Poisson–Lie Groups and Gauge Theory

: We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r -matrices in the description of moduli spaces of ﬂat connections and the Chern–Simons gauge theory.


Poisson-Lie Groups
This review article originated from a lecture series on Poisson-Lie groups given at the 12th International ICMAT Summer School on Geometry, Mechanics and Control in Santiago de Compostela. It is meant to provide an introduction that is accessible to physicists and mathematicians with no background on this topic, while at the same time covering current results.
Due to the large body of work on Poisson-Lie groups and their numerous applications, it is impossible to do justice to all the work on this topic. For this reason, we focus on certain aspects and omit many others. For each result, we either cite the work where it was first discovered or, where appropriate and available, refer the reader to a textbook or review article and the references therein.
Short and accessible introductions to the topic with emphasis on different aspects are given in lecture notes by A. G. Reyman [1], by Y. Kosmann-Schwarzbach [2], by M. Audin [3], which also contain detailed lists of references. A good introduction to Poisson geometry and Poisson-Lie groups is the textbook [4]. Other textbooks that cover Poisson-Lie groups are [5,6]. For a textbook presenting Poisson-Lie groups from the point of view of integrable systems, see [7]. For an accessible introduction to dynamical r-matrices, see the textbook [8].

Motivation
A Poisson-Lie group is a Lie group that is also a Poisson manifold in such a way that its multiplication is a Poisson map. Poisson-Lie groups arise in the description of integrable systems and in the context of two-and three-dimensional gauge theories. They can be viewed as the classical counterparts of quantum group symmetries in certain quantum systems, such as quantum integrable systems and quantised Chern-Simons or BF-Theories. Both, quantum groups and Poisson-Lie groups, admit an infinitesimal description in terms of Lie algebras with additional structure that is the infinitesimal counterpart of the Poisson structure. Such a Lie algebra is called a Lie bialgebra.
One motivation for Poisson-Lie groups arises from Lie group actions on Poisson manifolds. Phase spaces of physical systems are usually Poisson manifolds, and Lie group actions on these manifolds arise from physical or gauge symmetries of these systems. This includes integrable systems and constrained Hamiltonian systems.

5.
For every smooth manifold M, the cotangent bundle T * M has a canonical symplectic structure. If we interpret functions on M and vector fields on M as functions on T * M, their Poisson brackets are given by where f 1 , f 2 ∈ C ∞ (M), X 1 , X 2 ∈ Vec(M) and [ , ] denotes the Lie bracket of vector fields We now consider Lie group actions on Poisson manifolds. In the following, we assume that all Lie groups are finite-dimensional. Definition 2.

1.
A Lie group G is a smooth manifold G with a group structure such that the multiplication map µ : G × G → G, (g, h) → gh and the inversion i : G → G, g → g −1 are smooth.

2.
A Lie group action of G on a smooth manifold M is a smooth map • The set G m = {g m | g ∈ G} for m ∈ M is called the orbit of m ∈ M.

•
The set G\M = {G m | m ∈ M} of orbits is called the orbit space.

•
The set of invariant functions on M is denoted Example 2.

1.
If G and H are Lie groups, then G × H is a Lie group with (g, h) · (g , h ) = (gg , hh ). It is called the direct product of G and H.

2.
R n and C n with the usual addition are Lie groups.

3.
Any closed subgroup of the group GL(n, C) of invertible n × n-matrices is a Lie group. A Lie group of this form is called a matrix Lie group. This includes the Any Lie group acts on itself by left multiplication : G × G → G, g h = gh, by right multiplication with inverses : G × G → G, g h = hg −1 , and by conjugation : G × G → G, g h = ghg −1 .

5.
Any matrix Lie group G ⊂ GL(n, C) acts on C n by : G × C n → C n , M v = M · v and any matrix Lie group G ⊂ GL(n, R) acts on R n by : G × R n → R n , M v = M · v. 6.
The orbits of the group action : SO(n) × R n → R n , M v = M · v are the origin and the (n − 1)-dimensional spheres of radius r > 0 The orbits of the group action : O(1, n − 1) × R n → R n , M v = M · v are the origin, the timelike or two-sheeted hyperboloids the spacelike or one-sheeted hyperboloids and the light cone

8.
If G and H are Lie groups and : G × H → H is a smooth group action such that g − : H → H is a homomorphism of Lie groups for all g ∈ G, then G × H is a Lie group with the group multiplication (g, h) · (g , h ) = (gg , h · (g h )).
It is called the semidirect product of G and H and denoted G H. 9.
Examples are the n-dimensional Euclidean group E n = SO(n) R n and the n-dimensional Poincaré group P n = SO(1, n − 1) R n .
Leaving aside questions of smoothness, one sees that invariant functions on M correspond to functions on orbit space. Every invariant function on M is constant on each orbit and defines a function on the orbit space. Similarly, every function on the orbit space can be lifted to an invariant function on M. This raises the question:

Given a Poisson manifold M with a Lie group action
: G × M → M, what is a practical condition on the group action that ensures that the Poisson bracket of two invariant functions is again invariant?
It seems plausible to impose that the maps g − : M → M, m → g m are Poisson maps for all g ∈ G. Indeed, this condition implies { f 1 , f 2 } ∈ C ∞ (M) G for all f 1 , f 2 ∈ C ∞ (M) G . However, it turns out that this is too restrictive and excludes many interesting examples.
To relax this requirement, we impose that the Lie group G is equipped with a Poisson structure and that the action map : G × M → M is a Poisson map with respect to the Poisson structure on M and the product Poisson structure on G × M from Example 1, 2.
Due to the Leibniz identity and antisymmetry, any Poisson structure on G is of the form { f 1 , f 2 }(x, y) = C(x, y)(∂ x f 1 ∂ y f 2 − ∂ y f 1 ∂ x f 2 )(x, y) with C ∈ C ∞ (R 2 ). (

7)
A short computation shows that the Jacobi identity is satisfied for all brackets of the form (7). The compatibility (6) between Poisson structure and group multiplication translates into the requirement C(x + x , y + e x y ) = e x C(x , y ) + C(x, y), which is satisfied if and only if C(x, y) = a(e x − 1) + by for some constants a, b ∈ R. Hence, all Poisson-Lie structures on G are of the form As the determinant function det = a 11 a 22 − a 12 a 21 satisfies {det, a ij } k = 0 for all i, j ∈ {1, 2} and k ∈ {1, 2, 3}, they induce Poisson-Lie structures on the subgroup SL(2, R).
We now consider smooth group actions of Poisson-Lie groups G on Poisson manifolds M, for which the action maps : G × M → M are Poisson. Definition 4. Let G be a Poisson-Lie group.

