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 ass...
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irk-123456789-1487522019-02-19T01:27:13Z The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix Mouquin, V. 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 Ševera under an equivalence of categories between Poisson and quasi-Poisson spaces. 2017 Article The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix / V. Mouquin // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D17; 53D30; 17B62 DOI:10.3842/SIGMA.2017.063 http://dspace.nbuv.gov.ua/handle/123456789/148752 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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 Ševera under an equivalence of categories between Poisson and quasi-Poisson spaces. |
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Mouquin, V. The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix Symmetry, Integrability and Geometry: Methods and Applications |
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Mouquin, V. |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix |
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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix / V. Mouquin // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 10 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 063, 13 pages
The Fock–Rosly Poisson Structure
as Defined by a Quasi-Triangular r-Matrix
Victor MOUQUIN
University of Toronto, Toronto ON, Canada
E-mail: mouquinv@math.toronto.edu
Received March 26, 2017, in final form August 01, 2017; Published online August 09, 2017
https://doi.org/10.3842/SIGMA.2017.063
Abstract. 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 Ševera under an
equivalence of categories between Poisson and quasi-Poisson spaces.
Key words: flat connections; Poisson Lie groups; r-matrices; quasi-Poisson spaces
2010 Mathematics Subject Classification: 53D17; 53D30; 17B62
1 Introduction
Let G be a connected complex Lie group with Lie algebra g, and let s ∈ S2g 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 πr on the corresponding moduli space A(Σ, V ),
which depends on a quasitriangular r-matrix rv for every v ∈ V such that all rv’s have symmetric
part s. Under the quotient by the group of lattice gauge transformations GV , πr descends
to the Atiyah–Bott Poisson structure on the full moduli space, and quantizations of πr play
a fundamental role in quantum gravity (see [10] and references therein).
The bivector field πr 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 πr is a Poisson structure, by
viewing it in the framework of Poisson structures defined by a Lie algebra action and a quasitri-
angular r-matrix. Recall that given an action ρ : h→ X1(Y ) of a Lie algebra h on a manifold Y
and a quasitriangular r-matrix r ∈ h⊗ h, if the pushforward πY = ρ(r) is a bivector field, it is
automatically Poisson, and one says that πY 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 σΓ of the Lie algebra gΓ1/2 on A(Σ, V ), where Γ1/2 is the set of half edges of Γ, and
a quasitriangular r-matrix rΓ ∈ gΓ1/2 ⊗ gΓ1/2 , such that σΓ(rΓ) is a Poisson structure. Both σΓ
and rΓ depend on Γ, but one proves that πr = σΓ(rΓ) 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].
mailto:mouquinv@math.toronto.edu
https://doi.org/10.3842/SIGMA.2017.063
2 V. Mouquin
On the other hand, A(Σ, V ) carries a canonical quasi-Poisson structure Qs, 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 G-
connections 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, φs) quasi-Poisson spaces and the
category of (g, r) Poisson spaces, where r is a quasitriangular r-matrix whose symmetric part is
s, and φs ∈ ∧3g is the Cartan element associated to s (see (2.1)). We show in this paper that
πr corresponds to Qs 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 πr 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 πr and the quasi-Poisson structure Qs under an equivalence
of categories between Poisson and quasi-Poisson spaces is proven.
1.1 Notation
Throughout this paper, vector spaces are understood to be over R or C.
If Γ is a finite set and {Xγ : γ ∈ Γ} a family of sets indexed by Γ, for x ∈
∏
γ∈ΓXγ and
γ ∈ Γ, xγ ∈ Xγ denotes the γ-component of x. If {Vγ : γ ∈ Γ} is a family of groups and v ∈ Vγ ,
(v)γ ∈
⊕
γ∈Γ Vγ is the image of v under the embedding of Vγ into
⊕
γ∈Γ Vγ as the γ-component.
When the Vγ ’s are vector spaces, we extend this notation to tensor powers. Namely, for an integer
k ≥ 1 and v ∈ V ⊗kγ , (v)γ is the image of v under the embedding of V ⊗kγ into (
⊕
γ∈Γ Vγ)⊗k as
the γ-component.
