aa r X i v : . [ m a t h - ph ] A ug POISSON BRACKETS IN KONTSEVICH’S ”LIE WORLD”
FLORIAN NAEF
Abstract.
Abstract.
In this note the notion of Poisson brackets in Kontsevich’s ”Lie World” isdeveloped. These brackets can be thought of as ”universally” defined classical Poisson structures,namely formal expressions only involving the structure maps of a quadratic Lie algebra. We prove auniqueness statement about these Poisson brackets with a given moment map. As an application we getformulae for the linearization of the quasi-Poisson structure of the moduli space of flat connections ona punctured sphere, and thereby identify their symplectic leaves with the reduction of coadjoint orbits.Equivalently, we get linearizations for the Goldman double Poisson bracket, our definition of Poissonbrackets coincides with that of Van Den Bergh [2] in this case. This can furthermore be interpreted asgiving a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces. Introduction
The motivation of this note was originally to give another proof of the result Theorem 6.6 in [1].Theorem 6.6 states that the moduli space of flat g -connections on a surface of genus 0 with prescibedmonodromy around the punctures is symplectomorphic to the symplectic reduction of the product ofcoadjoint orbits, at least if the prescribed monodromies are sufficiently close to the identity. The proce-dure is as follows. We identify the relevant moduli space with a reduction of a Hamiltonian quasi-Poissonspace whose underlying manifold is the product of a number of G ’s. Using the exponential map we canfurthermore pull the situation back to a product of g ’s. Summarizing, we get a quasi-Poisson structureon g × · · · × g together with a moment map, which we with to compare to the standard Kostant-Kirillov-Souriau structure. Moreover, all those structures are defined by ”formulae” only involving the lie bracketand the inner product of g . A precise definition of this is given below. It turns out that for such uni-versally defined Hamiltonian (quasi-)Poisson structures the moment map uniquely defines the bivectorfield and vice versa. 2. Lie spaces
We recall some definitions from [5]. Let
Lie denote the category of free complete graded (super-)Liealgebras, where morphisms are continuous Lie algebra morphisms. We define the category
LieSp of(formal affine) Lie spaces as the opposite category of Lie. This definition is very much in analogy withthe equivalence of (commutative) affine schemes and
Ring op . Much of the language that follows ismotivated by this analogy. By definition, there is a canonical contravariant functor O : LieSp = Lie op − → Lie . More concretely, a Lie space L is nothing but a (graded) Lie algebra, which we choose to call thecoordinate Lie algebra of the space and denote it by O ( L ) . Morphisms between Lie spaces are maps ofLie algebras in the opposite direction. A choice of free homogenous generators of O ( L ) shall be called acoordinate system, or just coordinates. Let us denote by L ( z , · · · , z n ) ∈ Lie , the completed free graded (super-)Lie algebra in generators z , · · · , z n , where each generator has possiblynon-zero degree, and the completion is taken with respect to the lower central series. Let furthermore L ( z , · · · , z n ) ∈ LieSp , denote the Lie space whose coordinate Lie algebra is L ( z , · · · , z n ) . Thus L ( z , · · · , z n ) and L ( z , · · · , z n ) are the same objects, the only difference is in the direction we choose to write morphisms, and of coursein our interpretation. Using the language introduced above, the z , · · · , z n are coordinates on the space L ( z , · · · , z n ) . And elements of O ( L ( z , · · · , z n )) are Lie series in the the coordinates z , · · · , z n . Let ow L n := L ( x , · · · , x n ) L n := L ( x , · · · , x n ) denote the above with all generators x i of degree 0.In this context, L n is nothing but the product of n copies of the affine line L , since products in LieSp are coproducts in
Lie that is completed free products. In what follows, we wish to do differentialgeometry on these Lie spaces. Guiding our intuition is the fact, that each element of L n induces aformal g -valued function on g × n . Abstractly this follows from the fact that g × n = Hom ( L n , g) , but moreconcretely is it seen by just interpreting elements in L n as formulae. Take for instance [ x , x ] , it can beseen as a function taking as inputs two elements x , x ∈ g and giving as output another element of g .In this sense, the space L n can be thought of as a ”universal version” of g × n . If we want to produce a k -valued function, one possibility is to take the product of two g -valued functions with respect to someinner product on g . Let us from now on assume that g is a quadratic Lie algebra, i.e. there is a chosennon-degenerate invariant inner product. The definition of functions on a Lie space is then chosen suchthat it induces k -valued functions on g × n , that is F ( L ) := O ( L ) ⊗ O ( L ) / { a ⊗ b − ± b ⊗ a, [ a, b ] ⊗ c − a ⊗ [ b, c ] } , or in other words the object in vector spaces representing the functor of symmetric invariant innerproducts on L . We will denote the universal inner product by O ( L ) ⊗ O ( L ) − → F ( L ) a ⊗ b → h a, b i . Remark . Note that there is a difference between the space of functions and the coordinate algebra.Whereas the latter carries the structure of a Lie algebra, the former is merely a vector space, that isfunctions cannot be multiplied. To get an algebra, one might choose instead to work with the symmetricalgebra over F ( L ) , however, we choose not to do so. Remark . In terms of graphical calculus, elements of the coordinate Lie algebra can be seen as rootedJacobi tree, whereas functions are simply Jacobi trees, where the leaves are labeled by generators of theLie algebra. This picture will in particular explain later, why we cannot contract arbitrary forms withpolyvectorfields, since this would generate loops, and thus leave the world we choose to work in.