1.
A Poisson-G space is a Poisson manifold M with a smooth group action : G × M → M that is a Poisson map with respect to the Poisson structure on M and the product Poisson structure on G × M:

2.
A homomorphism of Poisson-G spaces from M to N is a Poisson map φ : M → N that intertwines the G-actions on M and M: Example 4.

1.
Let (G, { , } G ) be a Poisson-Lie group and G = (G, −{ , } G ). Then G is a Poisson space over itself and over G with the group actions As these group actions commute, this gives G the structure of a Poisson G × G-space.

2.
A Poisson-Lie subgroup of a Poisson-Lie group G is a closed subgroup H ⊂ G that is a Poisson-Lie group and for which the inclusion ι : If H ⊂ G is a Poisson-Lie subgroup, then there is a unique Poisson structure on the homogeneous space G/H for which the projection π : G → G/H, g → gH is a Poisson map. The homogeneous space G/H with this Poisson structure and the canonical G-action where we used first that π is Poisson, then that G is a Poisson-Lie group to pass to the second line and then the identity π(−g ) = − g H and that π is Poisson to pass to the third line.

Poisson-Lie Groups and Lie Bialgebras
In general, it is not easy to construct or classify Poisson-Lie groups. To systematically investigate Poisson-Lie groups, one uses the same strategy as for Lie groups, namely to consider the associated infinitesimal structures. For every Lie group the tangent space at the unit element is a Lie algebra and every finite-dimensional Lie algebra exponentiates to a unique connected and simply connected Lie group. To extend these statements to Poisson-Lie groups, one determines the infinitesimal structures induced by their Poisson brackets.

Lie Bialgebras
We first introduce some background on Lie algebras. In the following, all Lie algebras will be real and finite-dimensional unless stated otherwise. Definition 5.

1.
A Lie algebra g is a real vector space g together with an antisymmetric linear map [ , ] : g⊗g → g, X⊗Y → [X, Y], the Lie bracket, that satisfies the Jacobi identity

2.
A Lie algebra homomorphism from g to h is a linear map φ : 3. Every associative algebra A becomes a Lie algebra with the commutator In particular, this applies to gl(n, C) = Mat(n × n, C) and to every R-linear subspace Examples of the latter are the following Lie algebras with the commutator brackets: If g, h are Lie algebras then the vector space g ⊕ h becomes a Lie algebra with This is called the direct sum of the Lie algebras g and h. 6.
If g, h are Lie algebras and : then the vector space g ⊕ h becomes a Lie algebra with the Lie bracket It is called the semidirect product of the Lie algebras g and h and denoted g h.
To introduce the additional structure on a Lie algebra that is the counterpart of the Poisson bracket on a Poisson-Lie group, we require the adjoint action of a Lie algebra g on itself and on its n-fold tensor products ad X : g ⊗n → g ⊗n (10) For an element X = X 1 ⊗ . . . ⊗X n ∈ g ⊗n , we set ∑ cyc X := X 1 ⊗ . . . ⊗X n + X 2 ⊗ . . . ⊗X n ⊗X 1 + . . . + X n ⊗X 1 ⊗ . . . ⊗X n−1 and extend this linearly to g ⊗n . With these definitions, we can introduce the Lie algebra counterpart of a Poisson-Lie group.

2.
A homomorphism of Lie bialgebras from a Lie bialgebra g to a Lie bialgebra h is a linear map φ : g → h that satisfies Example 6.

1.
Every Lie algebra g becomes a Lie bialgebra with δ = 0 : g → g⊗g. This corresponds to the trivial Poisson structure on a Lie group G with Lie algebra g.
( [10] (Example 1)) Consider the two-dimensional Lie algebra g = R R with basis {X, Y} and Lie bracket Every antisymmetric map δ : g → g⊗g defines a cocommutator on g. Up to isomorphisms of Lie bialgebras, there are exactly two non-trivial Lie bialgebra structures on g with

Remark 2.
Condition (C1) in Definition 6 is called cocycle condition, because it states that δ : g → g⊗g is a 1-cocycle with values in g⊗g in the sense of Lie algebra cohomology. If we equip g⊗g with the g-module structure X (Y⊗Z) = ad X (Y⊗Z) = [X, Y]⊗Z + Y⊗[X, Z], then the cocycle condition takes the form In Section 3, we investigate the case, where δ is not just a 1-cocycle, but a 1-coboundary.
The concept of a Lie bialgebra is symmetric with respect to vector space duals. For every Lie algebra g consider the dual vector space g * and the pairing or evaluation map , : g * ⊗n × g ⊗n → R, The dual f * : h * ⊗k → g * ⊗n of a linear map f : g ⊗n → h ⊗k is then characterised by the condition In particular, the cocommutator δ : g → g⊗g defines an antisymmetric map δ * : g * ⊗g * → g * , and the Lie bracket [ , ] : g⊗g → g defines an antisymmetric linear map [ , ] * : g * → g * ⊗g * . Together, these maps define a Lie bialgebra structure on g * . Lemma 1. If (g, [ , ], δ) is a Lie bialgebra, then (g * , δ * , [ , ] * ) is a Lie bialgebra. It is called the dual Lie bialgebra to (g, [ , ], δ). Proof. With the notation δ := [ , ] * , [ , ] := δ * and ad α (β⊗γ) := [α, β] ⊗γ + β⊗ [α, γ] for all α, β, γ ∈ g * , we have by definition and this implies The first identity and the Jacobi identity for the Lie bracket guarantees the coJacobi identity for δ . The second identity and the coJacobi identity for δ guarantee the Jacobi identity for [ , ] . The last two identities show that the cocycle condition is self-dual.

1.
If g is a Lie bialgebra with a trivial cocommutator, then g * is abelian and vice versa.

2.
We consider the real Lie algebra g = R R with basis {X, Y} and Lie bracket from Example 6, 3. with the two Lie bialgebra structures Denoting by {x, y} the basis dual to {X, Y}, we find that the dual Lie bialgebra structures have the cocommutator and the Lie brackets By exchanging the basis elements, we can transform the Lie bracket [ , ] 2 into [ , ] 1 , and we find that the Lie bialgebras (g, [ , ], δ 1 ) and (g, [ , ], δ 2 ) are self-dual.

