The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular $r$-Matrix

We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces.


Introduction
Let G be a connected complex Lie group with Lie algebra g, and let s ∈ S 2 g be a g-invariant element. The moduli space of flat G-connections on a Riemann surface Σ has the well known [3] canonical Atiyah-Bott Poisson structure. If one "marks" finitely many points V ⊂ ∂Σ in the boundary of Σ and consider only gauge transformations which are trivial over V , Fock and Rosly have constructed in [4,5] a Poisson structure π r on the corresponding moduli space A(Σ, V ), which depends on a quasitriangular r-matrix r v for every v ∈ V such that all r v 's have symmetric part s. Under the quotient by the group of lattice gauge transformations G V , π r descends to the Atiyah-Bott Poisson structure on the full moduli space, and quantizations of π r play a fundamental role in quantum gravity (see [10] and references therein).
The bivector field π r was given in [4,5] by a formula, of which the proof that it defines a Poisson structure was left as a computation. In this paper, as an application of the methods developed in [8], we give a simpler and more conceptual proof that π r is a Poisson structure, by viewing it in the framework of Poisson structures defined by a Lie algebra action and a quasitriangular r-matrix. Recall that given an action ρ : h → X 1 (Y ) of a Lie algebra h on a manifold Y and a quasitriangular r-matrix r ∈ h ⊗ h, if the pushforward π Y = ρ(r) is a bivector field, it is automatically Poisson, and one says that π Y is defined by the action ρ and the r-matrix r.
More precisely, given an oriented skeleton Γ of a marked surface (Σ, V ), one has a natural action σ Γ of the Lie algebra g Γ 1/2 on A(Σ, V ), where Γ 1/2 is the set of half edges of Γ, and a quasitriangular r-matrix r Γ ∈ g Γ 1/2 ⊗ g Γ 1/2 , such that σ Γ (r Γ ) is a Poisson structure. Both σ Γ and r Γ depend on Γ, but one proves that π r = σ Γ (r Γ ) does not.
Marked surfaces can be fused at their marked points. One also has the notion introduced in [8] of fusion of Poisson spaces admitting a Poisson action by a quasitriangular Lie bialgebra, and we show that the Poisson structures on the associated moduli spaces correspond under these constructions. In particular, the Fock-Rosly Poisson structure is an example of a mixed product Poisson structure associated to pairs of Poisson actions introduced in [8].
On the other hand, A(Σ, V ) carries a canonical quasi-Poisson structure Q s , first discovered in [9] when V is a singleton, and further studied in [7] for general V 's, which can be obtained by reduction of the canonical symplectic structure on the infinite-dimensional affine space of Gconnections on Σ. Quasi-Poisson manifolds were introduced in [1, 2] as a way to obtain a unified picture of various notions of moment maps. It is shown in [1,6,8] (see also Section 5.1) that one has an equivalence of categories between the category of (g, φ s ) quasi-Poisson spaces and the category of (g, r) Poisson spaces, where r is a quasitriangular r-matrix whose symmetric part is s, and φ s ∈ ∧ 3 g is the Cartan element associated to s (see (2.1)). We show in this paper that π r corresponds to Q s under this equivalence of categories.
An interesting project would be to develop a theory of quantizations of Poisson structures defined by actions of Lie algebras and quasitriangular r-matrices. This paper provides the setting to study the quantization of the Fock-Rosly Poisson structure from this point of view.
The paper is organized as follows. In Section 2 we recall the basic facts on quasitriangular r-matrices which will be needed later, and in Section 3 we recall the fusion of ciliated graphs and marked surfaces. The Poisson structure π r on the moduli space A(Σ, V ) is defined in Section 4, where we prove that it is independent of the choice of an oriented skeleton of (Σ, V ), and that fusion of marked surfaces corresponds to fusion of the associated Poisson structures. In Section 5, the equivalence between π r and the quasi-Poisson structure Q s under an equivalence of categories between Poisson and quasi-Poisson spaces is proven.