If ρ : Y × G → Y (resp. λ : G × Y → Y ) is a right (resp. left) action of a Lie group G on
a manifold Y , we will denote by ρ : g → X1(Y ) (resp. λ : g → X1(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 xR and xL 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, δg), where g is a Lie algebra, and δg : g→ ∧2g the
cocycle map. Recall that δg satisfies
δg([x, y]) = [x, δg(y)] + [δg(x), y], x, y ∈ g,
and that the dual map δ∗g : ∧2g∗ → g∗ is a Lie bracket on g∗.
2 Poisson structures defined 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.
2.1 Quasitriangular r-matrices
Let g be a finite-dimensional Lie algebra, and let r = s+Λ ∈ g⊗g, with s ∈ (S2g)g and Λ ∈ ∧2g.
One says that r is a quasitriangular r-matrix on g if it satisfies the classical Yang–Baxter equation
[Λ,Λ] + φs = 0,
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 3
where [ , ] : ∧k g⊗∧lg→ ∧k+l−1g is the Schouten bracket on the exterior powers of a Lie algebra,
and φs ∈ ∧3g is defined by
φs(ξ, η, ζ) = 2〈ξ, [s#(η), s#(ζ)]〉, ξ, η, ζ ∈ g∗, (2.1)
where s] : g∗ → g is given by 〈s](ξ), η〉 = s(ξ, η), ξ, η ∈ g∗. If r =
∑
i xi ⊗ yi ∈ g ⊗ g is
a quasitriangular r-matrix, it defines a Lie bialgebra structure
δr : g→ g ∧ g, δr(x) =
∑
i
[x, xi]⊗ yi + xi ⊗ [x, yi], x ∈ g,
and one calls the pair (g, r) a quasitriangular Lie bialgebra.
Let (g, δg) be a Lie bialgebra and (Y, πY ) a Poisson manifold. A (right) Poisson action of (g, δg)
on (Y, πY ) is a Lie algebra morphism ρ : g→ X1(Y ) satisfying
[ρ(x), πY ] = ρ(δg(x)), x ∈ g,
and one also says that (Y, πY , ρ) is a right (g, δg)-Poisson space.
Let g be a finite-dimensional Lie algebra, Y a manifold and ρ : g→ X1(Y ) a right action of g
on Y . For r =
∑
i xi ⊗ yi ∈ g⊗ g, one has the 2-tensor field
ρ(r) =
∑
i
ρ(xi)⊗ ρ(yi) ∈ X1(Y )⊗ X1(Y ),
and writing r = s+ Λ, with s ∈ S2g and Λ ∈ ∧2g, it is clear that ρ(r) is a bivector field on Y if
and only if ρ(s) = 0.
Proposition 2.1 ([8, Proposition 2.18]). If r is a quasitriangular r-matrix and if ρ(r) is a bivec-
tor 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 xi ⊗ yi ∈ g ⊗ g, define
Mixn(r) ∈ ∧2(gn) by
Mixn(r) =
∑
1≤j<k≤n
(
Mixn(r)
)
j,k
, (2.2)
where(
Mixn(r)
)
j,k
=
∑
i
(yi)j ∧ (xi)k, 1 ≤ j < k ≤ n,
and for any sign function ε : {1, . . . , n} → {1,−1}, let
rε,n = (ε(1)s, . . . , ε(n)s) + (Λ, . . . ,Λ) ∈ gn ⊗ gn,
where r = s+ Λ with s ∈ S2g and Λ ∈ ∧2g, and let
r(ε,n) = rε,n −Mixn(r) ∈ gn ⊗ gn. (2.3)
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 gn, and the Lie bialgebra
structure
δ(n)
r = δr(ε,n) (2.4)
is independent of ε. Moreover, the map
diagn : (g, δr)→
(
gn, δ(n)
r
)
, diagn(x) = (x, . . . , x), x ∈ g,
is an embedding of Lie bialgebras.
4 V. Mouquin
For any r ∈ g⊗ g and any sign function ε, denote by Λ
(n)
r the anti-symmetric part of r(ε,n).
Writing r =
∑
i xi ⊗ yi = s + Λ, with 2s =
∑
i(xi ⊗ yi + yi ⊗ xi) and 2Λ =
∑
i xi ∧ yi, one has
explicitly
Λ(n)
r = (Λ, . . . ,Λ)−Mixn(r) =
1
2
n∑
j=1
∑
i
(xi)j ∧ (yi)j −
∑
1≤j<k≤n
∑
i
(yi)j ∧ (xi)k.