3.
We consider the Lie algebra sl(2, R) with the basis and the three Lie bialgebra structures on sl(2, R) from Example 6, given in (11). Denoting by {j 0 , j 1 , j 2 } the dual basis with j i , J k = δ i k , we find that the cocommutator of sl(2, R) * is given by and the Lie brackets by

Tangent Lie Bialgebras of a Poisson-Lie Group
We now show that Lie bialgebras are the infinitesimal structures associated with Poisson-Lie groups. This requires some preliminaries about the relation between Lie groups and Lie algebras. Every Lie group defines a Lie algebra, which is given as its tangent space at the unit element. Its Lie bracket can be characterised in terms of the exponential map exp : T 1 G → G. If G is a matrix Lie group, this is just the usual matrix exponential. The relation between Lie groups and Lie algebras is then summarised by the following theorem. Theorem 1. Let G be a Lie group. Then: 1.
The tangent space g = T 1 G is a Lie algebra with the Lie bracket where exp : g → G, X → e X is the exponential map.

2.
For every smooth group homomorphism φ : G → H, the tangent map T 1 φ : g → h is a homomorphism of Lie algebras.

3.
For every finite-dimensional real Lie algebra g, there is a unique connected and simply connected Lie group G with T 1 G = g.

4.
If G and H are connected and simply connected with Lie algebras T 1 G = g and T 1 H = h, then, for every Lie algebra homomorphism φ : g → h, there is a unique smooth group homomorphism φ : There is a canonical group action of G on g, the adjoint action For all g ∈ G, the maps Ad(g) = g Ad − : g → g are Lie algebra homomorphisms. Differentiating the adjoint action yields the Lie algebra homomorphisms Ad(e tX )Y.

1.
The Lie algebras of the matrix Lie groups GL(n, C) and GL(n, R) are gl(n, C) = Mat(n × n, C) and gl(n, R) = Mat(n × n, R) with the matrix commutator as the Lie bracket.

2.
For every matrix Lie group G ⊂ GL(n, C), the associated Lie algebra is an R-linear subspace V ⊂ Mat(n × n, C) that is closed under the matrix commutator. The Lie algebras for the Lie groups from Example 2, are the Lie algebras in Example 5:

3.
If G and H are Lie groups with Lie algebras g and h, then the Lie algebra of the direct product G × H is the direct sum g ⊕ h.

4.
If G, H are Lie groups and : G × H → H a smooth group action such that g − : H → H is a group homomorphism for all g ∈ G, then the Lie algebra of the semidirect product G H from Example 2 is the semidirect product g h with To describe the Poisson bracket on a Poisson-Lie group concretely, it is useful to work with distinguished vector fields, namely the vector fields associated with the group action of the Lie group on itself by left and right multiplication, the right and left invariant vector fields.

Definition 7.
Let G be a Lie group and : G × M → M a smooth group action of G on a smooth manifold M.

1.
The action vector field X ∈ Vec(M) for X ∈ g is given by

The right invariant and left invariant vector fields on G are the action vector fields for the action of G on itself by left and right multiplication
Remark 3.

1.
The vector fields X L are called right invariant and the vector fields X R left invariant, since they commute with the right and left multiplication maps R h : G → G, g → gh and L h : G → G, g → hg for h ∈ G: This implies that they are determined uniquely by their value at the unit element 1 ∈ G: For any basis {T a } of g and any g ∈ G, the sets {T L a (g)} and {T R a (g)} are bases of T g G.

2.
The left-and right invariant vector fields are related by 3. The vector fields associated with a group action : G × M → M form a Lie subalgebra of the Lie algebra of vector fields on M, since we have

4.
In particular, the right and left invariant vector fields on a Lie group G form Lie subalgebras of Vec(G) isomorphic to g.
To relate Poisson-Lie groups and Lie bialgebras, we describe the Poisson bracket of a Poisson-Lie group G in terms of a Poisson bivector B G ∈ Λ 2 (TG) that assigns to every point g ∈ G an antisymmetric element B(g) ∈ T g G⊗T g G. The Poisson bracket is then given by By Remark 3, 1. the right and left invariant vector fields associated with a basis {T a } of g = T 1 G form bases of T g G at every point g ∈ G. We can therefore describe the Poisson bivector and the Poisson-Lie structure by its coefficient functions B ab ∈ C ∞ (G) with respect to one of these bases: where we use Einstein summation conventions and sum over repeated upper and lower indices. As the Poisson bracket is antisymmetric, we have B ab = −B ba . The action of the left and right invariant vector fields on the Poisson bivector at the unit element then defines the cocommutator on the Lie algebra g = T 1 G.

1.
Let G be a Poisson-Lie group with Poisson bivector B = B ab T L a ⊗T L b . Then, its Lie algebra g is a Lie bialgebra with cocommutator given by It is called the tangent Lie bialgebra of G.

3.
For every Lie bialgebra g, there is a unique connected and simply connected Poisson-Lie group G with tangent Lie bialgebra g.

4.
Every homomorphism of Lie bialgebras φ : g → h lifts to a unique Lie group homomorphism φ : G → H with T 1 φ = φ between the associated connected and simply connected Poisson-Lie groups G, H.
Proof. We prove the Statements 1. and 2. Statements 3. and 4. then follow by integrating the structures on the Lie bialgebras to the associated connected and simply connected Lie group. 1. As the Poisson bracket of a Lie group vanishes at the unit element by Remark 1, we have B ab (1) = 0 for all coefficient functions B ab of the Poisson bivector. This implies The antisymmetry of the Poisson bracket guarantees the antisymmetry of δ. The cocycle condition follows because the multiplication is a Poisson map. To see this, we express this condition in terms of the Poisson bivector. Using the identities that follow directly from the definitions of the right and left invariant vector fields, we obtain Hence, we have shown that Together with the formulas for the left invariant vector fields and their Lie bracket, this implies The coJacobi identity follows from the Jacobi identity for the Poisson bracket on G. With formula (23) we obtain As the Poisson bivector vanishes at the unit element, this yields and by applying the Jacobi identity for the Poisson bracket, we obtain . This proves the coJacobi identity. 2. It is a standard result from the Lie theory that for any homomorphism of Lie groups φ : G → H the tangent map T 1 φ : g → h is a homomorphism of Lie algebras and satisfies If φ : G → H is a homomorphism of Poisson-Lie groups, it is also a Poisson map, and from formula (23), we obtain for all X ∈ g

Example 9.
We consider the Poisson-Lie group G = R R from Example 3, 3. with By identifying G with the matrix Lie group we obtain the following basis of the Lie algebra T 1 G = g The left invariant vector fields on g are given by This implies and shows that the cocommutator is given by This shows that the tangent Lie bialgebra of G is the Lie bialgebra (R R, aδ 2 + bδ 1 ) from Example 6.
Theorem 2 generalises the relation between Lie groups and Lie algebras. It shows that the additional structure of a Poisson-Lie group, the Poisson bracket, corresponds to the additional structure of the Lie bialgebra, the cocommutator. The ambiguity in exponentiation is the standard one between Lie groups and Lie algebras. The Lie (bi)algebras are determined uniquely by the (Poisson-)Lie groups, while uniqueness in passing from Lie (bi)algebras to (Poisson-)Lie groups requires connectedness and simply-connectedness.
Keeping this ambiguity in mind, one can straightforwardly extend concepts from Lie bialgebras to Poisson-Lie groups. In particular, this applies to the notion of a dual. The relation between Poisson-Lie groups and Lie bialgebras implies that the Poisson bracket and multiplication of a Poisson-Lie group define the multiplication and Poisson bracket of another Poisson-Lie group. If G is a Lie group with the trivial Poisson bracket, then g is a Lie bialgebra with the trivial cocommutator, and g * is abelian. In this case, the connected and simply connected dual is the abelian Lie group G * = g * with the vector addition as the group multiplication the Poisson bracket 2.
The Poisson-Lie group (R R, δ) from Example 3 and Example 9 is self-dual, since by Example 7, its Lie bialgebra is self-dual.