Remark . As in [5] everything works analogously if one replaces Lie algebras by associativealgebra. Instead of developing the theory in parallel, the differences are pointed out in remarks. Inthe associative world F also goes under the name of HH ( A ) = A/ [ A, A ] , that is the zero-th Hochschildhomology. Moreover, by embedding a free Lie algebra into its universal envelopping algebra, whichis a free associative algebra, all ”Lie” functions embed into ”Ass” functions. The last part can beseen from the Cartan-Eilenberg isomorphism HH ( U (g)) = H Lie (g , ( U g) ad ) , which applied to our casesays HH ( U ( L n )) = ( U ( L n )) L n ∼ = S ( L n ) L n , namely that ”Ass” functions are the L n -coinvariants of thesymmetric algebra over L n . In particular, we see that the quadratic part coincides with the definition of”Lie” functions. Graphically, we are replacing Jacobi trees with ribbon trees.In order to get forms and polyvector fields, we introduce the odd tangent and cotangent bundle,respectively, T [ ] ( L ( z , · · · , z n )) := L ( z , · · · , z n , dz , · · · , dz n ) , | dz i | = | z i | + ∗ [ ] ( L ( z , · · · , z n )) := L ( z , · · · , z n , ∂ , · · · , ∂ n ) , | ∂ i | = − | z i | + and T [ ] L n := L ( x , · · · , x n , dx , · · · , dx n ) , | dx i | = ∗ [ ] L n := L ( x , · · · , x n , ∂ , · · · , ∂ n ) , | ∂ i | = in the non graded case. Their functions are then denoted by Ω ( L ) := F ( T [ ] L ) , X( L ) := F ( T ∗ [ ] L ) . Both are graded vector spaces and by the usual formulae Ω ( L n ) can be endowed with a differential ofdegree 1. After some preparation, the usual formulae can be used to define a Lie bracket on X( L n ) witha Lie bracket analogous to the Schouten bracket. The Schouten bracket can be interpreted as induced y the canonical odd symplectic stucture on T ∗ [ ] L n . It will be shown that the bracket also defines anaction of polyvectorfields on the coordinate Lie algebra of T ∗ [ ] L n . These structures are compatible withspecialization, that is for any quadratic Lie algebra g we get canonical maps Ω ( L n ) → Ω (g × n ) , X( L n ) → X(g × n ) , of complexes and Lie algebras, respectively. More concretely, let e α be a basis of g . Let t αβ = h e α , e β i be the coefficients of the inner product and t αβ its inverse. Let x α denote the dual basis of e α and hencea coordinate system on g . The above maps are then induced by O ( T [ ] L n ) − → Ω (g × n ) ⊗ g x i → x αi ⊗ e α dx i → dx αi ⊗ e α and O ( T ∗ [ ] L n ) − → X(g × n ) ⊗ g x i → x αi ⊗ e α ∂ i → t αβ ∂∂x αi ⊗ e β . To descend to functions, the inner product on g is applied on the g factor. The invertibility of the innerproduct on g is only used in the second map. A form, polyvectorfield or g -valued function on g × n inducedby an object on L n will be called universal . For example, the KKS Poisson bivector on g , h x, [ ∂ x , ∂ x ] i ,is a universal bivector field. Let us explicitly compute the image of this bivector field under the abovemap as follows, D x α ⊗ e α , h t βγ ∂∂x βi ⊗ e γ , t δǫ ∂∂x δi ⊗ e ǫ iE = t αη c ηγǫ t βγ t δǫ x α ∂∂x β ∂∂x δ = c βδα x α ∂∂x β ∂∂x δ , where c ηγǫ are the structure constants of g and in the last step we raised and lowered indices usingthe inner product. The adjoint action on g , seen as a g -valued vector field using the inner product,is universal, as it is induced by [ x, ∂ x ] . Moreover, these objects get represented faithfully that way, asshown by Lemma 2.4.
Let f ∈ Ω ( L n ) , X( L n ) or L n . If f = then f induces a non-zero object on sl( N ) with itsKilling form for N sufficiently large.Proof. Using polarization, one can reduce to the case where f is linear in each coordinate. Any formor vector field that is multi-linear in the odd variables, can be seen as an ordinary multi-linear functionon twice as many variables, by identifying T g × n ∼ = g × and T ∗ g × n ∼ = g × . By embedding into theassociative world, the problem is reduced to showing that the set of functions on sl( N ) n of the formtr ( ad x σ ( ) · · · ad x σ ( n ) ) + (− ) n tr ( ad x σ ( n ) · ad x σ ( ) ) for σ ∈ S n , are linearly independent. This can be seen by direct computation. (cid:3) Remark . One can also use the double of the truncated free Lie algebra (with zero cobracket) to showfaithfulness.It is clear that on any given quadratic Lie algebra we only get a comparatively small amount offunctions, forms and vector fields, in particular all the objects are g -invariant. The way one can use lemma2.4 is that whenever we have a construction or operation on a Lie space that induces a correspondingconstruction or operation on a concrete Lie algebra, one can use lemma 2.4 to show that identities thathold on all lie algebras also hold on the Lie space. The theory of lie spaces can thus be thought ofstudying structures on g × n that are of a particularly natural type, that is in the image of the abovespecialization maps.The usual yoga using contraction with the Euler vectorfield shows that Ω ( L n ) is acyclic. Moreover,there is a simple description Ω ( L n ) and Ω ( L n ) . emma 2.6. The following maps are isomorphisms of vector spaces. L n × n − → Ω ( L n )( α i ) → Σ h dx i , α i i u ( n, U ( L n )) = { ( a ij ) ∈ U ( L n ) , a ij + ∗ a ji = } − → Ω ( L n )( a ij ) → Σ (cid:10) dx i , Ad a ij dx j (cid:11) , where ∗ is the cannonical antipode on the universal enveloping algebra of L n . In words, the lemma says that the space of 1-forms is given by an n -tuple of Lie series, whereas thespace of 2-forms is given by a skew-symmetric matrix of associative series, where the antipode is usedfor the skew-symmetry.Using the lemma we define the maps ∂∂x i : F ( L n ) → L n as the composition of d , the inverse of theabove map, and projection onto the i th component, or equivalently such that dα = Σ D dx i , ∂α∂x i E for α ∈ F ( L n ) . Remark . The same is true in the ”associative” world, where the role of U ( L n ) is now played by A ⊗ A , where A is the underlying free associative algebra, because these objects encode functions thatare linear in two additional ”separator” variables. Proof.