The following lemma will be used in the proof of Proposition 5.2.
Lemma 2.3. Let r = s+ Λ ∈ g⊗ g, with s ∈ S2g and Λ ∈ ∧2g. Then
Λ(n)
r − diagn(Λ) = −Mixn(s).
Proof. A straightforward calculation shows that diagn(Λ) = (Λ, . . . ,Λ)−Mixn(Λ). Thus
Λ(n)
r − diagn(Λ) = −Mixn(r) + Mixn(Λ) = −Mixn(s). �
The following lemma will be used in the proof of Theorem 4.10.
Lemma 2.4. Let r ∈ g⊗ g. For integers m,n ≥ 0, one has(
Λ(m)
r ,Λ(n)
r
)
− (diagm,diagn)
(
Mix2(r)
)
= Λ(m+n)
r ∈ ∧2
(
gm+n
)
.
Proof. Indeed, writing r =
∑
i xi⊗ yi and letting Λ ∈ ∧2g be the anti-symmetric part of r, one
has
(diagm,diagn)
(
Mix2(r)
)
=
∑
1≤k≤m, 1≤l≤n
(yi)k ∧ (xi)l,
hence(
Λ(m)
r ,Λ(n)
r
)
− (diagm,diagn)
(
Mix2(r)
)
= (Λ, . . . ,Λ)−
∑
1≤k<l≤m+n
(yi)k ∧ (xi)l = Λ(m+n)
r . �
2.2 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, πY )
be a Poisson manifold with a right Poisson action ρ : gn → X1(Y ) of (g, r)n, and a right Poisson
action ψ : h→ X1(Y ) of a Lie bialgebra (h, δh), so that (Y, πY , ρ× ψ) is a right (g, r)n × (h, δh)-
Poisson space. By [8, Lemma 2.16] and Theorem 2.2, the triple
Fus(g,r)n(Y, πY , ρ× ψ) :=
(
Y, πY − ρ
(
Mixn(r)
)
, (ρ ◦ diagn)× ψ
)
(2.5)
is a right (g, r)× (h, δh)-Poisson space, which we call the fusion at (g, r)n of (Y, πY , ρ× ψ). As
a particular case, suppose that h = 0, that
(Y1, πY 1 , ρ1), . . . , (Yn, πY n , ρn)
are right (g, r)-Poisson spaces, that Y = Y1×· · ·×Yn is equipped with the direct product Poisson
structure πY = πY 1 × · · · × πY n , and that ρ : gn → X1(Y ) is given by
ρ(x1, . . . , xn) = (ρ1(x1), . . . , ρn(xn)), xj ∈ g.
The (g, r)-Poisson space in (2.5) is called in [8] the fusion product of (Yj , πY j , ρj), 1 ≤ j ≤ n.
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 5
3 Ciliated graphs and marked surfaces
In this section, we review the fusion of marked surfaces and ciliated graphs. Our main references
are [5, 7].
3.1 Ciliated graphs and marked surfaces
A marked surface (Σ, V ) is a compact Riemann surface, together with a non-empty finite col-
lection 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.
Proposition 3.1 ([7, Section 4]). Any marked surface (Σ, V ) admits a skeleton, and skeletons
of (Σ, V ) are unique up to isomorphisms and local changes
←→ (3.1)
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.
Figure 1. An annulus with four marked points and two non-isomorphic skeletons with their cilium
structure.
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 Γ1. Denote by Γ1/2 the set of half edges of Γ, and note
that Γ1/2 comes with a natural involution with no fixed points α 7→ α̌, mapping a half edge to
the opposite half edge, and for α ∈ Γ1/2 we write [α, α̌] for the edge composed of the two half
edges α and α̌. For every v ∈ V , let Γv ⊂ Γ1/2 be the set of half edges incident to v, so that
Γ1/2 =
⊔
v∈V Γv and Γv is a linearly ordered set for each v ∈ V .