Notation
Throughout this paper, vector spaces are understood to be over R or C.
If Γ is a finite set and {X γ : γ ∈ Γ} a family of sets indexed by Γ, for x ∈ γ∈Γ X γ and γ ∈ Γ, x γ ∈ X γ denotes the γ-component of x. If {V γ : γ ∈ Γ} is a family of groups and v ∈ V γ , (v) γ ∈ γ∈Γ V γ is the image of v under the embedding of V γ into γ∈Γ V γ as the γ-component. When the V γ 's are vector spaces, we extend this notation to tensor powers. Namely, for an integer is a right (resp. left) action of a Lie group G on a manifold Y , we will denote by ρ : g → X 1 (Y ) (resp. λ : g → X 1 (Y )) the induced right (resp. left) Lie algebra action of the Lie algebra g of G on Y . If x ∈ g ⊗k , k ≥ 1, we denote respectively by x R and x L the right and left invariant k-tensor field on G whose value at the identity e ∈ G is x.
Lie bialgebras will be denoted as pairs (g, δ g ), where g is a Lie algebra, and δ g : g → ∧ 2 g the cocycle map. Recall that δ g satisfies x, y ∈ g, and that the dual map δ * g : ∧ 2 g * → g * is a Lie bracket on g * .

Poisson structures def ined by quasitriangular r-matrices
We recall in this section basic facts about quasitriangular r-matrices and refer to [8] for a detailed exposition on Poisson Lie groups and Lie bialgebras.

Quasitriangular r-matrices
Let g be a finite-dimensional Lie algebra, and let r = s+Λ ∈ g⊗g, with s ∈ (S 2 g) g and Λ ∈ ∧ 2 g. One says that r is a quasitriangular r-matrix on g if it satisfies the classical Yang-Baxter equation where [ , ] : ∧ k g⊗∧ l g → ∧ k+l−1 g is the Schouten bracket on the exterior powers of a Lie algebra, and φ s ∈ ∧ 3 g is defined by where s : g * → g is given by s (ξ), η = s(ξ, η), ξ, η ∈ g * . If r = i x i ⊗ y i ∈ g ⊗ g is a quasitriangular r-matrix, it defines a Lie bialgebra structure and one calls the pair (g, r) a quasitriangular Lie bialgebra. Let (g, δ g ) be a Lie bialgebra and (Y, π Y ) a Poisson manifold. A (right) Poisson action of (g, δ g ) on (Y, π Y ) is a Lie algebra morphism ρ : x ∈ g, and one also says that (Y, π Y , ρ) is a right (g, δ g )-Poisson space. Let g be a finite-dimensional Lie algebra, Y a manifold and ρ : g → X 1 (Y ) a right action of g on Y . For r = i x i ⊗ y i ∈ g ⊗ g, one has the 2-tensor field and writing r = s + Λ, with s ∈ S 2 g and Λ ∈ ∧ 2 g, it is clear that ρ(r) is a bivector field on Y if and only if ρ(s) = 0.
. If r is a quasitriangular r-matrix and if ρ(r) is a bivector field, it is a Poisson bivector field, and (Y, ρ(r), ρ) is a right (g, r)-Poisson space.
In the context of Proposition 2.1, one says that ρ(r) is a Poisson structure def ined by the quasitriangular r-matrix r and the action ρ.
Let g be a Lie algebra and n ≥ 1 an integer. For any r = i x i ⊗ y i ∈ g ⊗ g, define Mix n (r) ∈ ∧ 2 (g n ) by and for any sign function ε : where r = s + Λ with s ∈ S 2 g and Λ ∈ ∧ 2 g, and let Theorem 2.2 ([8, Theorem 6.2]). If r ∈ g ⊗ g is a quasitriangular r-matrix on g, then for any n ≥ 1 and any sign function ε, r (ε,n) is a quasitriangular r-matrix on g n , and the Lie bialgebra structure is independent of ε. Moreover, the map is an embedding of Lie bialgebras.
For any r ∈ g ⊗ g and any sign function ε, denote by Λ (n) r the anti-symmetric part of r (ε,n) .
The following lemma will be used in the proof of Proposition 5.2.
Proof . Indeed, writing r = i x i ⊗ y i and letting Λ ∈ ∧ 2 g be the anti-symmetric part of r, one has