Coboundary and Quasitriangular Lie Bialgebras
Theorem 2 largely reduces the task of finding Poisson-Lie structures on a Lie group G to finding Lie bialgebra structures on its Lie algebra g. However, this is still difficult, since a cocommutator has to obey two nontrivial equations simultaneously, the cocycle condition and the coJacobi identity. The latter corresponds to the Jacobi identity of a Lie bracket on the dual vector space g * . The former is a compatibility condition between the Lie algebra structures on g and g * .
The name cocycle condition and Remark 2 suggest a way of replacing this condition by a simpler one. This is to consider coboundaries, linear maps δ : g → g⊗g, X → ad X (u), with a fixed element u ∈ g⊗g. As every cocycle is a coboundary, the cocycle condition is then satisfied automatically. The antisymmetry of δ follows if we require u to be antisymmetric and the coJacobi identity translates into the following condition. Proposition 1 ([9]). Let g be a Lie algebra and u ∈ g⊗g antisymmetric. Then δ : g → g⊗g, Proof. A direct computation shows that the map δ : g → g⊗g, X → ad X (u) satisfies the cocycle condition for any element u ∈ g⊗g. To show that the coJacobi identity is satisfied if and only if [[u, u]] is ad-invariant, we choose a basis {T a } of g. Then, u takes the form Using the antisymmetry of u together with the antisymmetry and the Jacobi identity of the Lie bracket, one obtains after some computations Given an element u ∈ g⊗g that defines a cocommutator on g, we can modify u by adding an ad-invariant element of t ∈ g⊗g without changing its cocommutator. If additionally t is symmetric, this does not affect the ad-invariance of the Schouten bracket.

Lemma 2.
Let g be a Lie algebra.

1.
If r = t + u ∈ g⊗g with u ∈ g⊗g antisymmetric and t ∈ g⊗g ad-invariant, one has Proof. We choose a basis {T a } of g and express u and t as linear combinations of basis elements: u = u ab T a ⊗T b and t = t ab T a ⊗T b . Then, the antisymmetry of u implies u ab = −u ba , and the Ad-invariance of t implies ad The claims then follow by a direct computation using these identities together with the antisymmetry and the Jacobi identity for the Lie-bracket.
Lemma 2 shows that an element r ∈ g⊗g defines a cocommutator on g if and only if its symmetric component is ad-invariant and its antisymmetric component has an adinvariant Schouten bracket. One possibility to satisfy the second condition is to require that the Schouten bracket [[r, r]] vanishes. Such an element is called a classical r-matrix for g. Definition 9. Let g be a Lie algebra with basis {T a }. A classical r-matrix for g is an element r = r ab T a ⊗T b ∈ g⊗g that satisfies the following conditions 1.
its symmetric component r the classical Yang-Baxter equation Corollary 2. If r ∈ g⊗g is a classical r-matrix for a Lie algebra g, then δ : g → g⊗g, X → ad X (r) is a cocommutator for g.

Proof.
As any classical r-matrix satisfies ad X (r (s) ) = 0, the map δ : g → g⊗g, X → ad X (r) depends only on the antisymmetric component of r. By Lemma 2, we have Hence r (a) satisfies the conditions in Proposition 1 and defines a cocommutator δ : g → g⊗g, X → ad X (r) on g.

1.
As the CYBE is quadratic in r and invariant under the reversal of the factors in the tensor product, for any solution r = r ab T a ⊗T b ∈ g⊗g, the elements r 21 = r ab T b ⊗T a and λr for λ ∈ R are solutions as well.

2.
Every non-degenerate ad-invariant symmetric bilinear form κ on g determines an adinvariant symmetric element t ∈ g⊗g, the Casimir element associated to κ.
It is given by t = κ ab T a ⊗T b for any basis {T a } of g, where κ ab = κ(T a , T b ) are the entries of the coefficient matrix of κ with respect to this basis and κ ab the entries of the inverse matrix with κ ab κ bc = δ a c . By Lemma 2, the element r = t + u with antisymmetric u is a classical r-matrix if and only if u satisfies the modified classical Yang-Baxter equation Lie bialgebras with coboundary cocommutators δ : g → g⊗g, X → ad X (r) are particularly easy to construct and can be classified with methods from Lie algebra cohomology. Among these Lie bialgebras, the ones whose cocommutators are given by a classical rmatrix play a special role. We will see in the following sections that their Poisson-Lie structure is particularly simple and that they define integrable systems with Lax pairs.
coboundary if its cocommutator is of the form δ : g → g⊗g, X → ad X (r) with an antisymmetric element r ∈ g⊗g; 2.
quasitriangular if its cocommutator is of the form δ : g → g⊗g, X → ad X (r) with a classical r-matrix r ∈ g⊗g; 3.
triangular if its cocommutator is of the form δ : g → g⊗g, X → ad X (r) with an antisymmetric classical r-matrix r ∈ g⊗g.

1.
Consider the real Lie algebra g = R R with basis {X, Y} and Lie bracket from Example 6 with the Lie bialgebra structures Any antisymmetric element of g⊗g is of the form r (a) = λX ∧ Y with λ ∈ R and satisfies This shows that the Lie bialgebra structure on g with cocommutator δ 2 is triangular, while the Lie bialgebra structure on g with cocommutator δ 1 is not even coboundary.

