = } , which is found by differentiating the condition [ π, π ] = T ∗ π P ( M ) , which we write as D π P ( M ) . Lemma 7. D π P ( M ) ≅ Ω ( M )/ δ µ Ω ( M ) . Proof. T π P ( M ) is given by 2-vectors X satisfying [ π, X ] =
0. At some point x ∈ M , the dual to Λ T x M is Λ T ∗ x M , so that a general smooth linear functional β F ∈ T π P ( M ) is a 2-form, with the inner product given by ∫ M ι π β F µ = ∫ M β F ∧ ι π µ = ( β F , π ) µ POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 9 clearly the particular form β F depends on the choice of volume form µ . The inte-grability condition [ X, π ] = D π P ( M ) will begiven by Ω ( M )/ Ξ π , where Ξ π is the space of 2-forms ˜ β satisfying ( ˜ β, ˙ π ) µ = , for all choices of ˙ π = dπ / dt ∣ t = . By the fundamental lemma of the calculus ofvariations the condition [ ˙ π, π ] = ( γ, [ π, ˙ π ]) µ = , for all γ ∈ Ω ( M ) . By the definition of δ µ this is equivalent to the statement that ( ˙ π, δ µ γ ) µ = . (cid:3) The space of µ -unimodular Poisson structures. A natural subspace of P ( M ) is P ( M ) µ , the space of Poisson structures on M which are unimodular withrespect to the volume form µ . This is defined as follows P ( M ) µ = { X ∈ A ( M ) ∣ [ X, X ] = , dι X µ = } . We then have the following characterisation of the tangent space T π P ( M ) µ = { X ∈ A ( M ) ∣ [ π, X ] = , dι X µ = } , and the smooth, regular part of the cotangent space is given by the following result. Lemma 8. D π P ( M ) µ ≅ Ω ( M )/( δ µ Ω ( M ) ∪ d Ω ( M )) .Proof. As before D π P ( M ) µ ≅ Ω ( M )/ Ξ ( M ) . By identical arguments to Lemma7, the condition [ X, π ] = T π P ( M ) µ implies a 2-form β ∈ Ω ( M ) representingan element in D π P ( M ) µ is defined up to addition of any δ µ -exact 3-form. Nowconsider the condition dι X µ =
0. By Stokes’ theorem, this further implies β isdefined up to addition of any d -exact 1-form (note this mirrors the construction ofphase space in the Hamiltonian formulation of ideal fluid motion [1, 14]). (cid:3) The Poisson bracket { ⋅ , ⋅ } µ We define a functional F ∶ P ( M ) → R as admissible if it can be written as anintegral F = ∫ M f µ, where f is a smooth function of π and its derivatives (up to arbitrary order). Nowlet π t , t ∈ R , be a smooth 1-parameter family of Poisson structures on M . Then thefunctional derivative β F ∈ D π P ( M ) ≅ Ω ( M )/ δ µ Ω ( M ) is defined by the condition(10) dFdt ∣ t = = ( β F , ˙ π ) µ . where ˙ π = dπ / dt ∣ t = ∈ T π P( M ) . We then define the Poisson bracket on admissiblefunctions. Definition 2. (11) { F, G } µ = ( δ µ β F ∧ δ µ β G , π ) µ . Which is once again admissible. { F, G } µ does not depend on the choice ofrepresentative 2-forms β F or β G , since any two choices differ by δ µ -exact 2-forms,and δ µ = Theorem 1.
The operation { F, G } µ defines a Poisson bracket on P( M ) . To prove we show (11) obeys the axioms of a Poisson bracket. R -bilinearityfollows from bilinearity of the functional derivative. Anti-commutativity followsfrom the properties of the wedge product. The Leibniz rule follows from the chainrule for the functional derivative. All that remains is to establish the Jacobi identity.A key role throughout is played by a family of vector fields associated to smoothfunctionals. Definition 3.
For an admissable functional F on P( M ) with volume form µ , thevector field V F is given by V F = π ( δ µ β F , ⋅) . Proposition 3.
These vector fields have the following properties: V F is tangent to the symplectic foliation defined by π , V F is volume preserving with respect to µ ( dι V F µ = ), [ V F , π ] = π ( dδ µ β F ) π ( ⋅ ) − π ∧ π ( dδ µ β F , ⋅ ) , [ V F , φ µ ] = − π ( L φ µ δ µ β F , ⋅ ) ,For two functionals F , G , the corresponding vector fields satisfy [ V F , V G ] = − π ( δ µ ( δ µ β F ∧ δ µ β G ) , ⋅ ) . Proof.