Fusion products of Poisson spaces
Let n ≥ 1 be an integer, r ∈ g ⊗ g a quasitriangular r-matrix on a Lie algebra g, and let (Y, π Y ) be a Poisson manifold with a right Poisson action ρ : g n → X 1 (Y ) of (g, r) n , and a right Poisson is a right (g, r) × (h, δ h )-Poisson space, which we call the fusion at (g, r) n of (Y, π Y , ρ × ψ). As a particular case, suppose that h = 0, that are right (g, r)-Poisson spaces, that Y = Y 1 ×· · ·×Y n is equipped with the direct product Poisson structure π Y = π Y 1 × · · · × π Y n , and that ρ : g n → X 1 (Y ) is given by

Ciliated graphs and marked surfaces
In this section, we review the fusion of marked surfaces and ciliated graphs. Our main references are [5,7].

Ciliated graphs and marked surfaces
A marked surface (Σ, V ) is a compact Riemann surface, together with a non-empty finite collection of points V ⊂ ∂Σ lying in the boundary of Σ. A skeleton of a marked surface (Σ, V ) is a graph Γ embedded in Σ, with set of vertices V and such that Σ deformation retracts onto Γ. Let Γ be a skeleton of a marked surface (Σ, V ). For every v ∈ V , the orientation of Σ induces a linear ordering of the half edges incident to v, thus one is led to formulate the following Definition 3.2 ( [5,7]). A ciliated graph is a graph in which each vertex is equipped with a linear order of the half edges incident to it.
The name is inspired by the fact that when drawing a ciliated graph, one can specify the linear order of half edges at each vertex by drawing a small "cilium" between the minimal and maximal half edge. We introduce further notations in order to discuss ciliated graphs. Let Γ be a ciliated graph with set of vertices V and set of edges Γ 1 . Denote by Γ 1/2 the set of half edges of Γ, and note that Γ 1/2 comes with a natural involution with no fixed points α →α, mapping a half edge to the opposite half edge, and for α ∈ Γ 1/2 we write [α,α] for the edge composed of the two half edges α andα. For every v ∈ V , let Γ v ⊂ Γ 1/2 be the set of half edges incident to v, so that Γ 1/2 = v∈V Γ v and Γ v is a linearly ordered set for each v ∈ V .

Fusion of ciliated graphs and marked surfaces
We recall now the fusion of marked surfaces and ciliated graphs.
Let (Σ, V ) be a marked surface. Since Σ is oriented, every v ∈ V defines a piece of arc in ∂Σ starting at v and a piece of arc in ∂Σ ending at v. For a pair (v 1 , v 2 ) of distinct elements of V , the fusion of Σ at (v 1 , v 2 ) is the marked surface (Σ (v 1 ,v 2 ) , V v 1 =v 2 ) obtained by gluing a piece of arc ending in v 1 with a piece or arc starting at v 2 , so that v 1 and v 2 are identified. The set of Let Γ be a ciliated graph with set of vertices V and edges Γ 1 , and let (v 1 , v 2 ) be a pair of distinct vertices, with Γ v 1 = {α 1 < · · · < α k } and Γ v 2 = {α k+1 < · · · < α l }. The fusion of Γ at (v 1 , v 2 ) is the ciliated graph Γ (v 1 ,v 2 ) obtained by identifying v 1 and v 2 , and with linear order on the set Γ v 1 =v 2 of half edges incident to v 1 = v 2 given by α 1 < · · · < α k < α k+1 < · · · < α l .
Note that the fusion of marked surfaces and ciliated graphs are associative operations, but not necessarily commutative. The following lemma is straightforward.
Since is a skeleton for a disk with two marked points, and since every ciliated graph can be obtained by successive fusion of copies of , every marked surface can be obtained by successive fusion of disks with two marked points. Conversely, a ciliated graph Γ with set of edges V is the skeleton of a marked surface (Σ Γ , V ), well defined up to isomorphism, obtained by fusing marked disks corresponding to the edges of Γ. Thus the map Γ → (Σ Γ , V ) gives a bijective correspondence between isomorphism classes of ciliated graphs up to local changes in (3.1) and isomorphism classes of marked surfaces.