2.
Consider the Lie algebra sl(2, R) with the basis (14) and the three Lie bialgebra structures on sl(2, R) from Example 6 in (11). From the Lie bracket of sl(2, R), it follows directly that the three cocommutators are all given by antisymmetric elements of sl(2, R)⊗sl(2, R) and hence all three Lie bialgebra structures are coboundary. To determine if they are quasitriangular, we look for an ad-invariant symmetric element of sl(2, R)⊗sl(2, R) whose Schouten bracket is minus the Schouten bracket of the elements in the last equation. Up to multiplication by scalars, there is a unique ad-invariant symmetric bilinear form on sl(2, R), the Killing form given by J 0 , J 0 = −1 and J 1 , J 1 = J 2 , J 2 = 1. It follows that every ad-invariant symmetric element of sl(2, R)⊗sl(2, R) is of the form This is the only candidate for the symmetric component of the classical r-matrix. A straightforward but lengthy computation shows that it satisfies Similarly, we compute the Schouten brackets for J 2 ∧ J 0 , J 1 ∧ J 2 and J 1 ∧ (J 2 − J 0 ): This shows that the first Lie bialgebra structure is quasitriangular with r-matrix the second is coboundary, but not quasitriangular, and the third is triangular.

3.
For complex semisimple Lie algebras and their compact and normal real forms, one can show that every Lie bialgebra structure is coboundary. This follows from an argument based on Lie algebra cohomology and allows one to classify their Lie bialgebra structures.
At this stage, it is not apparent what is the advantage of a quasitriangular Lie bialgebra compared to a coboundary one. As the cocommutator depends only on the antisymmetric component of r, the symmetric part of a classical r-matrix is irrelevant for the Lie bialgebra structure. Its advantage is that it defines Lie bialgebra homomorphisms between g and its dual g * . If the symmetric part is non-degenerate, this gives rise to a decomposition of the dual Lie bialgebra into Lie subbialgebras. Lemma 3. Let g be a Lie algebra with basis {T a } and r = r ab T a ⊗T b a classical r-matrix for g. Denote by g * cop the Lie bialgebra with the same Lie algebra bracket as g * but the opposite cocommutator. Then, the maps are Lie bialgebra homomorphisms: If the symmetric component of r is non-degenerate, one has g = σ + (g * ) ⊕ σ − (g * ) as a vector space with Lie subbialgebras σ ± (g * ) ⊂ g.

Proof.
This follows by a direct computation. The cocommutator of g is given by and this implies for the Lie bracket and the cocommutator The computations for σ − are analogous.
Quasitriangular Lie bialgebras can be constructed systematically in a very simple way. In fact, every Lie bialgebra g defines a canonical quasitriangular Lie bialgebra structure on the vector space g ⊕ g * . This construction is due to Drinfeld [9] and can be viewed as the Lie bialgebra counterpart of a a corresponding construction for quantum groups and monoidal categories.

Theorem 3 ([9]
). Let g be a Lie bialgebra with dual g * and g * cop the Lie bialgebra with opposite cocommutator.

1.
There is a unique quasitriangular Lie bialgebra structure on the vector space g ⊕ g * such that the inclusions ι : g → D(g) and ι : g * cop → D(g) are Lie bialgebra homomorphisms. This Lie bialgebra is called the classical double D(g).

2.
If {T a } is a basis of g with dual basis {t a }, then the classical r-matrix of D(g) is r = T a ⊗t a and the Lie algebra structure of D(g) reads where f c ab and C bc a are the structure constants of g and g * with respect to {T a } and {t a }.
Proof. Suppose that the Lie bracket and the cocommutator of g take the form with structure constants f c ab , C bc a . The antisymmetry of the commutator and cocommutator reads f c ab = − f c ba and C bc a = −C cb a . The Jacobi identity, the coJacobi identity and the cocycle condition for δ take the form The condition that the inclusion maps are homomorphisms of Lie bialgebras then implies that the Lie bracket and cocommutator of D(g) satisfy Hence, the cocommutator of D(g) is determined uniquely by the cocommutator and Lie bracket of g and satisfies the coJacobi identity. If D(g) is quasitriangular with r-matrix r = T a ⊗t a , then From these conditions, one finds that the Lie bracket of D(g) must take the form (31). By definition of D(g), the Jacobi identity holds for Lie brackets involving only elements of g or of g * . For brackets involving elements of g and of g * , it can be established by a direct computation using expressions (31) and the identities (32). A similar computation proves the ad-invariance of the symmetric component of r and for the CYBE.

1.
If g is a Lie algebra with the trivial cocommutator δ = 0, then the classical double D(g) is a semidirect product g g * with Lie-bracket and cocommutator given by

2.
We consider the Lie algebra sl(2, R) with the basis and the second cocommutator from Example 6, Then, by Example 7, the dual Lie algebra has the Lie bracket The associated classical double D(sl(2, R)) has the Lie brackets (33), (34) and Using these expressions for the Lie bracket, one can show that D(sl(2, R)) ∼ = sl(2, C).

Application: Integrable Systems from Quasitriangular Lie Bialgebras
Quasitriangular Lie bialgebras have important applications in integrable systems. A quasitriangular Lie bialgebra g with a non-degenerate symmetric component of the classical r-matrix allows one to construct a Hamiltonian system with a Lax pair and conserved quantities in involution on the dual vector space g * . In fact, the role of Poisson-Lie and Lie bialgebra symmetries in integrable systems was one the origins of these structures.  • The Poisson manifold M stands for the phase space of the system and the Hamiltonian H for its total energy. The time evolution equation is then the usual time evolution equation in the Hamilton-Jacobi formalism, and the conserved quantities are the constants of motion. To solve the equations of motion, one is interested in having as many independent conserved quantities in involution as possible. By taking such functions as coordinates, one can reduce the number of degrees of freedom in the equations of motion and solve these equations efficiently.
The condition that conserved quantities f 1 , f 2 , . . . , f n ∈ C ∞ (M) are independent is usually stated as the requirement that they combine into the function f = ( f 1 , . . . , f n ) : M → R n such that rank(d f ) = n almost everywhere.
It is often also required that the Poisson bracket of a Hamiltonian system is symplectic, which requires M to be even-dimensional. On a 2n-dimensional symplectic manifold M, there can be at most n independent conserved quantities in involution. If there are n independent conserved quantities in involution, one calls the system integrable. The name integrable is motivated as follows. If f 1 , . . . , f n are independent conserved quantities in involution, then the Hamiltonian is a conserved quantity as well, it can be expressed as a function of the variables f 1 , . . . , f n . One usually takes it as one of the conserved quantities and sets H = f 1 . One may then introduce n additional phase space coordinates q 1 , . . . , q n , that are conjugate to the variables f 1 , . . . , f n with respect to the symplectic structure, and their equations of motion readq i = {H, q i } = ∂H/∂ f i . The right-hand side is a function of the conserved quantities f i , and thus, the equations of motion may be solved explicitly by integration.
It is often not easy to find conserved quantities. If the Hamiltonian system has obvious symmetries, then by Noether's theorem, these symmetries usually give rise to conserved quantities, and one may determine conserved quantities by studying the symmetries of the system. An important method to find conserved quantities of a Hamiltonian system are Lax pairs. This concept goes back to Lax [11], for the construction of Lax pairs from classical r-matrices, see [12][13][14][15]. For an accessible and broad treatment of Lax pairs and classical r-matrices in different integrable systems, see the textbook [7].
The left-hand side and the right-hand-side of equation (35) are understood as equations for the coefficient functions of L, P with respect to a basis. If {T a } is a basis of g with associated structure constants [T a , T b ] = f c ab T c , then one can write L = L a T a , P = P a T a with functions L a , P a ∈ C ∞ (M), and equation ( The advantage of a Lax pair is that it gives rise to many conserved quantities for the underlying Hamiltonian system. The Lax equations guarantee that for any ad-invariant function f on g, the function f • L ∈ C ∞ (M) is a conserved quantity. The simplest adinvariant functions on g are the trace polynomials in representations of g.
where we used the cyclic invariance of the trace. As the eigenvalues of ρ(L) are functions of the trace polynomials, they are conserved as well.
Lemma 4 allows one to construct conserved quantities from a Lax pair. However, in general, it is not guaranteed that they are conserved quantities in involution. Their Poisson brackets are given by the Poisson brackets of the matrix elements of L in different representations, and to ensure that these matrix elements Poisson commute, one needs additional conditions on L.
Quasitriangular Lie bialgebras give a systematic way of constructing Hamiltonian systems with Lax pairs whose conserved quantities are in involution. This construction makes use of the symmetric part of the r-matrix and requires that its symmetric part is non-degenerate.
The key observation is that in that situation, we can use the symmetric part of r to pull back the Lie bracket on g * defined by its antisymmetric part to a Lie bracket on g.