The first property follows as V F is of the form π ( α, ⋅ ) for some 1-form α .The second property is a consequence of Lemma 4. To see the third observe that [ V F , π ] = [ π ( δ µ β F , ⋅ ) , π ] , Now for
P, Q ∈ A ( M ) , α ∈ Ω ( M ) , the following identity holds [ P, Q ]( α ) = − [ P ( α ) , Q ] − [ Q ( α ) , P ] − ( P ∧ Q )( dα ) + P ( dα ) Q + P Q ( dα ) . Along with the fact that [ π, π ] =
0, this gives the third property. The fourth followsfrom observing that ι [ V F ,φ µ ] µ = −L φ µ ι V F µ = L φ µ ( δ µ β F ∧ ι π µ ) = ι − π ( L φµ δ µ β F , ⋅ ) µ, along with the property L φ µ π =
0. The fifth is a consequence of Lemma 5 and thefact that δ µ = (cid:3) With these vector fields we may rewrite the Poisson bracket in a number ofdifferent ways(12) { F, G } µ = ( δ µ β F ∧ δ µ β G , π ) µ = ( δ µ β G , V F ) µ = ( β G , [ π, V F ]) µ = ( L V F β G , π ) µ . Now we compute the functional derivative of { F, G } µ . Lemma 9.
On a closed manifold, the functional derivative of { F, G } µ with respectto π is represented by the 2-form β { F,G } µ , given by β { F,G } µ = ( δ µ β F ∧ δ µ β G ) + ( L V F β G − L V G β F ) + ( γ G ([ π, V F ] , ⋅ ) − γ F ([ π, V G ] , ⋅ )) , (13) where the tensor γ F ∈ Ω ( M ) ⊙ Ω ( M ) is rank-4 covariant, with symmetries (incoordinates) γ ijkl = γ klij = − γ jikl . POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 11
Proof.
Using dot to denote time derivative, we have d { F, G } µ dt ∣ t = = ∫ M ι ˙ π µ ∧ δ µ β F ∧ δ µ β G + ι π µ ∧ δ µ β F ∧ ( ˙ δ µ β G + δ µ ˙ β G ) + . . . , where the additional terms are the derivatives of δ µ β F . The first term in the aboveequation gives the first term in (13). Now we compute the ˙ δ µ term. This is givenby ( δ µ F ∧ ˙ δ µ G, π ) µ = ( ˙ δ µ β G , V F ) µ = ( β G , [ ˙ π, V F ]) µ = ( L V F β G , ˙ π ) µ . Finally we must compute ˙ β G , the second variation of G with respect to π . This willbe a tensor γ G ∈ Ω ( M ) ⊙ Ω ( M ) , and the derivative will be equal to ( γ G ( ˙ π, ⋅ ) , [ π, V F ]) µ using the symmetries of γ gives ( γ G ([ π, V F ] , ⋅ ) , ˙ π ) µ , and yields the final term. Antisymmetry in F, G gives all terms in (13). (cid:3)
We can now establish the Jacobi identity for the Poisson bracket.
Lemma 10.
The bracket (11) satisfies the Jacobi identity.Proof.
We first deal with γ terms. Considering {{ F, G } , H } + {{ G, H } , F } + {{ H, F } , G } = {{ F, G } , H } + ↻ we find ( γ G ([ π, V F ] , [ π, V H ]) − γ F ([ π, V G ] , [ π, V H ])) + ↻ = , where we use the symmetries of γ . Now consider the first line of (13). Using (12)this is given by ( δ µ β H , π ( δ µ ( δ µ β F ∧ δ µ β G ) , ⋅ )) µ + ↻ = ( δ µ β H , [ V F , V G ]) µ + ↻ , where we have used Proposition 3. Cyclic permutations of the second line of (13)give ( L V H L V G β F − L V H L V F β G , π ) µ + ↻ = ( L [ V G ,V F ] β H , π ) + ↻ , and we have used L X L Y − L Y L X = L [ X,Y ] . Manipulating the above expressiongives ( δ µ β H , − [ V F , V G ]) µ + ↻ , which equals the contribution from the first line of (13). Using Lemma 5 we canwrite the remaining expression as2 ( δ µ β H , − [ V F , V G ]) µ + ↻ = ∫ M π ( δ µ β F ∧ δ µ β G , δ µ β H ) µ + ↻ . We then use Lemma 6, finding2 ∫ M π ( δ µ β F ∧ δ µ β G , δ µ V H ) µ + ↻ = − ( δ µ ( β F ∧ β G ∧ β H ) , π ) µ . Using the definition of δ µ this is given by − ( β F ∧ β G ∧ β H , [ π, π ]) µ , which vanishes by the integrability of π . (cid:3) This completes the proof of Theorem 1.
Varying the volume form.