3.2 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 (v1, v2) of distinct elements of V ,
the fusion of Σ at (v1, v2) is the marked surface (Σ(v1,v2), Vv1=v2) obtained by gluing a piece of
arc ending in v1 with a piece or arc starting at v2, so that v1 and v2 are identified. The set of
marked points Vv1=v2 on Σ(v1,v2) is then obtained from V by identifying v1 and v2.
6 V. Mouquin
v2 v1 fusion−→
v1 = v2
Let Γ be a ciliated graph with set of vertices V and edges Γ1, and let (v1, v2) be a pair of
distinct vertices, with Γv1 = {α1 < · · · < αk} and Γv2 = {αk+1 < · · · < αl}. The fusion of Γ at
(v1, v2) is the ciliated graph Γ(v1,v2) obtained by identifying v1 and v2, and with linear order on
the set Γv1=v2 of half edges incident to v1 = v2 given by α1 < · · · < αk < αk+1 < · · · < αl.
Note that the fusion of marked surfaces and ciliated graphs are associative operations, but
not necessarily commutative. The following lemma is straightforward.
Lemma 3.3. Let (Σ, V ) be a marked surface with skeleton Γ, and let v1, v2 ∈ V be distinct
points. Then the image of Γ under the fusion map (Σ, V ) → (Σ(v1,v2), Vv1=v2) is isomorphic
to Γ(v1,v2), and is a skeleton for (Σ(v1,v2), Vv1=v2).
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 (ΣΓ, V ), well defined up to isomorphism, obtained
by fusing marked disks corresponding to the edges of Γ. Thus the map Γ 7→ (ΣΓ, V ) gives
a bijective correspondence between isomorphism classes of ciliated graphs up to local changes
in (3.1) and isomorphism classes of marked surfaces.
fusion−→
Figure 2. An annulus with three marked points obtained by fusing three disks with two marked points
each.
4 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.
4.1 The moduli space of flat connections over a marked surface
For a marked surface (Σ, V ), let Π1(Σ, V ) be the fundamental groupoid of Σ over the set of base
points V , and consider
A(Σ, V ) = Hom(Π1(Σ, V ), G),
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 7
the moduli space of flat connections on G-principal bundles over Σ which are trivialized over V .
The group GV naturally acts on the right on A(Σ, V ) by gauge transformations. For p ∈
Π1(Σ, V ), denote by evp : A(Σ, V ) → G the evaluation at p, by θ(p), τ(p) ∈ V the respective
source and target of p, and if g ∈ GV and v ∈ V , recall from Section 1.1 that gv is the v’th
component of g. The action of GV on A(Σ, V ) is then given by
ρV : A(Σ, V )×GV → A(Σ, V ), evp(ρV (y, g)) = g−1
θ(p) evp(y)gτ(p), (4.1)
where g ∈ GV , y ∈ A(Σ, V ), and p ∈ Π1(Σ, V ). Given a skeleton Γ of (Σ, V ) and an orientation
of each edge of Γ, Π1(Σ, V ) is then the free groupoid generated by Γ, and thus one has a natural
diffeomorphism
IΓ : GΓ1
∼=−→ A(Σ, V ). (4.2)
Now, choose a pair of distinct marked points (v1, v2). The fusion map (Σ, V )→ (Σ(v1,v2), Vv1=v2)
induces a map
ϕ(v1,v2) : A
(
Σ(v1,v2), Vv1=v2
)
−→ A(Σ, V ), (4.3)
and by choosing a skeleton for (Σ, V ), one easily sees that ϕ(v1,v2) is a diffeomorphism. The
next lemma is straightforward.
Lemma 4.1. Let diag(v1,v2) : GVv1=v2 → GV be the canonical embedding. Then for any g ∈
GVv1=v2 , one has
ϕ(v1,v2)
(
ρV v1=v2
(y, g)
)
= ρV
(
ϕv1,v2(y), diag(v1,v2)(g)
)
, y ∈ A(Σ(v1,v2), Vv1=v2).