Lemma 5.
Let g be a Lie algebra, κ a non-degenerate ad-invariant symmetric bilinear form on g and K : g → g * , X → κ(X, −) the associated linear isomorphism.
Then, for any antisymmetric element u = u ab T a ⊗T b ∈ g⊗g with [[u, u]] ad-invariant, the associated Lie bracket on g * is given by Proof. By Proposition 1, the map δ : g → g⊗g, X → ad X (u) is a cocommutator on g and hence defines a Lie bracket on g * . In terms of a basis {T a } of g, this Lie bracket reads for all α, β ∈ g * and basis elements T a . The ad-invariance of κ reads Using the definition of K, the antisymmetry of u and the Lie bracket and the ad-invariance of κ, we then obtain for all basis elements T a We can now consider the Lie algebra g with either its original Lie bracket [ , ] or with the Lie bracket [ , ] u from (36) and with the trivial cocommutator. Then, its dual Lie bialgebra is the vector space g * with the trivial Lie bracket and the cocommutator δ = [ , ] * : g * → g * ⊗g * or δ u = [ , ] * u : g * → g * ⊗g * . By Example 10, the associated Poisson-Lie group is the vector space g * with the usual vector space addition and with the Poisson bracket from (26) For these Poisson brackets, it is straightforward to construct quantities in involution. It turns out that any function f ∈ C ∞ (g * ) that is invariant under the coadjoint action of G on g * is a conserved quantity. The coadjoint action * : G × g * → g * and its infinitesimal counterpart ad * : g × g * → g * are given in terms of the adjoint action by Proposition 2. Let g be a Lie algebra, κ a non-degenerate ad-invariant symmetric bilinear form on g and u ∈ g⊗g antisymmetric with [[u, u]] ad-invariant. Then, the functions f ∈ C ∞ (g * ) that are invariant under the coadjoint action * : G × g * → g * are in involution for the Poisson-brackets { , } and { , } u on g * from (39).
Proof. If f ∈ C ∞ (g * ) is invariant under the coadjoint action of G on g * , then we have f (g * α) = f (α) for all g ∈ G and α ∈ g * , and this implies for all X ∈ g For the Poisson brackets of two functions f 1 , f 2 ∈ C ∞ (g * ) that are invariant under the coadjoint action, this implies { f 1 , (39).
So far, we considered the Poisson brackets { , } u for any non-degenerate ad-invariant symmetric bilinear form κ on g and any antisymmetric element u ∈ g⊗g with [[u, u]] ad-invariant. The ad-invariance of [[u, u]] was needed to obtain a Lie bracket on g * and the non-degenerate ad-invariant symmetric bilinear form κ to pull back this Lie bracket to g.
Given a classical r-matrix r ∈ g⊗g whose symmetric component is non-degenerate, we can take its antisymmetric component u = r (a) for u and use its symmetric component to define a non-degenerate ad-invariant symmetric bilinear form κ on g. If {T a } is a basis of g, in which r (s) = r ab (s) T a ⊗T b , we define the coefficient matrix of κ by By considering the Poisson bracket { , } u for the antisymmetric component of r and any smooth function that is invariant under the coadjoint action as a Hamiltonian, we then obtain a Hamiltonian system on g * that is equipped with a Lax pair whose trace polynomials are conserved quantities in involution.

Theorem 4.
Let r ∈ g⊗g be a classical r-matrix for g with a non-degenerate symmetric component r (s) and antisymmetric component u.

2.
For any representation ρ : g → End R (V), the trace polynomials f ρ k are conserved quantities in involution.

Proof. 1. We consider the maps
where φ : g → g is defined as in (36). Then, we have L = L a T a with L a (α) = r ab (s) α, T b , which implies d α L a = r ab (s) T b . By (41) the invariance of H under the coadjoint action implies α, [X, d α H] = 0 for all X ∈ g. Inserting this in the equations of motion for L, we obtaiṅ where we used the ad-invariance of r (s) in (iii). This shows that L and P form a Lax pair.
To prove the formula for the Poisson brackets of L, we compute 2. Let now ρ : g → End R (V) and τ : g → End R (W) be representations of g. Their trace polynomials are conserved quantities by Lemma 4. Using the cyclic invariance of the trace, one obtains for the associated trace polynomials The ad-invariant symmetric bilinear form defined by the symmetric part of the r takes the form The map φ : sl(2, R) → sl(2, R) and the Lie bracket [ , ] u on sl(2, R) from (36) are given by If we interpret the elements H, J ± as functions on sl(2, R) * , the associated Poisson structure on sl(2, R) * reads This allows us to restrict this Poisson bracket to the linear subspace ker(J + − J − ) ⊂ sl(2, R), which is invariant under the coadjoint action of sl(2, R) on sl(2, R) * . As a Hamiltonian, it is natural to choose the function H = H 2 + J 2 + , which is invariant under the coadjoint action. If we set p = H and J + = J − = e −q , we obtain the usual symplectic structure on R 2 and the integrable Hamiltonian system with Hamiltonian H = p 2 + e −2q .
Although this example is very simple and can be solved explicitly, it illustrates the general method. It can be generalised to normal real forms of complex semisimple Lie algebras, for details see [5] [2.3D]. The associated integrable systems are the Toda lattice systems.