The Poisson bracket { F, G } µ depends on thevolume form, which we have thus far held fixed. We now consider varying thevolume form. Let ν = f µ , for non-zero f ∈ C ∞ ( M ) , be an alternative volume form.The modular vector field φ ν satisfies dι π ν = ι φ ν ν , and is related to φ µ by φ ν = φ µ − H log f , where H f is the Hamiltonian vector field of f . The functional derivate β F is notinvariant under this change. The value of the functional F does not change underthis transformation, and an examination of (10) shows that the functional derivativeis replaced with β ′ F = β F / f . One may also transform F so that the smooth integralkernel is left unchanged, however we do not consider such transformations. Wethen find the following. Lemma 11. δ ν β ′ F = f − δ µ β F . Proof. δ ν β ′ F = δ µ β ′ F − ι H log f β ′ F = δ µ β ′ F + ι H / f β F . We also have δ µ β ′ F = f − δ µ β F − ι H / f β F . (cid:3) We then have the following characterisation of the brackets for different choicesof volume form.
Theorem 2.
The family of brackets on P ( M ) given by ∫ M g (( δ µ β F ∧ δ µ β G ) ∧ ι π µ ) , with g ∈ C ∞ ( M ) a non-zero function are all Poisson.Proof. Using Lemma 11, under a change of volume form µ → ν = f µ the Poissonbracket { F, G } ν is given by { F, G } ν = ∫ M ( δ ν β ′ F ∧ δ ν β ′ G ) ∧ ι π ν = ∫ M f (( δ µ β F ∧ δ µ β G ) ∧ ι π µ ) . Setting g = / f gives the result. (cid:3) Poisson submanifolds of { ⋅ , ⋅ } µ . The bracket { ⋅ , ⋅ } µ restricts to several nat-ural subsets of P ( M ) . Theorem 3.
The following submanifolds and their intersections are Poisson sub-manifolds of the bracket { ⋅ , ⋅ } µ :Regular Poisson structures of rank r ,Poisson structures with a fixed symplectic foliation,Unimodular Poisson structures, µ -unimodular Poisson structures. To prove we first give the following proposition
Proposition 4.
The Hamiltonian flow on P ( M ) of the bracket { ⋅ , ⋅ } µ with Hamil-tonian functional F is given by ∂ t π = [ V F , π ] = L V F π, POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 13 and alternatively written ∂ t π = π ( dδ µ β F ) π ( ⋅ ) − π ∧ π ( dδ µ β F , ⋅ ) . Proof.
This follows from (12) and Proposition 3. (cid:3)
Proof of Theorem 3.
It then suffices to show that all properties are preserved bythe flow for any admissible functional F . Since the flow acts by diffeomorphismsof π the first and third properties hold. The vector field V F is tangent to thesymplectic foliation by definition, so the second property holds. Finally, note thatfor µ -unimodular Poisson structures, choosing µ as the volume form implies theflow is µ -preserving. Furthermore we have d∂ t ι π µ = d L V F ι π µ = L V F dι π µ = , which gives the result. (cid:3) Note that the set of µ -unimodular Poisson structures is not, in general, a Poissonsubmanifold of the bracket { ⋅ , ⋅ } ν for ν ≠ µ .5. Casimirs
A Casimir of the bracket is a functional C satisfying { C, F } µ = F . From (12) we see that the condition for C to be a Casimiris [ V C , π ] = , for all π . We first give the following geometric characterisation of certain Casimirinvariants. Proposition 5.
Let C be a smooth admissible functional on P ( M ) depending onlyon the symplectic foliation of M defined by π . Then C is a Casimir of the bracket { ⋅ , ⋅ } µ .Proof. This follows from Theorem 3. (cid:3)
We can consider the change in a Casimir (of the form δ µ β C =
0) under a defor-mation of π not arising from the bracket { ⋅ , ⋅ } µ . This is given by ( β C , ˙ π ) µ . Now ˙ π defines a class in the second Poisson cohomology group [ ˙ π ] ∈ H ( M, π ) , and [ β C ] is a class in the homology group H ( M, ( π, µ )) , and we have the following result. Proposition 6.