Let Γ be a skeleton of (Σ, V ), and for any γ ∈ Γ1, let (Dγ , Vγ) be a disk marked with two
points, where Vγ ⊂ Γ1/2 consists of the two half edges of γ, so that tγ∈Γ1Vγ = Γ1/2. As (Σ, V )
is a fusion of (tγ∈Γ1Dγ ,Γ1/2), one has a diffeomorphism
ϕΓ : A(Σ, V )
∼=−→ A
(
tγ∈Γ1 Dγ ,Γ1/2
)
,
and thus an action of GΓ1/2 on A(Σ, V ) via ϕΓ. Choosing an orientation for each edge of Γ and
identifying A(Σ, V ) ∼= GΓ1 as in (4.2), the action of GΓ1/2 on GΓ1 is given by
σΓ : GΓ1 ×GΓ1/2 → GΓ1 ,
σΓ(g, h)γ = h−1
αγ gγhα̌γ , h ∈ GΓ1/2 , g ∈ GΓ1 , γ ∈ Γ1, (4.4)
where for γ ∈ Γ1, αγ ∈ Γ1/2 is the source half edge of γ.
4.2 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 ∈
(
S2g
)g
.
For every v ∈ V , let Λv ∈ ∧2g be such that rv = s + Λv is a quasitriangular r-matrix. Identi-
fying gΓv with g|Γv | using the linear order on Γv, let r
(εv ,Γv)
v ∈ gΓv ⊗ gΓv be as in (2.3), where
εv : Γv → {−1, 1} is defined as
εv(α) =
{
1, α is a source half edge,
−1, α is an end half edge,
α ∈ Γv.
8 V. Mouquin
Since Γ1/2 =
⊔
v∈V Γv, one has gΓ1/2 =
⊕
v∈V gΓv , and recalling our notation in Section 1.1, let
rΓ =
∑
v∈V
(
r(εv ,Γv)
v
)
v
∈ gΓ1/2 ⊗ gΓ1/2 , (4.5)
a quasitriangular structure for the Lie bialgebra⊕
v∈V
(
gΓv , δ(Γv)
rv
)
,
where δ
(Γv)
rv is as in (2.4). Using the notational convention in Section 1.1, recall from (4.4) the
right Lie algebra action σΓ : gΓ1/2 → X1(GΓ1).
Theorem 4.2. The bivector field
πΓ = σΓ(rΓ) ∈ X2
(
GΓ1
)
is a Poisson structure on GΓ1.
Proof. The symmetric part of rΓ is
sΓ =
∑
γ∈Γ1
(s)αγ − (s)α̌γ ,
where for γ ∈ Γ1, αγ ∈ Γ1/2 is the source half edge of γ. By (4.4), one has σΓ(sΓ) = 0, thus
Theorem 4.2 follows from Proposition 2.1. �
Remark 4.3. Let ΛΓ be the anti-symmetric part of the quasitriangular r-matrix rΓ. The
bivector field
πΓ = σΓ(ΛΓ)
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 πΓ is a Poisson structure.
Consider the quasitriangular Lie bialgebra(
gV , r
)
=
⊕
v∈V
(g, rv), where r =
∑
v∈V
(rv)v ∈ gV ⊗ gV . (4.6)
For v ∈ V , let diagv : g→ gΓv , diagv(x) =
∑
γ∈Γv
(x)γ , for x ∈ g, and define the map
diagΓ : gV → gΓ1/2 =
⊕
v∈V
gΓv , diagΓ(x) =
∑
v∈V
(diagv(xv))v, x ∈ gV . (4.7)
By Theorem 2.2, diagΓ : (gV , δr) → (gΓ1/2 , δrΓ) is an embedding of Lie bialgebras, and by
Lemma 4.1, one has
ρV = IΓ ◦ σΓ ◦ diagΓ, (4.8)
where ρV : gV → X1(A(Σ, V )) is the derivative of the action by gauge transformations in (4.1).
Thus as an immediate consequence, one has
Corollary 4.4. The triple (A(Σ, V ), IΓ(πΓ), ρV ) is a right (gV , r)-Poisson space.
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 9
v1 v2
Figure 3. The marked surface (Σ3, V ).