Surjectivity follows easily from the defining relations in both cases. For injectivity one defines theoperation of contracting with the coordinate vectorfields ∂ i as follows. Let ι ∂ i denote the derivation ofdegree -1 on O ( T [ ] L n ) with values in its universal enveloping algebra with module structure given myleft multiplying by specifying ι ∂ i ( dx j ) = δ ij , ι ∂ i ( x j ) = One checks that the following map is well-defined Ω ( L n ) ι ∂i − → O ( T [ ] L n ) = L ( x , · · · , x n , dx , · · · , dx n ) h α, β i 7− → ad ∗ ι ∂i α β + (− ) | α || β | ad ∗ ι ∂i β α. This operation has the following defining property. Let φ be the derivation of degree -1 on O ( T [ ] L n ) with values in L ( x , . . . , x n , dx , . . . , dx n , t ) , where t has degree 0, defined by φ ( dx j ) = δ ij t, φ ( x j ) = which straightforwardly extends to Ω ( L n ) φ − → F ( L ( x , · · · , x n , dx , · · · , dx n , t ) . The maps ι ∂ i and φ are then adjoint to each other in the following sense, that for any ω ∈ Ω ( L n ) wehave φ ( ω ) = h t, ι ∂ i ( ω ) i . Now the lemma follows by applying the operator ι ∂ i to the elements of the specified form. Moreprecisely, for forms the maps ι ∂ i for i =
1, . . . , n are an inverse to the map in the lemma. And in the caseof two forms it follows from the observation, that any degree element in L ( x , · · · , x n , dx , · · · , dx n ) isof the form P ad π i dx i for uniquely determined associative series π i . (cid:3) Using the above lemma we can construct the Schouten bracket. We define the following map of degree-1 X • [ · , · ] − → Der ( T ∗ [ ] L n ) α → [ α, x i ] = (− ) | ∂ i | ( | α | − | ∂ i | ) ∂α∂ ( ∂ i ) [ α, ∂ i ] = −(− ) | x i | ( | α | − | x i | ) ∂α∂x i . Remark . Note that the gory signs would disappear if we choose right partial derivatives instead ofleft partial derivatives. The formula can then be seen as induced by [ · , · ] = ← −−− ∂∂ ( ∂ i ) − → ∂∂x i − (− ) | x i || ∂ i | ← − ∂∂x i −−− → ∂∂ ( ∂ i ) t is straightforward to check that this operation descends to a bracket on X • and the above mapintertwines this bracket with the commutator of derivations. In particular, the bracket on X • satisfiesthe Jacobi identity.Moreover, the degree one part identifies vector fields with derivations, that is X − → Der ( L n ) X h α i , ∂ i i 7− → ( x i → α i ) . As the construction of Ω • is covariant, vectorfields also act on forms, we denote this action by the usualsymbol for Lie derivatives, that is by L X for X ∈ X Poisson bivector fields.Definition 2.9.
A bivector field Π ∈ X is called a Poisson structure if [ Π, Π ] = . It is called non-degenerate if the matrix representing it by lemma 2.6 is not a zero-divisor.Any bivector field Π induces a bracket by the formula { α, β } Π = [ α, [ Π, β ]] , where α, β are either bothfunctions, or one of the is a function and the other one an element of the coordinate algebra. If Π isPoisson, then this defines a Lie bracket on the space of functions and an action of this Lie algebra offunctions onto the underlying Lie algebra. However, the bivector field Π carries more information thanthese operations, and more precisely, there exists Π = Π distinct bivector fields that induce the sameoperations, but only Π is Poisson, whereas Π is not (see examples of quasi-Poisson structures later). Example 2.10.
Any bivector field of the form * α i , X j [ x j , ∂ j ] + for arbitrary α i defines trivial operations. Remark . It is possible to view a bivector field as a kind of biderivation on the Lie algebra withvalues in its universal envelopping algebra. To make this precies, one can add two ”placeholder” symbols s , t as generators of the Lie algebra. The formula ( a, b ) → { h a, s i , h b, t i } Π defines a bracket with valuesin functions linear in s and t , which we can identify with the universal envelopping algebra similar asin lemma 2.6 and in particular in its proof. Bivector fields are in one-to-one correspondence with suchmappings that are biderivations in a suitable sense. Remark . Using the same formula ( a, b ) → { h a, s i , h b, t i } Π for a, b ∈ A in the associativeworld, gives rise to a map A ⊗ A → A ⊗ A . One checks that the theory of Poisson brackets in theassociative world is equivalent to the theory of double Poisson brackets of van den Bergh [2] in the casewhere the underlying algebra is free. Moreover, the embedding of the ”Lie” world into the ”Ass” worldpreserves non-degeneracy as will be clear from the proof of lemma 2.16 below. Remark . Note that the notion of non-degeneracy does not descend to specialization and is easierto satisfy.
Example 2.14.
The following two bivector fields can readily be seen to be non-degenerate Poissonstructures. Π KKS = h ∂ x , [ x, ∂ x ] i Π symp = h ∂ x , ∂ y i More examples are constructed by taking direct sums of those, which will also be denoted by the samesymbol if only one type is used.
Lemma 2.15 (”Weinstein’s splitting theorem”) . Given a Poisson structure Π , one finds coordinates,i.e. free generators x i , y i , z j of the underlying Lie algebra, such that Π = P h ∂ x i , ∂ y i i + ˜ Π , where ˜ Π doesonly contain terms in the variables z j and does not contain any constant terms.Proof. Let Π denote the constant part of Π . We are trying to classify deformations of Π that aregoverned by the dg Lie subalgebra of the Schouten Lie algebra (X • , [ · , · ] , d = [ Π , · ]) where vector fieldshave components at least quadratic and bivector fields are at least linear. One checks that this is actuallya direct summand. One checks that (X • , d ) deformation retracts onto (X( z , · · · , z n ) , d = ) where the z j form a basis of the null space of Π . The result now follows since the gauge group in this case ispronilpotent. (cid:3) ach bivector field Π = Σ (cid:10) ∂ i , Ad Π ij ∂ j (cid:11) ∈ X defines a map T ∗ [ ] L (−) Π → T [ ] L by the formula ( dx i ) Π = [ Π, x i ] = ι dx i Π = Ad Π ij ∂ j Lemma 2.16.