The rate of change of C under the deformation ˙ π is given by thepairing between [ ˙ π ] and [ β C ] defined in (9) . We now give two examples of Casimir invariants. Notably, the second defines aclass in H ( M, ( π, µ )) for all π , the first does not. Example 1. (Symplectic volume). Let M be a 2 m -dimensional manifold, and takethe bracket on the Poisson submanifold S ( M ) , the space of symplectic structureson M , i.e . non-degenerate closed 2-forms ω . This defines the Poisson structurevia the relation ω ( π ( ⋅ )) = − Id, where ω ∶ T M → T ∗ M is the map induced bythe symplectic form. The symplectic volume form ν = ω m /( m ! ) = f µ . Then thesymplectic volume is given as S = ∫ M ω m ( m ! ) . Now the equation ι π ( ω ) = Tr ( ω ( π ( ⋅ ))) = − m implies ι ˙ π ω + ι ˙ π ω =
0. Finallynote that ι π ω m = − m ω m − . This implies that the functional derivative is givenby β S = f ω / m . Then a short computation gives δ ν β S = − df / m , so that V S is aHamiltonian vector field, hence [ V S , π ] = S is a Casimir. Example 2. (Godbillon-Vey invariant). Let M be a 3-manifold, and P ( M ) thespace of regular (rank 2) Poisson structures on M . Then the non-zero 1-form α = ι π µ defines the symplectic foliation, satisfying the integrability condition α ∧ dα = η such that dα = α ∧ η and then GV = ∫ M η ∧ dη, is the Godbillon-Vey invariant [3], depending only on the foliation defined by α .One may show [11] that the functional derivative gives a 2-form σ = ( dγ − η ∧ γ ) ,where γ is defined by dη = α ∧ γ . σ satisfies dσ = η ∧ σ and α ∧ σ =
0. These twofacts imply δ µ σ = , and hence that GV is a Casimir. Note that while the Godbillon-Vey invariant canbe defined for any corank 1 Poisson structure, it gives an element in the third deRham cohomology group in all cases. Integrated against an arbitrary class in H ,one finds a functional that does not have a smooth functional derivative exceptin three dimensions. There are further Godbillon-Vey invariants of codimension q foliations, living in the de Rham cohomology group H + q . For these to giveCasimirs of Poisson structures we require n = + q , and n − q = s . Hence we findCasimirs of regular rank 2 s Poisson structures on manifolds of dimension 4 s − The isochoric bracket on µ -unimodular structures From Proposition 4 it is natural to consider the subset of µ -unimodular Poissonstructures P ( M ) µ , and define the bracket { ⋅ , ⋅ } µ on it. We call this the isochoric(same volume) bracket. This proceeds along the same lines as before but with sev-eral changes. First, note that the cotangent space in this case is given by Lemma 8as Ω ( M )/ δ µ Ω ( M ) ∪ d Ω ( M ) . We then define the bracket as before { F, G } µ = ∫ M δ µ β F ∧ δ µ β G ∧ ι π µ, however δ µ β F is now no longer independent of the representative 2-form. We mayadd any exact 2-form to β F , meaning that δ µ β F → δ µ β F + δ µ dα , for some 1-form α . Lemma 12.
The isochoric bracket { F, G } µ on the space of µ -unimodular Poissonstructures P ( M ) µ does not depend on the representative 2-forms β F and β G .Proof. Since the structures are µ -unimodular, the modular vector field vanishes.Hence δ µ on 2-forms is given by ι π d − dι π . Now suppose we add dα to β F , then δ µ dα = dδ µ α = − d ( ι π dα ) , and we can compute ( dδ µ α ∧ δ µ β G , π ) µ = ( δ µ β G , H ι π dα ) µ , where H ι π dα is the Hamiltonian vector field of the function ι π dα . Then we find ( δ µ β G , H ι π dα ) µ = ( β G , [ π, H ι π dα ]) µ = . (cid:3) POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 15
The proof of the Jacobi identity runs as before. The flow equation remains ∂ t π = [ V F , π ] , but now V F is not uniquely specified by the functional F , V F is defined only upto Hamiltonian vector fields associated to functions of the form ι π dα , for arbitrary1-forms α , which does not affect the flow. This becomes clearer from the alternateform of the flow equation ∂ t π = π ( dδ µ β F ) π − π ∧ π ( dδ µ β F ) , as δ µ β F is defined only up to the exact 1-form dδ µ α = δ µ dα , the 1-form dδ µ β F doesnot depend on α . 7. The invariants
A basic invariant of a Poisson structure at a point x on a manifold M is itscorank at x . This is the dimension of the kernel of the map π ∶ T ∗ x M → T x M (a symplectic structure has corank 0 everywhere). An alternative description canbe phrased in terms of steady flows. On a Poisson manifold, the Hamiltonian flowinduced by a function f is given by Lie derivative along the Hamiltonian vector field H f , at a point x this is determined by the local value of df ∈ T ∗ x M . A point x is afixed point of this flow iff df is in the kernel of π at x . The dimension of ker π canbe therefore be said to count the number of non-trivial solutions (non-zero 1-forms)to the fixed-point equations at x .We can generalise this idea to define invariants of a Poisson structure using theflow equation of the bracket { ⋅ , ⋅ } µ . For a given Poisson structure π , the smoothpart of the cotangent space is given by D π P ( M ) ≅ Ω ( M )/ δ µ Ω ( M ) . Given anelement β F ∈ D π P ( M ) there is a map T ∶ D π P ( M ) → T π P ( M ) given by T ∶ β F ↦ [ π ( δ µ β F , ⋅ ) , π ] = π ( dδ µ β F ) π ( ⋅ ) − π ∧ π ( dδ µ β F , ⋅ ) The kernel of this map, ker T is then the subspace of D π P ( M ) giving steadysolutions to the flow equation. T however is naturally composed of two maps, T = [ ⋅ , π ] ○ ˜ T . The second [ ⋅ , π ] ∶ A ( M ) → A ( M ) , is equal to − δ L , the Lichnerowicz differential. The first˜ T ∶ D π P ( M ) → A ( M ) , is of the form ¯ T ∶ β F ↦ V F and gives the vector field V F of the functional F . Thequotient, P ( π, µ ) = ker ˜ T / ker T gives the set of distinct Poisson vector fields (vectorfields X satisfying [ X, π ] =
0) that may be arise from the flow equations of thebracket { ⋅ , ⋅ } µ . The set P ( π, µ ) satisfies the following relation ( c.f . Hamiltonianvector fields). Lemma 13. If [ V F , π ] = [ V G , π ] = , then [ V F , V G ] = V { F,G } µ Proof.