Example 4.5. For any integer n ≥ 1, let (Σn, V ) be a disk with n − 1 inner disks removed
and two marked points v1, v2, such as in Fig. 3, and let Γ be the skeleton in Fig. 3 with edges
oriented from v1 to v2. Let ri = s+Λi, be the r-matrix associated to vi, i = 1, 2. The orientation
of Σ induces an isomorphism gΓvi ∼= gn, and one has
r
(εv1 ,Γv1 )
1 = (s, . . . , s) + (Λ1, . . . ,Λ1)−Mixn(r1) = (s, . . . , s) + Λ(n)
r1
r
(εv2 ,Γv2 )
2 = (−s, . . . ,−s) + (Λ2, . . . ,Λ2)−Mixn(r2) = (−s, . . . ,−s) + Λ(n)
r2 ,
and
IΓ(πΓ) =
(
Λ(n)
r1
)R
+
(
Λ(n)
r2
)L
.
In particular, if Λ2 = −Λ1, the Poisson structure IΓ(πΓ) =
(
Λ
(n)
r1
)R − (Λ(n)
r1
)L
is multiplicative,
and the Poisson Lie group(
Gn,
(
Λ(n)
r1
)R − (Λ(n)
r1
)L)
is then called a polyuble in [4], and has Lie bialgebra
(
gn, δ
(n)
r1
)
. When n = 2 and r1 is factorizable,
that is when s] : g∗ → g is invertible,
(
G2, (Λ
(2)
r1 )R − (Λ
(2)
r1 )L
)
is isomorphic to the double of the
Poisson Lie group
(
G,ΛR1 − ΛL1
)
[8]. Thus polyubles are generalizations of doubles of Poisson
Lie groups.
4.3 Independence of choice of skeleton
Continuing with the setup and notation of Section 4.2, one has a Poisson structure IΓ(πΓ)
on A(Σ, V ). The goal of this subsection is to show that IΓ(πΓ) 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
I−1
Γ′ ◦ IΓ :
(
GΓ1 , πΓ
)
−→
(
GΓ′
1 , πΓ′
)
(4.9)
is a Poisson isomorphism.
Lemma 4.6. The Poisson structure IΓ(πΓ) 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 ∈ Γ1,
which is given the opposite orientation. The map I−1
Γ′ ◦ IΓ : GΓ1 → GΓ′
1 = GΓ1 is thus the group
inversion in the factor γ, and the identity on all other factors, hence for any x ∈ gΓ1/2 , one has
I−1
Γ′ ◦ IΓ(σΓ(x)) = σΓ′(x),
which implies that I−1
Γ′ ◦ IΓ(πΓ) = πΓ′ . �
10 V. Mouquin
Lemma 4.7. Consider the following two oriented skeletons
v1v2
v3
γ1
γ2
Γ
v1v2
v3
γ′1γ2
Γ′
of a disk with three marked points. Then the map (4.9) is a Poisson isomorphism.
Proof. Identifying GΓ1 and GΓ′
1 with G2 and writing I = I−1
Γ′ ◦IΓ, one has I(g1, g2) = (g1g2, g2),
g1, g2 ∈ G, and
πΓ =
(
ΛRv1
,ΛLv3
)
+
(
ΛLv2
,ΛRv2
)
+
∑
i
(
0, yRi
)
∧
(
xLi , 0
)
,
where rv2 =
∑
i xi ⊗ yi. A direct calculation using Adg(s) = s for any g ∈ G, shows that(
ΛRv1
, 0
)
= I
(
ΛRv1
, 0
)
,(
ΛLv3
,ΛLv3
)
−Mix2(Λv3)L = I
(
0,ΛLv3
)
,(
0,ΛRv2
)
−Mix2(s)L = I
((
ΛLv2
,ΛRv2
)
+
∑
i
(
0, yRi
)
∧
(
xLi , 0
))
,
from which one gets I(πΓ) =
(
ΛRv1
,ΛRv2
)
+
(
ΛLv3
,ΛLv3
)
−Mix2(rv3)L = πΓ′ . �
We return to the case of a general marked surface (Σ, V ).
Proposition 4.8. Let Γ,Γ′ be oriented skeletons of (Σ, V ). Then the map (4.9) is a Poisson
isomorphism.
Proof. By Lemma 4.6 and Proposition 3.1, one can assume that Γ and Γ′ have the following
form:
v1v2
v3
γ1
γ2
Γ
v1v2
v3
γ′1γ2
Γ′
Using Lemma 4.7, a straightforward calculation, the details of which are left to the readers,
shows that I−1
Γ′ ◦ IΓ(πΓ) = πΓ′ . �
Recall the quasitriangular r-matrix r on gV defined in (4.6) and let
πr = IΓ(πΓ) ∈ X2(A(Σ, V )),
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.