Let Π ∈ X be non-degenerate. Then the induced map Ω n (−) Π − → X n is injective for n ≥ .Proof. Let A denote the universal enveloping algebra of the free Lie algebra, that is the free associativealgebra. Let DA denote the free A -bimodule generated by the symbols ∂ i . By symmetrizing over thepermutation group in n symbols the space of n-forms Ω n can be embedded into DA ⊗ n ⊗ A ⊗ A op A . Themap (−) Π , for simplicity seen as an endomorphism of Ω n , extends as follows to an injective map. Thenon-degeneracy assumption is equivalent to Π defining an injective map of a free right- A module andconsequently (using skew-adjointness of Π ) also an injective map φ of a free left- A module A m where m is the dimension. More concretely, let e i be the canonical basis of A m , then A m φ − → A m α i e i → α i Π ij e j is an injective map of free left A modules. Consider A ⊗ A as a right- A module using the following A -bimodule structure ( α ⊗ β ) .a = αa ′ ⊗ ∗ a ′′ β, where Sweedler’s notation is used, and ∗ denotes the antipode. This defines a free A -module, as theinvertible map α ⊗ β → α ′ ⊗ β ′′ gives a map to an obviously free A -module. The left- A ⊗ A op comingfrom the outer A -bimodule structure is left as it is. The natural map DA → DA determined by Π isnow given by id ⊗ φ , that is DA ∼ = ( A ⊗ A ) ⊗ A A m id ⊗ φ −−−− → ( A ⊗ A ) ⊗ A A m ∼ = DAα∂ i β ∼ = ( α ⊗ β ) ⊗ e i → ( α ⊗ β ) ⊗ Π ij e j = ( αΠ ′ ij ⊗ ∗ Π ′′ ij β ) ⊗ e j ∼ = α ad Π ij ( ∂ j ) β, and is injective since we tensor an injective map with a free module. To finish the proof, one noticesthat DA ⊗ k is a free A -bimodule for all k ≥ and DA ⊗ n ⊗ A ⊗ A op A ∼ = DA ⊗ n − k ⊗ A ⊗ A op DA ⊗ k for any k (here DA ⊗ n − k being an A -bimodule is naturally a right- A ⊗ A op module. (cid:3) Lemma 2.17. If Π is Poisson, there is a canonical Lie bracket on Ω • such that (−) Π is a map of dgLAs.Proof. By lemma 2.4 it is enough to define the bracket on the left hand side that specializes to theclassical one. To that purpose we define an odd Poisson bivector field on T ∗ [ ] L n as follows e Π = D ∂∂dx i ad Π ij ∂∂x j E + D ∂∂x i ad dΠ ij ∂∂x j E , which satisfies the required properties. (cid:3) For later convenience we spell out the bracket on 1-forms, that is of the Lie algebra of 1-forms denotedby Ω . Let α = h α i , dx i i β = h β j dx j i be 1-forms, then [ h α i , dx i i , h β j , dx j i ] Π = (cid:10) [ α Π , β i ] − [ β Π , α i ] , dx i (cid:11) + (cid:10) α i , Ad dΠ ij β j (cid:11) . The dgLA X • is canonically filtered by (polynomial degree − + form degree − ). This also defines afiltration on Ω • Π , and by the splitting lemma it is clear that the zeroth associated graded component isa copy of a gl , where is the dimension of the symplectic vector space determined by the constantterm of Π . In particular, Ω is then an extension of a gl by a pronilpotent Lie algebra, and hence it’seasily integrated to a group Exp ( Ω ) together with its actions on forms and polyvector fields. Moreover, Exp ( Ω ) is an central extension of a Lie algebra of derivations of L n by a factor, which in the non-degenerate case can be identified with the space of Casimir functions, that is { f ∈ F ( L n ) | [ Π, f ] = } . Remark . The bivector field ˜ Π is actually Poisson and the map (−) Π is a Poisson map, for astraightforward definition of Poisson maps. However, we shall have no use for this slightly strongerfact. emark . A more geometric construction goes as follows. Let C = T ∗ [ ] T [ ] L n = T [ ] T ∗ [ ] L n denote the standard Courant algebroid. Let Q ∈ F ( C ) denote the Euler vectorfield on T [ ] M , which is aHamiltonian for the de Rham differential on C . The map (−) Π can be seen as the composition T ∗ [ ] L n → C exp ( { Π, · } ) → C → T [ ] L n , where the first and third maps are canonical. Thus the Courant bracet { · , { Q, · }} gets twisted to { · , { e Q, · }} ,where e Q = exp ( { Π, · } )( Q ) . This is indeed a Poisson bracket on T [ ] L n iff it is at most quadratic, i.e. iff { Π, { Π, Q }} = , i.e. iff Π is Poisson. In that case e Q = Q + e Π .Maurer-Cartan elements in Ω • thus inject to the ones in X • . Namely, a Maurer-Cartan element σ ∈ Ω defines a new Poisson bracket Π + σ Π . In terms of matrices (cf lemma 2.6) this Poisson structureis given by Π − ΠσΠ = Π ( − Πσ ) . We call σ ∈ Ω non-degenerate if the matrix ( − Πσ ) is invertible.This is in particular sufficient for Π + σ Π to be a non-degenerate Poisson structure. Let us denote by Π and σ the constant terms, then the condition is equivalent to ( − Π σ ) being non-degenerate, i.e.invertible. Corollary 2.20.