Using Proposition 3 we have [ V F , V G ] = − π ( δ µ ( δ µ β F ∧ δ µ β G ) , ⋅ ) . Now we have [ V F , π ] = [ V F , φ µ ] = [ δ µ , L V F ] =
0, and that [ V F , π ( α, ⋅ )] = π ( L V F α, ⋅ ) . This allows us to write [ V F , V G ] = π ( L V F δ µ β G , ⋅ ) = π ( δ µ L V F β G , ⋅ ) = − π ( δ µ L V G β F , ⋅ ) , where the last equality follows from antisymmetry of the Lie bracket. Finally, usingLemma 9 we have V { F,G } µ = π ( δ µ ( δ µ β F ∧ δ µ β G + L V F β G − L V G β F ) , ⋅ ) . Which is rewritten as V { F,G } µ = − [ V F , V G ] + [ V F , V G ] + [ V F , V G ] = [ V F , V G ] . (cid:3) We may then define the invariant R ( π, µ ) of the pair ( π, µ ) . We note here thatthis invariant is essentially a decomposition of ker T , and there may well be othersuch decompositions that are as, or more, useful. Definition 4.
The invariant R ( π, µ ) of a Poisson structure π and volume form µ is the short exact sequence0 ÐÐÐÐ→ ker ˜ T ÐÐÐÐ→ ker T ÐÐÐÐ→ P ( π, µ ) ÐÐÐÐ→ µ -unimodular Poissonstructures. The difficulty here is that for the isochoric bracket, V F is no longeruniquely defined for a functional F , so we need to be a little more careful. Let ˜ V F denote the equivalence class of vector fields associated to F ( i.e . vector fields of theform π ( δ µ β F + dδ µ α, ⋅ ) for arbitrary α ∈ Ω ( M ) . Then we have the following result,which is the counterpart to Lemma 13 Lemma 14.
Suppose the functionals F and G give fixed points of the flow of the µ -unimodular bracket for π . Then any representatives of ˜ V F and ˜ V G are Poissonvector fields, and [ V F , V G ] ∈ ˜ V { F,G } µ for any such representatives V F and V G .Proof. Consider [ V F , H δ µ dα ] , for an arbitrary 1-form α . Now, since [ V F , π ] =
0, wehave [ V F , H δ µ dα ] = π ( L V F δ µ dα, ⋅ ) , and using the fact that [ L V F , d ] = [ V F , H δ µ dα ] = π ( dδ µ L V F α, ⋅ ) . Hence changing δ µ by some dδ µ -exact 1-form, changes the Lie bracket by the vectorfields by π of some dδ µ -exact 1-form. Since ˜ V { F,G } µ is defined only up to dδ µ exact1-forms, using Lemma 13 we have the result. (cid:3) Now we may extend our definition to the isochoric bracket. The smooth partof the cotangent space is of the form D π P ( M ) µ ≅ Ω ( M )/( δ µ Ω ( M ) ∪ d Ω ( M )) ,and the map T µ ∶ D π P ( M ) µ → T π P ( M ) µ is as before (and single valued). Now let¯ P ( π ) denote the equivalence classes of Poisson vector fields that arise as the kernelof T µ , i.e. all such Poisson vector fields modulo Hamiltonian vector fields of theform π ( dδ µ α, ⋅ ) . Then ker ˜ T µ is now the set of elements of ker T µ which give thezero vector field, modulo the Hamiltonian vector fields. Definition 5.
The isochoric invariant ¯ R ( π ) of a µ -unimodular Poisson structureis the short exact sequence0 ÐÐÐÐ→ ker ˜ T µ ÐÐÐÐ→ ker T µ ÐÐÐÐ→ ¯ P ( π ) ÐÐÐÐ→ POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 17
Both these invariants describe, or count, the steady solutions to the flow equa-tions for the brackets on P ( M ) and P ( M ) µ . In the following sections we computethem for regular Poisson structures on 2- and 3-manifolds, as well as for symplecticstructures. 8. Rank-2 Poisson structures
Poisson structures with rank (at most) 2 satisfy π ∧ π =
0. In this case, the flowequation is simply ∂ t π = π ( dδ µ β F ) π, giving a time-dependent rescaling of π . Using δ µ β F = ∂ t π = − ( ι φ δ µ β F ) π, and we obtain the following result. Proposition 7.