Theorem 4.9. The Poisson structure πr on A(Σ, V ) is independent of the choice of Γ, and
(A(Σ, V ), πr, ρV ) is a right (gV , r)-Poisson space.
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 11
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 Λv ∈ ∧2g such that rv = s + Λv is a quasitriangular r-matrix, and one considers the
quasitriangular r-matrix r ∈ gV ⊗ gV defined in (4.6).
Suppose one has rv1 = rv2 for two distinct elements v1, v2 ∈ V , and consider the fused surface
(Σ′, V ′) = (Σ(v1,v2), Vv1=v2) with Poisson structure πr′ , where r′ ∈ gV
′⊗gV
′
is defined as in (4.6),
and let v′ ∈ V ′ be the vertex obtained by fusing v1 and v2. Recall the fusion of Poisson spaces
discussed in Section 2.2.
Theorem 4.10. One has
Fus(g,rv1 )×(g,rv2 )(A(Σ, V ), πr, ρV ) =
(
A(Σ′, V ′), πr′ , ρV ′
)
.
Proof. We identify A(Σ, V ) and A(Σ′, V ′) using the natural diffeomorphism ϕ(v1,v2) in (4.3),
and let (A(Σ′, V ′), π′, ρ′) be the fusion at (g, rv1) × (g, rv2) of (A(Σ, V ), πr, ρV ). Let Γ be an
oriented skeleton of (Σ, V ) and recall from Lemma 3.3 that Γ′ = Γ(v1,v2) is an oriented skeleton
of (Σ′, V ′), and that one has a natural identification Γ1
∼= Γ′1. Recall the map diagΓ : gV → gΓ1/2
defined in (4.7) and consider its restriction diagΓ |g{v1,v2} to g{v1,v2}. Using (4.8) and Lemma 2.4,
one has
π′ = πr − ρV |g{v1,v2}
(
Mix2(rv1)
)
= IΓ ◦ σΓ
(
rΓ − diagΓ |g{v1,v2}
(
Mix2(rv1)
))
= IΓ′ ◦ σΓ′(rΓ′) = πr′ ,
and by Lemma 4.1, one has ρ′ = ρV ′ . �
Example 4.11. Let (Σ, V ) be a disk with two marked points v1, v2 and assume that the r-
matrices r1 and r2 associated to v1 and v2 are equal. Let the edge of Γ be oriented from v1 to v2
and identify A(Σ, V ) ∼= G via IΓ, so that one has
πr = ΛR + ΛL,
where Λ is the anti-symmetric part of r1 = r2.
v2
v1
fusion−→ v′
The fused surface (Σ′, {v′}) is an annulus with one marked point, and one has
πr′ = ΛR + ΛL − ρV
(
Mix2(r1)
)
= ΛR + ΛL +
∑
i
yRi ∧ xLi
=
∑
i
1
2
xRi ∧ yRi +
1
2
xLi ∧ yLi + yRi ∧ xLi ,
where r1 = r2 =
∑
i xi ⊗ yi.
12 V. Mouquin
5 Quasi Poisson geometry
5.1 Quasi Poisson spaces and fusion of quasi Poisson spaces
Let g be a Lie algebra, s ∈ (S2g)g, and recall the element φs ∈ (∧3g)g defined in (2.1). Recall
from [2] that a right (g, φs)-quasi Poisson space is a triple (Y,QY , ρ), where Y is a manifold,
ρ : g → X1(Y ) a right Lie algebra action, and QY ∈ X2(Y ) is a g-invariant bivector field on Y ,
such that
[QY , QY ] = ρ(φs).
We denote by QP(g, φs) the category of right (g, φs)-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.
Proposition 5.1 ([1, 6, 8]). Let Λ ∈ ∧2g such that r = s + Λ is a quasitriangular r-matrix.
Then one has an equivalence of categories
P(g, r)←→ QP(g, φs), (Y, πY , ρ) 7→ (Y, πY − ρ(Λ), ρ),
where the functor on the level of morphisms is the identity map.