The Fock-Rosly Poisson structure
In this section, we introduce a Poisson structure, first discovered by Fock and Rosly, on the moduli space of flat connections over a marked surface, which is defined by an action of a Lie algebra and a quasitriangular r-matrix.
Throughout Section 4, G is a connected complex Lie group, and g is its Lie algebra.

The moduli space of f lat connections over a marked surface
For a marked surface (Σ, V ), let Π 1 (Σ, V ) be the fundamental groupoid of Σ over the set of base points V , and consider the moduli space of flat connections on G-principal bundles over Σ which are trivialized over V . The group G V naturally acts on the right on A(Σ, V ) by gauge transformations. For p ∈ Π 1 (Σ, V ), denote by ev p : A(Σ, V ) → G the evaluation at p, by θ(p), τ (p) ∈ V the respective source and target of p, and if g ∈ G V and v ∈ V , recall from Section 1.1 that g v is the v'th component of g. The action of G V on A(Σ, V ) is then given by where g ∈ G V , y ∈ A(Σ, V ), and p ∈ Π 1 (Σ, V ). Given a skeleton Γ of (Σ, V ) and an orientation of each edge of Γ, Π 1 (Σ, V ) is then the free groupoid generated by Γ, and thus one has a natural diffeomorphism and by choosing a skeleton for (Σ, V ), one easily sees that ϕ (v 1 ,v 2 ) is a diffeomorphism. The next lemma is straightforward.

The Fock-Rosly Poisson structure
Let (Σ, V ) be a marked surface and Γ an oriented skeleton of (Σ, V ). From now till the end of Section 4, we fix an s ∈ S 2 g g .
For every v ∈ V , let Λ v ∈ ∧ 2 g be such that r v = s + Λ v is a quasitriangular r-matrix. Identifying g Γv with g |Γv| using the linear order on Γ v , let r (εv,Γv) v ∈ g Γv ⊗ g Γv be as in (2.3), where Since Γ 1/2 = v∈V Γ v , one has g Γ 1/2 = v∈V g Γv , and recalling our notation in Section 1.1, let a quasitriangular structure for the Lie bialgebra is as in (2.4). Using the notational convention in Section 1.1, recall from (4.4) the right Lie algebra action σ Γ : g Γ 1/2 → X 1 (G Γ 1 ).
first appeared in [4,5], where the proof that it is Poisson was left as a computation to be checked. Theorem 4.2 gives a simpler and more conceptual proof that π Γ is a Poisson structure.
Consider the quasitriangular Lie bialgebra For v ∈ V , let diag v : g → g Γv , diag v (x) = γ∈Γv (x) γ , for x ∈ g, and define the map By Theorem 2.2, diag Γ : (g V , δ r ) → (g Γ 1/2 , δ r Γ ) is an embedding of Lie bialgebras, and by Lemma 4.1, one has where ρ V : g V → X 1 (A(Σ, V )) is the derivative of the action by gauge transformations in (4.1). Thus as an immediate consequence, one has Example 4.5. For any integer n ≥ 1, let (Σ n , V ) be a disk with n − 1 inner disks removed and two marked points v 1 , v 2 , such as in Fig. 3, and let Γ be the skeleton in Fig. 3 with edges oriented from v 1 to v 2 . Let r i = s+Λ i , be the r-matrix associated to v i , i = 1, 2. The orientation of Σ induces an isomorphism g Γv i ∼ = g n , and one has and In particular, if Λ 2 = −Λ 1 , the Poisson structure I Γ (π Γ ) = Λ L is multiplicative, and the Poisson Lie group is then called a polyuble in [4], and has Lie bialgebra g n , δ (n) r 1 . When n = 2 and r 1 is factorizable, that is when s : g * → g is invertible, G 2 , (Λ