The Poisson bracket { ⋅ , ⋅ } µ is trivial on the space of µ -unimodularrank-2 Poisson structures. Poisson 2-manifolds.
Now we can further consider the special case of regularPoisson structures on 2-manifolds, we note that the analysis below carries throughmore or less identically for log-symplectic 2-manifolds. All regular two-dimensionalPoisson structures are unimodular, so that the modular vector is Hamiltonian. Inparticular, if π is ν -unimodular, then the modular vector field φ µ is the Hamiltonianvector field H log g where the function g = ( ν / µ ) . In this case δ µ β F = − dι π β F − ι H log g β F , so that(14) ∂ t π = ( L H log g π ( β F )) π = { π ( β F ) , log g } π π, where { ⋅ , ⋅ } π is the Poisson bracket on M defined by π . Proposition 8.
Let π be a ν -unimodular regular Poisson structure on a 2-manifold M . Let g = ( ν / µ ) and C g the Poisson subalgebra of the bracket { ⋅ , ⋅ } π consisting offunctions commuting with g , i.e. C g ≅ { f ∈ C ∞ ( M ) ∣ { f, g } π = } , then the invariant R ( π, µ ) is given by ÐÐÐÐ→ H ( M, R ) ≅ R ÐÐÐÐ→ C g ÐÐÐÐ→ gH C g ÐÐÐÐ→ , where H C g is the set of all Hamiltonian vector fields arising from functions in C g .Proof. First observe that the cotangent space D π P ( M ) for a 2-manifold M is justΩ ( M ) , since Ω ( M ) ≅ ∅ . Furthemore, since π ≠
0, we may use the isomorphism ι π ∶ Ω ( M ) → C ∞ ( M ) given by γ ↦ ι π γ along with the fact that g ≠ π ( β F ) = − gf , where f ∈ C ∞ ( M ) is arbitrary. Then we compute(15) δ µ β F = d ( f g ) − ι φ µ β F = d ( f g ) − f gd log g = gdf. Then ker ˜ T consists of those functions f such that df =
0, hence ker ˜ T ≅ H ( M, R ) ≅ R . Now we have [ V F , π ] = [ gH f , π ] , which can be computed as [ gH f , π ] = { f, g } π π, hence ker T is all functions f such that { f, g } π vanishes, this is the set C g . Nowany function f ∈ C g yields the vector field gH f , and the set of all such is gH C g which is P ( π, µ ) . (cid:3) Remark . Note that if ν / µ is constant (for example if ν = µ ), then C g ≅ C ∞ ( M ) ,and P ( π, µ ) comprises all Hamiltonian vector fields.In the case ν = µ it is instead profitable to study the isochoric invariants ¯ R ( π ) and ¯ P ( π ) . Proposition 9.
Let π be a ν -unimodular regular Poisson structure on a 2-manifold M . Then the isochoric invariant ¯ R ( π ) is given by ÐÐÐÐ→ R ÐÐÐÐ→ R ≅ H ( M, R ) ÐÐÐÐ→ ÐÐÐÐ→ , in particular ¯ P ( π ) = . Proof.
First recall that D π P ( M ) µ ≅ Ω ( M )/ d Ω ( M ) . Now set f = ι π β F , thenobserve that δ µ β F = − df and hence that V F is Hamiltonian, so that all elements of D π P ( M ) µ are in the kernel of T µ . Now, finally observe ( M ) µ ≅ Ω ( M )/ d Ω ( M ) ≅ R , whose elements correspond to constant multiples of the volume form ν . Thedifference between any two elements yields a constant function via the map ι π , andhence gives the same (zero) vector field. (cid:3) We generalise this result to symplectic structures in Section 9.8.2.
Poisson 3-manifolds.
We can also consider the space of Poisson structureson 3-manifolds. In this case the 1-form α = ι π µ satisfies α ∧ dα = ι φ µ ι π µ =
0. Then we find the flow equation givenby(16) ∂ t π = ( ι φ µ dπ ( β F )) π = − ( L φ π ( β F )) π. Now consider the subset of regular Poisson structures on a 3-manifold M , so thatthe symplectic foliation is non-singular. We now seek steady solutions of the flowequation. This implies there is a function f = π ( β F ) satisfying L φ µ f =
0. Nowconsider the modular form [18] η , defined by ι φ µ µ = ι π µ ∧ η . The modular form isdefined only up to addition of a term of the form gα for a function g , and dη = α ∧ γ for some 1-form γ , so that the restriction of η to the leaves of the symplecticfoliation F π is closed. Hence η defines a class [ η ] in the foliated cohomology group H ( F π ) [3]. Now the existence of a non-trivial steady solution to the flow equationimplies f satisfies π ( df, η ) =
0, and furthermore that the Hamiltonian vector field H f commutes with the modular vector field, i.e. [ φ µ , H f ] =
0. In particular, thisimplies η = df + gα . We then find that dα = α ∧ df so that d ( e f α ) = df ∧ e f α + e f α ∧ df =
0, hence the form ι π e f µ is closed. This is equivalent to the following two statements:the Poisson structure is unimodular with respect to the volume form e f µ , and thesymplectic foliation of π admits a transverse measure. Then we have the followingresult. Proposition 10.