The set of non-degenerate Maurer-Cartan elements in Ω • is an E xp ( Ω ) homogeneousspace isomorphic to P := { Π + σ Π | [ Π + σ Π , Π + σ Π ] = ( − Π σ ) invertible } where the action on thelatter is by automorphism of the free Lie algebra. The stabilizer Lie algebra at σ is isomorphic to Ω with Lie bracket induced by the Poisson bracket Π + σ Π .Proof. The isomorphism alluded to is given by (−) Π , which is injective on forms of degree ≥ . Tran-sitivity of the E xp ( Ω ) action essentially follows from acyclicity of Ω • . More precisely, by the splittinglemma we can write E xp ( Ω ) as a prounipotent extension of GL k . The action of E xp ( Ω ) descend tothis GL k by only considering the constant terms. It is then just the usual action of GL k on symplecticforms on a vector space. The invertibility condition of the theorem ensures that we get a symplectic formagain, that is it does not become degenerate. Let now σ ∈ Ω be any non-degenerate Maurer-Cartanelement. By applying an element from GL k we can assume that Π + σ Π has the same constant term as Π , and thus σ lies in the pronilpotent part of Ω • . Now the standard argument in an acyclic pronilpotentdgLA shows that σ is gauge equivalent to .For the statement about the stabilizer, we only need to determine it at σ = by the transitivity ofthe E xp ( Ω ) action, where it is clear. (cid:3) Remark . If Π has no constant part, then the assumption on non-degeneracy is void. Lemma 2.22.
The set P is in bijection with non-degenerate closed 2-forms. More precisely, P = { Π ω = ( − Πω ) − Π | dω = ( − Π ω ) invertible } ∼ = { ω ∈ Ω | ( − Π ω ) invertible } . Proof.
One checks that each element in Π ∈ P is of the form ( − Πω ) − Π for a unique ω ∈ Ω , bysolving the equation of matrices with entries in the free associative algebra, Π − ΠσΠ = ( − Πω ) − Π, for ω , namely one gets ω = σ ( − Πσ ) − . (1)Uniqueness follows since one uses non-degeneracy of Π . Moreover, this is well-defined, since by definitionof P the matrix ( − Πσ ) is invertible. Moreover, since ( − Πσ ) = ( − Πω ) − , we see that the invertiblitycondition is the same as in the definition of P . It remains to check that dσ + [ σ, σ ] Π is equivalent to dw = . For this purpose, let us define the map C : E xp ( Ω ) → Ω , by the assignment C ( φ ) = ω for φ.Π = Π ω and φ ∈ E xp ( Ω ) . One checks that this is a group 1-cocyle. Since E xp ( Ω ) acts transitivelyon P , the image of C is exactly the image of P under the map defined by the formula (1). The associatedLie algebra 1-cocycle c is given by λ → dλ , which implies that any ω defined by (1) is indeed closed. Toshow that we get any non-degenerate closed two-form, we first reduce to the case where Π ω = byusing the GL action. By solving a Moser flow type equation one shows that there is a one-parameterfamily φ t ∈ E xp ( Ω ) such that C ( φ t ) = tω . More precisely let ω = dλ and ˙ φφ − = α t , the Moserequation is then c ( α t ) + ( α t ) . ( tω ) = ω, with a solution given by α t = λ ( + tΠω ) − . (cid:3) emark . The cocycle C : E xp ( Ω ) → Ω can be lifted to a group cocycle with values in Ω ifthere is a Liouville vector field for the Poisson structure, that is if there exists X such that [ Π, X ] = Π .This is in particular true for any homogeneous Poisson structure. For the KKS Poisson structure this isused in [9]. Remark . A different proof for the last steps can be obtained by showing that the formula [ Π ω , Π ω ] = ( dω ) Π ω holds for all non-degenerate two-forms ω . One quick way of seeing this is by using lemma 2.4 it can bereduced to the same formula in ”ordinary” differential geometry, where it follows for symplectic Π bythe usual formula [ Π, Π ] = − ( d ( Π − )) Π and for general Π by a density argument.2.2. Moment Maps.
Let ρ = P [ x i , ∂ i ] ∈ T ∗ [ ] L n denote the canonical action vector field. Note thatit is independent of the choice of coordinates. Remark . Viewed under the natural embedding of Lie into associative algebras, ρ corresponds to thecannonical element in Der ( A, A ⊗ A ) that maps a → ⊗ a − a ⊗ .One defines the operator ι ρ : Ω • → O ( T ∗ [ ] L n ) by the following procedure. Adjoin an extra variable t and define ρ t := h t, ρ i as in the proof of lemma 2.6. Now write ρ t as h ∂ i , ρ ti i and define a derivation ι ρ t of degree − on T ∗ [ ]( L n × L ( t ) by sending dx i → ρ ti , which descends to a derivation on Ω • ( L n × L ( t )) .The operator ι r ho is now defined by the formula ι ρ t α = h t, ι ρ ( α ) i for ∀ α ∈ Ω • ( L n ) Lemma 2.26. ι ρ t ( dα ) = [ α, t ] ∀ α ∈ F ( L n ) ι ρ (cid:10) dx i , Ad ω ij dx j (cid:11) = (cid:10) dx i , Ad x i Ad ω ij dx j (cid:11) ∀ (cid:10) dx i , Ad ω ij dx j (cid:11) ∈ Ω ( L n ) Definition 2.27.
An element µ ∈ L n is called a moment map for Π if [ Π, µ ] = ρ Remark . The existence of a moment map implies non-degeneracy.
Remark . Any linear Poisson structure, for instance the linear part of a splitting of an arbitraryPoisson structure, defines a Lie algebra in the ”Lie world”, which by Lazard duality is a commutativealgebra C . Non-degeneracy is equivalent to the absence of elements β ∈ C such that αβ = ∀ α ∈ C , andexistence of a moment map is equivalent to the existence of a unit element in C . Moreover, the Poissonstructure is rigid, if C is semi-simple, i.e. does not contain nilpotents, i.e. is the product of finite fieldextensions. Lemma 2.30.