Coboundary and Quasitriangular Poisson-Lie Groups
If a Poisson-Lie group has a Lie bialgebra that is coboundary or even quasitriangular, its Poisson structure becomes much simpler, and the same holds for its dual and its classical double. In the quasitriangular case, these simplifications also extend to Poisson G-spaces. They allow one to form products of Poisson G-spaces and to write down simple expressions for Poisson structures on G that give it the structure of a Poisson G-space.

Definition 14.
A Poisson-Lie group is called coboundary or (quasi)triangular if its tangent Lie bialgebra is coboundary or (quasi)triangular. Example 14.

1.
To every connected and simply connected Poisson-Lie group G, one can associate a quasitriangular Poisson-Lie group, its classical double D(G). This is the unique connected and simply connected Poisson-Lie group D(G) with tangent Lie bialgebra D(g).

2.
If G is a complex semisimple Lie group or its compact or normal real form, then G is coboundary. This follows from the corresponding statement for complex semisimple Lie algebras and their real forms in Example 11.
Just as the cocommutator of a coboundary Lie bialgebra, the Poisson bracket of a coboundary Poisson-Lie group has a particularly simple description. In this case, one can express the Poisson bracket in terms of the left and right invariant vector fields on G with structure constants that are the coefficients of the coboundary with respect to a basis.
Theorem 5 ([14,15]). Let G be a coboundary Poisson-Lie group with cocommutator δ : g → g⊗g, X → ad X (u) for an antisymmetric element u = u ab T a ⊗T b . Then, its Poisson bracket is given by the Sklyanin bracket on G Proof. As u ∈ g⊗g defines a cocommutator on the Lie algebra g, its Schouten bracket [[u, u]] is ad-invariant by Proposition 1. To show that the Sklyanin bracket satisfies the Jacobi identity, one computes where one uses the antisymmetry of u, the fact that left and right invariant vector fields on a Lie group G commute and formula (21) for their Lie bracket to pass to the second line. The last identity then follows from the relation (20) between left and right invariant vector fields on G and the ad-invariance of [[u, u]].
To show that this is a Poisson-Lie structure on G, one uses the maps R g : G → G, h → hg and L g : G → G, h → gh and the relation (20) between left and right invariant vector fields on G to compute To show that this coincides with the Poisson-Lie structure on G defined by u, we compute the cocommutator of the associated Lie bialgebra by differentiating the Poisson bivector. Using formulas (20) that relate the left and right invariant vector fields on G, we find that the Poisson bivector of the Sklyanin bracket is given by and formula (23) for the cocommutator yields The description in Theorem 5 has the advantages that it depends only on the choice of a Lie algebra basis and exhibits clearly the relation with the associated Lie bialgebra structures. Note in particular that the conditions in Theorem 5 are satisfied for any quasitriangular Poisson-Lie group, since in that case the Schouten bracket of the antisymmetric part of the r-matrix is ad-invariant by Lemma 2. Just as in the case of a quasitriangular Lie bialgebra, one might wonder what is the additional benefit of having a quasitriangular Poisson-Lie group and not just a coboundary one. The answer is that it allows one to describe the Poisson-Lie structure of the dual Poisson-Lie group G * and the classical double D(G) in terms of a Poisson bracket on G. This result is due to Semenov-Tian-Shansky [15] and Lu and Weinstein [16].
Recall from Lemma 3 that a classical r-matrix r = r ab T a ⊗T b ∈ g⊗g defines Lie bialgebra homomorphisms where g * cop denotes the Lie bialgebra g * with the opposite cocommutator, {T a } is a basis of g and {t a } the associated dual basis of g * . By Theorem 2, there are unique connected and simply connected Poisson-Lie groups G and G * with tangent Lie bialgebras g and g * . The Lie bialgebra g * cop is the tangent Lie bialgebra of the Poisson-Lie group G * cop with the group multiplication and minus the Poisson bracket of G * . Again by Theorem 2, the Lie bialgebra homomorphisms σ ± : g * cop → g lift to unique homomorphisms of Poisson-Lie groups S ± : G * cop → G with tangent maps T 1 S ± = σ ± . We can use these maps to describe the Poisson-Lie structures of the classical double D(G) in terms of a Poisson bracket on G × G.
In terms of the dual basis {j 0 , j 1 , j 2 } of g * with j a , J b = δ a b , the maps σ ± : g * cop → g are given by The map σ + − σ − is a linear isomorphism, and we have im(σ + ) = span{J 1 , Using the matrix expressions (14) for the basis elements, one obtains
Factorisability of a Poisson-Lie group G leads to many simplifications. In particular, the expression for the Poisson bracket { , } * in Theorem 6 allows one to determine the symplectic leaves of the dual Poisson-Lie group G * . A direct computation shows that any conjugation invariant function on G Poisson commutes with all functions on G with respect to the Poisson bracket { , } * on G. This shows that its symplectic leaves must be contained in the conjugacy classes of G. In fact, for a factorisable Poisson-Lie group G, the diffeomorphism S from Theorem 6 identifies the symplectic leaves of G * with the conjugacy classes of G. This simplifies a more general description of the symplectic leaves of a Poisson-Lie group in terms of dressing transformations. For more background on factorisable Poisson-Lie groups, symplectic leaves and dressing transformations, see the original articles [16][17][18], the textbook ([5] (Chapter 1.5)) and the references given therein.

Poisson Spaces from (dynamical) Classical r-Matrices 4.1. Poisson Spaces over Quasitriangular Poisson-Lie Groups
These simplifications of the Poisson structure for quasitriangular Poisson-Lie groups G also extend to the associated Poisson G-spaces. The (antisymmetric part of) the classical r-matrix defines several Poisson G-space structures on G. Examples are the dual Poisson-Lie structure from Theorem 6, which is a Poisson G-space with respect to the conjugation action, and the Heisenberg double Poisson structure on G that gives G the structure of a Poisson G × G-space. The Heisenberg double Poisson structure is also called Semenov-Tian-Shansky bracket and was first introduced in [15,19]. The term Heisenberg double is from [20], where it was related to a symplectic form and considered as a generalisation of a cotangent bundle symplectic structure.
Proposition 3. Let G be a Poisson-Lie group.