Let π be a regular non-unimodular Poisson structure on a 3-manifold. Then R ( π, µ ) is given by ÐÐÐÐ→ I ( F π )/ δ µ I ( F π ) ÐÐÐÐ→ I ( F π )/ δ µ I ( F π ) ÐÐÐÐ→ ÐÐÐÐ→ , where I ( F π ) is the ideal of Ω ( M ) defined by the symplectic foliation F π . In par-ticular, P ( π, µ ) = , moreover P ( π, µ ) = iff π is non-unimodular. POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 19
Proof.
The above arguments establish that a non-trivial solution to the flow equa-tion only exists if π is unimodular. Hence P ( π, µ ) = T will be given byelements β F ∈ Ω ( M )/ δ µ Ω ( M ) satisfying π ( δ µ β F , ⋅ ) =
0. This is equivalent to therequirement that δ µ β F = gι π µ for some function g . Since I ( F π ) is a differentialideal of Ω ( M ) with respect to δ µ , this is satisfied by all 2-forms in I ( F π )/ δ µ I ( F π ) (note that I ( F π ) = Ω ( M ) ). Now suppose β is not of this form. Then we maywrite β F = f σ where σ is an area form for F π , so that ι π σ =
1. Then observe that ι φ µ µ = ι φ µ α ∧ σ = − α ∧ ι φ µ σ , so that ι φ µ σ = − η , the modular form, hence we requirethat df − f η = gι π µ . This can only occur if L φ µ f =
0, but this would comprise anon-trivial steady solution to the flow equation, which does not exist. Hence thereis no such function f .Now we discuss the second part. Non-unimodularity implying P ( π, µ ) = P ( π, µ ) = (cid:3) Now recall Novikov’s theorem [15] stating that every codimension-1 foliation of S has a Reeb component, and hence does not admit a transverse measure. Thisimplies the following. Corollary 1.
Let π be a regular Poisson structure on S , then P ( π, µ ) = , i.e.there are no non-trivial steady solutions to the flow equation of the bracket { ⋅ , ⋅ } µ . Now recall the definition of the Godbillon-Vey invariant GV in Example 2, GV = ∫ M η ∧ dη, it is easy to see that GV = η = df + gα and we have the following. Proposition 11.
Let π be a regular Poisson structure on a closed 3-manifold M .If there is an admissible functional making π a non-trivial steady solution of theflow equation for the bracket { ⋅ , ⋅ } µ (i.e. P ( π, µ ) ≠ ), then both the class of η in H ( F π ) and the Godbillon-Vey invariant of F π vanish. It is known that the Godbillon-Vey invariant obstructs unimodularity on Poisson3-manifolds [5,18], here we find that it obstructs the existence of steady solutions ofthe flow equation. This mirrors its application in ideal fluids, where under certainconditions it provides an obstruction to steady flow [12].We now study unimodular regular Poisson structures on 3-manifolds. The resultsmirror those for two-dimensional manifolds, but with some additional informationcoming from the symplectic foliation.
Proposition 12.
For a ν -unimodular regular Poisson structure on a 3-manifold M with volume form µ , let g = ( ν / µ ) ∈ C ∞ ( M ) . Then R ( π, µ ) is given by ÐÐÐÐ→ R ⊕ I ÐÐÐÐ→ C g ⊕ I ÐÐÐÐ→ gH C g ÐÐÐÐ→ , where I = I ( F π )/ δ µ I ( F π ) , and C g and H C g are defined as in Proposition 8. Moreover, the invariant ¯ R ( π ) isgiven by ÐÐÐÐ→ Ω ( M )/ d Ω ( M ) ÐÐÐÐ→ Ω ( M )/ d Ω ( M ) ÐÐÐÐ→ ÐÐÐÐ→ . Proof.