Independence of choice of skeleton
Continuing with the setup and notation of Section 4.2, one has a Poisson structure I Γ (π Γ ) on A(Σ, V ). The goal of this subsection is to show that I Γ (π Γ ) does not depend on the choice of Γ, nor on the choice of an orientation of the edges of Γ. Letting Γ be another oriented skeleton of (Σ, V ), this is equivalent to proving that the map is a Poisson isomorphism.
Lemma 4.6. The Poisson structure I Γ (π Γ ) is independent of the orientation of the edges of Γ.
Proof . One can assume that Γ is the same oriented skeleton as Γ, except for an edge γ ∈ Γ 1 , which is given the opposite orientation. The map I −1 Γ • I Γ : G Γ 1 → G Γ 1 = G Γ 1 is thus the group inversion in the factor γ, and the identity on all other factors, hence for any x ∈ g Γ 1/2 , one has Lemma 4.7. Consider the following two oriented skeletons Γ of a disk with three marked points. Then the map (4.9) is a Poisson isomorphism.
Proof . Identifying G Γ 1 and G Γ 1 with G 2 and writing I = I −1 Γ •I Γ , one has I(g 1 , g 2 ) = (g 1 g 2 , g 2 ), g 1 , g 2 ∈ G, and where r v 2 = i x i ⊗ y i . A direct calculation using Ad g (s) = s for any g ∈ G, shows that We return to the case of a general marked surface (Σ, V ).
Proof . By Lemma 4.6 and Proposition 3.1, one can assume that Γ and Γ have the following form: Using Lemma 4.7, a straightforward calculation, the details of which are left to the readers, shows that I −1 Recall the quasitriangular r-matrix r on g V defined in (4.6) and let where Γ is any oriented skeleton of (Σ, V ). The following theorem is a consequence of Lemma 4.6, Proposition 4.8, and Lemma 4.4.

Fusion of Poisson spaces and marked surfaces
We continue with the setup of Section 4.2. In particular, (Σ, V ) is a marked surface, for v ∈ V , one has Λ v ∈ ∧ 2 g such that r v = s + Λ v is a quasitriangular r-matrix, and one considers the quasitriangular r-matrix r ∈ g V ⊗ g V defined in (4.6). Suppose one has r v 1 = r v 2 for two distinct elements v 1 , v 2 ∈ V , and consider the fused surface (Σ , V ) = (Σ (v 1 ,v 2 ) , V v 1 =v 2 ) with Poisson structure π r , where r ∈ g V ⊗ g V is defined as in (4.6), and let v ∈ V be the vertex obtained by fusing v 1 and v 2 . Recall the fusion of Poisson spaces discussed in Section 2.2.
Example 4.11. Let (Σ, V ) be a disk with two marked points v 1 , v 2 and assume that the rmatrices r 1 and r 2 associated to v 1 and v 2 are equal. Let the edge of Γ be oriented from v 1 to v 2 and identify A(Σ, V ) ∼ = G via I Γ , so that one has where Λ is the anti-symmetric part of The fused surface (Σ , {v }) is an annulus with one marked point, and one has

Quasi Poisson spaces and fusion of quasi Poisson spaces
Let g be a Lie algebra, s ∈ (S 2 g) g , and recall the element φ s ∈ (∧ 3 g) g defined in (2.1). Recall from [2] that a right (g, φ s )-quasi Poisson space is a triple (Y, Q Y , ρ), where Y is a manifold, ρ : g → X 1 (Y ) a right Lie algebra action, and Q Y ∈ X 2 (Y ) is a g-invariant bivector field on Y , such that We denote by QP(g, φ s ) the category of right (g, φ s )-quasi Poisson spaces, where the morphisms are g-equivariant smooth maps respecting the quasi Poisson bivector fields, and if r ∈ g ⊗ g is a quasitriangular r-matrix on g, denote by P(g, r) the category of right (g, r)-Poisson spaces, where the morphisms are g-equivariant Poisson maps.

A canonical quasi Poisson structure on A(Σ, V )
Let G be a connected complex Lie group, g its Lie algebra, and let s ∈ (S 2 g) g . Let (Σ, V ) be a marked surface and for v ∈ V , let Λ v ∈ ∧ 2 g be such that r v = s + Λ v is quasitriangular r-matrix, and let r ∈ g V ⊗ g V be as in (4.6). By Proposition 5.1, is a right (g, φ s ) V -quasi Poisson space.