Let Π ∈ X be a non-degenerate Poisson structure. If it admits a moment map, then itis unique.Proof. This can be seen by rewriting the moment map condition as dµ Π = ρ and using non-degeneracyof Π . (cid:3) The next theorem is the main result of this note, showing that Poisson structures are essentiallyuniquely determined by their moment maps. Let Π ∈ X be non-degenerate Poisson with moment map µ ∈ L n . Let Π be the constant terms of Π , in particular Π defines a pairing on an n-dimensional vectorspace V . Let Z denote the kernel of this pairing. One checks that the degree 2 part of µ , denoted by µ ,endows V/Z with a linear symplectic structure.
Theorem 2.31.
Given Π ∈ X a non-degenerate Poisson structure with moment map µ ∈ L n . Thenwe have the following. i) All elements in P admit a unique moment map. ii) Assigning the moment map to a given Poisson structure in P defines an injective map P → L n ,compatible with the transitive E xp ( Ω ) -action. iii) The image of this map is { φ.µ | φ ∈ E xp ( Ω ) } = { ˜ µ ∈ µ + L ≥ | ˜ µ is non-degenerate on V/Z } he theorem gives us a well-defined map { φ.µ | φ ∈ E xp ( Ω ) } − → Ω η → ω η , with the property that Π ω η = ( − Πω η ) − Π is Poisson with moment map η , which is a bijection in thecase where Π has no constant terms. Proof. i) Note that since each element in P is of the form φ.Π for some φ ∈ Aut n there is at leastone corresponding moment map, namely φ.µ , which is unique by the previous lemma.ii) For a µ + ˜ µ and a 2-form ω , one can rewrite the equation [ Π ω , µ + ˜ µ ] = ρ as d ˜ µ = ι ρ ω , orAd ˜ µ i = − X j Ad x j Ad ω ji , (2) where d ˜ µ = P Ad ˜ µ i dx i , ω = P (cid:10) dx i , Ad ω ij dx j (cid:11) . This shows injectivity, namely the equationuniquely determines ω ij . The compatibility with the E xp ( Ω ) -action is clear.iii) For surjectivity one can check directly that the form ω defined by equation 2 is indeed closed andhas the desired property. A more geometric construction suggested by ˇ S evera goes as follows.Let M denote the central extension of ( T [ ] L n ) deg ≤ by Ω , that is [ α, β ] = h α, β i for 1-forms α , β . One checks that elements of the form ˜ µ + d ˜ µ + ω with ω closed form a subalgebra. Thus ω can be constructed by taking the degree 2 part of ˜ µ ( x + dx , . . . , x n + dx n ) . To write downthis element one needs to write ˜ µ = P [ µ k1 , µ k2 ] to determine ω = (cid:10) dµ k1 , dµ k2 (cid:11) . One computes ι ρ t ω = (cid:10) ι ρ t dµ k1 , dµ k2 (cid:11) − (cid:10) dµ k1 , ι ρ t dµ k2 (cid:11) = (cid:10) [ µ k1 , t ] , dµ k2 (cid:11) − (cid:10) dµ k1 , [ µ k2 , t ] (cid:11) = − (cid:10) t, [ µ k1 , dµ k2 ] + [ dµ k1 , µ k2 ] (cid:11) = − h t, d ˜ µ i . By computing the constant terms one checks that the condition on the mompent map is equivalentto ω being non-degenerate. (cid:3) Remark . There is also a version of the theorem in the associative case. Here the coefficients ω ij lie in A ⊗ A op . Similar as in lemma 2.16, one can choose an automorphism of A ⊗ A such thatAd x j is represented by left multiplying on the left factor. Then one checks that for 2 to have a solution,necessarily ˜ µ ∈ [ A, A ] , which is also sufficient for the rest of the proof to go through. Example 2.33.
The theorem states, that a moment map uniquely determines a Poisson bracket ina given gauge class. However, there exist distinct Poisson brackets with the same moment map. Toconstruct an example, consider Φ ∈ Aut ( L ) given by x → x + [ x , x ] , x → x − [ x , x ] , x → x .Clearly, Φ ( x + x + x ) = x + x + x , however, one easily checks that Φ does not preserve h ∂ i , [ x i , ∂ i ] i . Remark . In the symplectic case, the theorem is essentially equivalent to the result of Massuyeau-Turaev about non-degenerate Fox pairings (cf. [6]), which strengthens an earlier result of Kawazumi-Kuno (cf. [7]).2.3.
Kirillov-Kostant-Souriau Poisson structure.
In this section the results are spelled out for thecase of the Kirillov-Kostant-Souriau bivector field given by Π := Σ h x i , [ ∂ i , ∂ i ] i . This induces the map T ∗ [ ] L n (−) Π − → T [ ] L n , dx i = [ x i , ∂ i ] , x i = x i . The space of Casimir functions is determined by the following
Lemma 2.35.
The kernel of the map (−) Π : Ω ( L n ) → X( L n ) is linearly spanned by h x i , dx i i . Identifying vectorfields with derivations we get a map Ω ( L n ) − → Der ( L n ) h α i , dx i i 7− → ( x i → [ x i , α i ]) . Let us denote the Lie algebra Ω ( L n ) by tder n . he bracket on tder n can be computed as follows. Let α = h α i , dx i i , β = h β i , dx i i ∈ Ω ( L n ) then [ α, β ] = (cid:10) α ♯ ( β i ) − β ♯ ( α i ) + [ α i , β i ] , dx i (cid:11) . Remark . Our definition of tder n differs by an n -dimensional abelian direct summand from the onein [8]. The same remark applies to TAut n .Let us denote the integrating Lie group of tder n by TAut n . The above map exponentiates to Exp ( Ω ( L n )) − → Diff ( L n ) = Aut ( L n ) e α i dx i → ( x i → e A i x i e − A i ) where A i = e α ♯ − ♯ ! ( α i ) . Using the e A i as components of a map, TAut n can be thought of as Map ( L n , E xp ( L n )) .The closed one forms Ω cl ( L n ) ∼ = Ω ( L n ) form a Lie subalgebra, whose Lie group H am ( L n ) can becharacterized by the following lemma that appears in [10], Lemma 2.37 (Drinfeld) . Let φ ∈ E xp ( Ω ( L n )) . Then φ ∈ H am ( L n ) ⇐⇒ φ ( Σx i ) = Σx i Proof.