As in Proposition 8, we may take the modular vector field to be φ = H log g .Then note that the modular form can be chosen as η = d log g . As in Proposition 12we may write β F = f gσ + ρ , where ρ ∈ I ( F π ) and ι π σ =
1. Then the flow equation(16) becomes ∂ t π = { f, g } π π which gives ker T . For ker ˜ T note that δ µ hσ = ι H h σ + dh − dh − hι φ µ σ = − dh + hdg .Choosing h = − f g , allows arguments from Proposition 8 to be applied directly.For the invariant ¯ R ( π ) . note that the flow equation (16) is trivial, hence ¯ R ( π ) is the entire cotangent space D π P ( M ) µ . Now observe that δ ν Ω ( M ) ⊂ d Ω ( M ) ,hence we have D π P ( M ) µ ≅ Ω ( M )/ d Ω ( M ) . Finally, since the bracket is trivial,¯ P ( π ) = (cid:3) The isochoric bracket on symplectic manifolds
In the preceding section we computed the R and ¯ R invariants for Poisson struc-tures on 2 and 3-manifolds. In the case of a regular Poisson structure on a 2-manifold we found a connection to the de Rham cohomology group H dδ µ ( M ) ≅ R .Regular Poisson structures on 2-manifolds are symplectic, which suggests that inthis case the invariant ¯ R is particularly well-behaved. This is indeed the case.On a n = m -dimensional manifold, a Poisson structure π is symplectic if it isnon-degenerate. In particular we define the symplectic form ω by ω ( π ( ⋅ )) = − Id , where ω ∶ T M → T ∗ M is the map induced by the symplectic form. All symplecticmanifolds are unimodular with respect to the symplectic volume form ω m /( m ! ) ,and it is natural, therefore, to study the bracket { ⋅ , ⋅ } µ on the space of symplecticstructures on M whose symplectic volume form is µ , which we denote S ( M ) µ .On a symplectic manifold, the operation ι π ∶ Ω p ( M ) → Ω p − ( M ) is twice the dualLefschetz operator Λ as defined by Tseng and Yau [16] (there are a variety of signand magnitude conventions). In particular δ µ = ( − ) p [ Λ , d ] = ( − ) p + d Λ , whichcan be defined using the symplectic star operator [9]. Now we have the followingresult. Proposition 13. On S ( M ) µ , the flow equation of the bracket { ⋅ , ⋅ } µ is (17) ∂ t ω = dd Λ β F . Proof.
Since the flow equation acts by diffeomorphisms preserving the symplecticvolume, we have ∂ t ω = L V F ω = dι V F ω. Now ι V F ω = ω ( π ( − d Λ β F , ⋅ ) , ⋅ ) = d Λ β F , which gives the result. (cid:3) The structure on the smooth part of the cotangent space D π S ( M ) µ = Ω ( M )/( d Ω ( M ) + d λ Ω ( M )) (given by Lemma 8) then implies the following result. POISSON BRACKET ON THE SPACE OF POISSON STRUCTURES 21
Theorem 4.
For a symplectic Poisson structure π , the invariant ¯ R ( π ) is given by T µ H dd Λ ( M ) ¯ P ( π ) ,H dd Λ ( M ) H d + d Λ ( M ) ⊃ ⊃ where the subset ¯ P ( π ) consists of d Λ exact elements of H d + d Λ ( M ) and the subset ker ˜ T µ consists of elements of H dd Λ ( M ) which have a d Λ closed representative. Thegroups H d + d Λ ( M ) and H dd Λ ( M ) are the symplectic cohomology groups defined byTseng and Yau [16].Proof. Let dd Λ k , d k and d Λ k be the differentials acting on k -forms. Then the group H kdd Λ ( M ) is defined as H kdd Λ ( M ) = ker dd Λ k Im d k − ∪ Im d Λ k + . The cotangent space at π is given by D π S ( M ) µ = Ω ( M )/( d Ω ( M ) + d λ Ω ( M )) .Along with the flow equation (17) this gives ker T µ ≅ H Λ ( M ) . To obtain ¯ P ( π ) wenote that the set of vector fields for a functional F are all of the form V F = π ( d Λ β F + dd Λ α, ⋅ ) where α is an arbitrary 1-form. Poisson vector fields are those for which d Λ β F is d -closed. But by construction it is also d Λ closed. Now recall the definition of H kd + d Λ H kd + d Λ ( M ) = ker d k ∩ ker d Λ k Im dd Λ k . ¯ P ( π ) is then given by those elements of H d + d Λ ( M ) which are d Λ exact. Finally, wenote that ker ˜ T µ is given by those elements β of ker T for which d Λ β can be chosento be zero. (cid:3) Using Theorem 3.16 of [16] we then have the following.
Corollary 2.
For a given symplectic Poisson structure, π , all the groups in ¯ R ( π ) are finite dimensional. Using the Proposition 3.13 of [16] we may also characterise the invariants forsymplectic structures satisfying the strong Lefschetz property.
Corollary 3.
If the symplectic structure defined by π satisfies the strong Lefschetzproperty (equivalently the dd Λ Lemma [13, 16]), then ker T µ = ker ˜ T µ = H DR ( M, R ) , ¯ P ( π ) = , where H DR is the second de Rham cohomology group.Proof. If π satisfies the strong Lefschetz property, d Λ exactness and dd Λ exactnessare equivalent. (cid:3) References [1] Vladimir I Arnold and Boris A Khesin.
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