Proof. Let (Y, πY , ρ) be a right (g, r)-Poisson space. Then
[πY − ρ(Λ), πY − ρ(Λ)] = −2[ρ(Λ), πY ] + ρ([Λ,Λ]) = −ρ([Λ,Λ]) = ρ(φs),
and for x ∈ g,
[ρ(x), πY − ρ(Λ)] = ρ(δr(x)− [x,Λ]) = 0,
hence (Y, πY − ρ(Λ), ρ) is a right (g, φs)-quasi Poisson space. One similarly checks that if
(Y,QY , ρ) is a (g, φs)-quasi Poisson space, (Y,QY + ρ(Λ), ρ) is a right (g, r)-Poisson space. �
5.2 A canonical quasi Poisson structure on A(Σ, V )
Let G be a connected complex Lie group, g its Lie algebra, and let s ∈ (S2g)g. Let (Σ, V )
be a marked surface and for v ∈ V , let Λv ∈ ∧2g be such that rv = s + Λv is quasitriangular
r-matrix, and let r ∈ gV ⊗ gV be as in (4.6). By Proposition 5.1,(
A(Σ, V ), Qs := πr − ρV
(∑
v∈V
(Λv)v
)
, ρV
)
is a right (g, φs)
V -quasi Poisson space.
Proposition 5.2 (see also [7]). For any oriented skeleton Γ of (Σ, V ), one has
I−1
Γ (Qs) = −σΓ
(∑
v∈V
(
MixΓv(s)
)
v
)
, (5.1)
where for v ∈ V , MixΓv(s) ∈ ∧2(gΓv) ∼= ∧2(g|Γv |) is the element defined in (2.2) using the linear
order of Γv. In particular, Qs depends only on s ∈ (S2g)g.
Proof. Let ΛΓ ∈ ∧2gΓ1/2 be the anti-symmetric part of the quasitriangular r-matrix rΓ in (4.5)
and recall the map diagΓ : gV → gΓ1/2 in (4.7). By Lemma 2.3, one has
I−1
Γ (Qs) = σΓ
(
ΛΓ − diagΓ
(∑
v∈V
(Λv)v
))
= −σΓ
(∑
v∈V
(MixΓv(s))v
)
. �
The Fock–Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix 13
Remark 5.3. Formula (5.1) appears in [7, Section 4], where Qs was defined via fusion of
quasi-Poisson spaces.
Note that both πr and Qs descend to the same Poisson structure on the moduli space
A(Σ, V )/GV of flat G-connections over Σ.
Example 5.4. Let (Σ′, {v′}) be the annulus with one marked point in Example 4.11, and let
the edge of its skeleton Γ′ be oriented in the anti-clockwise direction. Identify A(Σ′, {v′}) ∼= G
via IΓ′ and g
Γ′
1/2 ∼= g2. Writing r =
∑
i xi ⊗ yi, one has 2s =
∑
i(xi ⊗ yi + yi ⊗ xi), thus
Qs = −σΓ′
(
Mix2(s)
)
=
1
2
∑
i
xRi ∧ yLi + yRi ∧ yLi .
Let ρ be the (right) action of G on itself by conjugation. The right (g, φs)-quasi Poisson space
(G,Qs, ρ) is an example of the Hamiltonian quasi-Poisson spaces which were studied in [2].
Acknowledgements
The author wishes to thank Jiang-Hua Lu, Marco Gualtieri and Francis Bischoff for their help-
ful comments. The author also wishes to thank the anonymous referees and editors, whose
suggestions helped improve this paper.
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1 Introduction
1.1 Notation
2 Poisson structures defined by quasitriangular r-matrices
2.1 Quasitriangular r-matrices
2.2 Fusion products of Poisson spaces
3 Ciliated graphs and marked surfaces
3.1 Ciliated graphs and marked surfaces
3.2 Fusion of ciliated graphs and marked surfaces
4 The Fock–Rosly Poisson structure
4.1 The moduli space of flat connections over a marked surface
4.2 The Fock–Rosly Poisson structure
4.3 Independence of choice of skeleton
4.4 Fusion of Poisson spaces and marked surfaces
5 Quasi Poisson geometry
5.1 Quasi Poisson spaces and fusion of quasi Poisson spaces
5.2 A canonical quasi Poisson structure on A(, V)
References
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