This follows from theorem 2.31. For convenience we give a direct proof. It is enough to show thatfor α = h α i dx i i , α ( Σx i ) = implies that α is closed. We are going to use the fact that any Lie series α can be written as α = ∂α∂x i x i = x i (cid:18) ∂α∂x i (cid:19) ∗ ∈ k h x , . . . , x n i , where ∗ denotes the antipode in k h x , . . . , x n i . Using this we get α ( Σx i ) = ⇐⇒ [ α i , x i ] = ⇐⇒ x i α i = α j x j ⇐⇒ x i ∂α i ∂x j x j = x i (cid:18) ∂α j ∂x i (cid:19) ∗ x j ⇐⇒ ∂α i ∂x j = (cid:18) ∂α j ∂x i (cid:19) ∗ ∀ i, j ⇐⇒ dα = (cid:3) Lemma 2.38.
TAut n acts transitively on Σx i + L ≥ .Proof. Also follows from our main theorem 2.31. (cid:3) Hamiltonian spaces as a
TAut -algebra
As was shown above the groups TAut n act transitively on P x i + L ≥ . The corresponding groupoidsfit together to form an operad in groupoids which we denote again by TAut n . Instead of giving theoperadic compositions, a faithful (after taking a suitable limit) action on a category is constructed, fromwhich the operadic structure can be infered.Let C n denote the category of formal g n -Hamiltonian spaces, that is Poisson manifolds with a Poissonmap into g n (recall that g is quadratic). Using the canonical map C n → C , one sees that Fun ( C n , C ) form an operad in groupoids.A map of operads TAut n → Fun ( C n , C ) is defined as follows. X x i + L ≥ − → Fun ( C n , C ) µ → ( M, Π, h ) → ( M, Π ω µ , µ ◦ h ) where ( M, Π, h ) is a Hamiltonian g -space with Π its Poisson structure and h : M → g n its moment map.On arrows it is defined as follows. TAut n − → End ( Fun ( C n , C )) g : g n → G n → M → M ; m → g ( h ( m )) .m where some abuse of notation is committed and an element g ∈ TAut n is viewed as its induced map g n → G n . heorem 3.1. The above map is a well-defined map of operads
TAut n → Fun ( C n , C ) Proof.
The first point to notice is that µ ∈ P x i + L ≥ give well-defined functors C n → C . Here aPoisson map M → g n is gauge transformed by a closed two form on g n and then composed with aPoisson map µ : g n → g . For the second part one needs to check that a g ∈ TAut n indeed intertwinesthe respective Poisson structures. Note that the above formula defines an action of the group TAut n on M by diffeomorphisms. It remains to check that g intertwines ( M, Π ω µ , µ ◦ h ) g → ( M, Π ω g.µ , ( g.µ ) ◦ h ) . The moment map part is obvious. Moreover, the statement can be reduced to the case µ = P x i bytransitivity of the action. Thus the statement becomes g.Π − Π = Π ω g.µ − Π. Since both sides define group cocycle with values in bivector fields on M , it is enough to verify that thecorresponding Lie cocycles coincide. Let ( u , · · · , u n ) ∈ tder n and let moreover ρ i = [ h i , Π ] denote the i -th g -valued action vector field on M , the Lie cocycle of the left hand side computes to L P h u i ◦ h,ρ i i Π = X h [ u i ◦ h, Π ] , ρ i i = X D ∂u i ∂h j [ h j , Π ] , ρ i E = X D ∂u i ∂h j ρ j , ρ i E , which is by definition the same as the right hand side. (cid:3) Remark . Not that for each µ ∈ P x i + L ≥ we get a product of Hamiltonian spaces. The resulting G -action, however, is always the diagonal action. Remark . One can extend beyond formal Hamiltonian spaces by restricting to suitably convergentelements.3.1.
Application to Hamiltonian quasi-Poisson spaces.
Let us briefly recall the relevant definitionsfrom [3]. Let φ ∈ Λ g denote the Cartan three-form of the quadratic Lie algebra g . Definition 3.4.
A pair ( M, Π ) of a g -manifold together with a bivector field Π ∈ Γ ( Λ TM ) is called quasi-Poisson if [ Π, Π ] = φ M , where φ M denotes the tri-vector field on M induced by φ and the g -action on M. Definition 3.5.
A map µ : M → G is called a moment map if ( ⊗ µ ∗ df ) Π = ( ρ ⊗ df )( Z ) where Z ∈ g ⊗ X( G ) is the adjoint action of g on G , where g and g ∗ are identified.A tuple ( M, Π, µ ) is called a g -Hamiltonian quasi-Poisson space The category of g -Hamiltonian quasi-Poisson spaces admits a monoidal structure given by the following Definition 3.6 (Fusion) . Let ( M, Π ) be a g × g -Hamiltonian quasi-Poisson space, then Π fus = Π − ψ M gives a g -Hamiltonian quasi-Poisson space with the diagonal g -action and moment map defined by mul-tiplying the two factors.For two g -Hamiltonian quasi-Poisson spaces M and N , we define their fusion product by M ⊛ N := ( M × N, Π M + Π N − ψ M × N , µ · µ ) . Example 3.7.
The moduli space of flat g -connections on a surface Σ g,n of genus g with n boundarycomponents, and a marked point on the boundary is given by Hom ( π , G ) ∼ = G + n − , and carries anatural quasi-Poisson structure. It can be constructed by viewing it as DG ⊛ g ⊛ G ⊛ n − . quasi-Poisson bivector can in general be turned into a Poisson bivector by adding an r-matrixterm. One particular (dynamical) r-matrix is the Alekseev-Meinrenken dynamical r-matrix. Thus theconstructions goes as follows. Let ν ( z ) := − coth ( z2 ) = − z12 + z + · · · and define the followinguniversal two-form on g , T = h dx, ν ( ad x ) dx i . Then recall (cf. [3])
Proposition 3.8 (Exponentiation) . Let ( M, Π ) be a g -Poisson manifold with moment map µ : M → g .Consider T as a map g → g ∧ g . Then ( M, Π − ( µ ∗ T ) M ) is a g -Hamiltonian quasi-Poisson manifold withmoment map exp ◦ µ .The bivector field can also be written as Π − ( µ ∗ T ) ♯ by considering T as a two-form on g and usingthe morphism ♯ between forms and polyvector fields induced by Π . Let E xp denote the functor sending a g -Hamiltonian Poisson space to the g -Hamiltonian quasi-Poissonspace given by the last proposition. Example 3.9.
The standard g -Hamiltonian quasi-Poisson space G with moment map the identity,corresponds to g with its KKS structure under this functor.The theorem is equivalent to T ∈ Ω ( L ) satisfying the dynamical Yang-Baxter equation − + [ T, T ] Π KKS = h dx, [ dx, dx ] i . Pulling back the fusion product along the functor E xp one gets a second monoidal structure on thecategory of g -Hamiltonian spaces, which we denote again by ⊛ . It is given by ( M, Π M , µ M ) ⊛ ( N, Π N , µ N ) = ( M × N, Π M + Π N + (( µ M × µ N ) ∗ σ ) ♯ , log ( e µ M e µ N )) for σ = T − T − T + h dx, dy i ∈ Ω ( L n ) . This σ is thus a Maurer-Cartan element in Ω ( L n ) since it is so for any sl k . Setting ω = σ ( + Πσ ) − ∈ Ω L n , the above Poisson structure can be written as ( Π M × Π N ) ( µ M × µ N ) ∗ ω ) and ω = ω log ( e x1 e x2 ) . Inparticular, there two products are induced by x + x and log ( e x e x ) , respectively.Let now F ∈ TAut be an element intertwining those two structures, that is such that F ( log ( e x e x )) = x + x Remark . As shown in [4] one particular source of such F is Drinfeld associators. Namely, let Φ = exp ( φ ) for φ ∈ ^ Lie ( x, y ) be a Drinfeld associator. Then we associate to it the F Φ with components (cid:16) Φ ( x, − x − y ) , e − x + y2 Φ ( y, − x − y ) (cid:17) . As a consequence of the above discussion, we get the following.
Proposition 3.11.
Let
M, N be two g -Hamiltonian Poisson spaces. Then the following map is Poisson. M × N F M,N → E xp − ( E xp ( M ) ⊛ E xp ( N ))( a, b ) → ( F ( µ M ( a ) , µ N ( b )) .a, F ( µ M ( a ) , µ N ( b )) .b ) Moreover, the F M,N are a natural transformation.
The maps F M,N can now be interpreted as a monoidal structure on the functor E xp . Let C denotethe category of g -Hamiltonian Poisson spaces with monoidal product given by the product of Poissonspaces. Instead of the trivial associator isomorphism, let C be endowed with the associator derived from F . More precisely, define Φ F = F F F − F − ∈ TAut , and use it to define a diffeomorphism Φ FX,Y,Z for any triple
X, Y, Z of g -Hamiltonian Poisson spaces. Let D denote the category of g -Hamiltonian quasi-Poisson spaces. Then we get Proposition 3.12. An F ∈ TAut such that F ( log ( e x e x )) = x + x promotes the functor E xp to amonoidal equivalence ( C , × , Φ F ) E xp − → ( D , ⊛ , id ) orollary 3.13. An F ∈ E xp ( Ω ( L )) satisfying (3.1) gives a Poisson map O λ × · · · × O λ n (cid:14)(cid:14) G → M ( Σ , C , · · · , C n ) where O λ i are coadjoint orbits for given λ i ∈ g ∼ = g ∗ , and M ( Σ , C , · · · , C n ) is the moduli space of flatconnections on a surface of genus 0 with n punctures and monodromies around the punctures prescribedby conjugacy classes C i = G.exp ( λ i ) .Remark . Taking F = F Φ KZ to be associated to the Knizhnik-Zamolodchikov associator, the previousmap is given by a , a → d − (cid:18) a z + a z − (cid:19) dz References [1] L. Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298 (1994), no. 4, 667–692[2] M. van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5711–5769[3] A. Alekseev, Y. Kosmann-Schwarzbach, E. Meinrenken, Quasi-Poisson manifolds, Canad. J. Math. 54 (2002), no. 1,3–29[4] A. Alekseev, B. Enriques, C. Torossian, Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergneequations, Publ. Math. Inst. Hautes ´Etudes Sci. 112 (2010), 143–189[5] M. Kontsevich, Formal (non)commutative symplectic geometry, The Gel’fand Mathematical Seminars, 1990–1992,Birkh¨auser Boston, Boston, MA (1993), 173–187[6] G. Masuyeau, V. Turaev, Fox pairings and generalized Dehn twists, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6,2403–2456[7] N. Kawazumi, Y. Kuno, The logarithms of Dehn twists, Quantum Topol. 5 (2014), no. 3, 347-423[8] A. Alekseev, C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. 175 (2012), no.2, 415–463[9] A. Alekseev, F. Naef, X. Xu, C. Zhu, Chern-Simons, Wess-Zumino-Witten and other cocycles, in preparation[10] V. Drinfeˇld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal ( ¯ Q / Q ) , Algebrai Analiz 2 (1990), no. 4, 149–181., Algebrai Analiz 2 (1990), no. 4, 149–181.