1.
If G is a coboundary with δ : g → g⊗g, X → ad X (u) for an antisymmetric element u = u ab T a ⊗T b , then the Heisenberg double Poisson structure gives G the structure of a Poisson G × G-space with respect to If G is quasitriangular with classical r-matrix r = r ab T a ⊗T b ∈ g⊗g then G becomes a Poisson G-space with : G × G → G, g h = ghg −1 and the dual Poisson structure from Theorems 2 and 6.
Proof. That the Poisson brackets indeed satisfy the Jacobi identity follows by direct computations similar to the ones in the proof of Theorem 5. For the bracket in 1. one obtains from the Ad-invariance of [[u, u]] and the relation between left and right invariant vector fields The computations for the bracket in 2. are similar. The compatibility condition from Definition 4 between the Poisson structure and the group actions follows by a direct computation from the expressions for the Poisson brackets and formula (43) for the Sklyanin bracket.
Besides giving simple expressions for the Poisson brackets of certain Poisson G-spaces, the classical r-matrix of a quasitriangular Poisson-Lie group also has more conceptual implications. It allows one to form products of Poisson G-spaces that can be viewed as the Poisson-Lie counterpart of the braided tensor product of module algebras over a quasitriangular Hopf algebra.
For this, note that if M and N are Poisson G-spaces over a Poisson-Lie group G, then M × N has a canonical Poisson structure and a canonical G-action, namely the product Poisson structure { , } M×N and the diagonal action : G × M × N → M × N, (g, m, n) → (g m, g n). However, together they do not equip M × N with the structure of a Poisson G-space. This becomes obvious when one considers functions f 1 ∈ C ∞ (M), f 2 ∈ C ∞ (N) together with the projection maps π M : M × N → M, (m, n) → m and π N : M × N → N, (m, n) → n. Then, the definition of the product Poisson structure implies for all m ∈ M and n ∈ N. However, the condition that M × N is a Poisson G-space requires Hence, M × N with the product Poisson structure and the diagonal G-action is in general not a Poisson G-space, unless the Poisson-Lie structure on G is trivial. It turns out that if G is quasitriangular, one can modify the Poisson structure on M × N with the classical r-matrix to obtain a Poisson G-space structure. This result goes back to the work of Fock and Rosly [21][22][23], but was not stated explicitly there. The following theorem was given in [24,25] and independently in [26]. It was shown in [24,26] that this product of Poisson G-spaces gives the category of Poisson G-spaces over a quasitriangular Poisson-Lie group G the structure of a monoidal category and in [24] that this monoidal category is monoidally equivalent to the category of quasi-Hamiltonian G-spaces.

2.
Every manifold M with a G-action : Proof. Clearly, the brackets { , } r and { , } M are antisymmetric and satisfy the Leibniz identity. That they satisfy the Jacobi identity follows by direct computations that use the CYBE for r, and for the bracket { , } r additionally the fact that M and N are Poisson G-spaces. The compatibility conditions from Definition 4 between the brackets and the G-actions follow by a direct computation that makes use of expression (43) for the Sklyanin bracket, and for the bracket { , } r also of the definition of the product Poisson structure and the Ad-invariance of r.

Application: Moduli Spaces of Flat Connections
An important application of the product of Poisson G-spaces is the description of the canonical symplectic structure on the moduli space of flat G-connections on an orientable surface Σ. The moduli space of flat G-connections is given as the quotient Hom(π 1 (Σ), G)/G of the set Hom(π 1 (Σ), G) of group homomorphisms ρ : π 1 (Σ) → G modulo the conjugation action : G × Hom(π 1 (Σ), G) → Hom(π 1 (Σ), G), (g, ρ) → g · ρ · g −1 . The moduli space of flat connections has a canonical symplectic structure described by Atiyah and Bott [27] and Goldman [28,29] that depends only on the choice of a non-degenerate Ad-invariant symmetric bilinear form on g = Lie G.
From the physics perspective, the moduli space of flat G-connections is the gauge invariant or reduced phase space of a Chern-Simons gauge theory with gauge group G on R × Σ. The choice of a non-degenerate Ad-invariant symmetric bilinear form on g defines the Chern-Simons gauge functional, and its canonical symplectic structure arises from symplectic reduction of the induced symplectic structure on the space of gauge fields [27].
A choice of a quasitriangular Poisson-Lie group structure on the gauge group G allows one to give a simple description of the canonical symplectic structure on Hom(π 1 (Σ), G)/G due to Fock and Rosly [21][22][23] that resembles a lattice gauge theory. In this description, the orientable surface Σ is modelled by a directed graph Γ embedded in Σ. The graph needs to be sufficiently fine to resolve the topology of the surface, so one requires that Σ \ Γ is a disjoint union of discs. The connected components of Σ \ Γ are called faces of Γ.
We denote by V, E, F, respectively, the sets of vertices, edges and faces of Γ. For each edge e, we write s(e) for the starting and t(e) for the target vertex of e. The orientation of the surface induces a cyclic ordering of the edge ends incident at each vertex v ∈ V. We equip each vertex with a marking, the cilium, that transforms this cyclic ordering into a linear ordering. The order of the edges is taken counterclockwise from the cilium, and we write e < f if the edge end e is of lower order than the edge end f . The faces of Γ then correspond bijectively to closed paths in Γ, up to cyclic permutations, that follow the face counterclockwise, turn maximally left at each vertex in the path and traverse each edge at most once in each direction. Cyclic permutations of paths correspond to different choices of their starting and endpoints.
Given a Lie group G, one can define a graph connection as a map A : E → G, e → g e that assigns an element of G to each oriented edge of Γ. Reversing the orientation of an edge e corresponds to replacing g e by g −1 e . The set A(Γ) of graph connections on Γ is thus given by the E-fold product G ×E . We denote by π e : G ×E → G, (g 1 , ..., g n ) → g e the maps that project on the factor associated with the edge e ∈ E.
The appropriate notion of curvature in this context is obtained by considering the ordered product of the group elements taken along a path that borders a face of Γ. For a flat graph connection, this product must vanish for the paths around each face of Γ. One thus chooses for each face of Γ an associated path f in Γ that starts and ends at a vertex and defines a moment map that encodes the curvature of a graph connection at the face f . Here, the product is taken over all edges traversed by f , in the order in which they occur in f and e = 1 if e is traversed in its orientation and e = −1 if it is traversed against its orientation. Group elements of edges that are traversed twice by f occur twice in the product, but with opposite signs. Although µ f depends on the choice of the starting point of the path f , the preimage µ −1 f (1) does not, since different choices of paths f for a given face are related by conjugation. The space of flat graph connections is thus independent of the choices of paths Graph gauge transformations are modelled by group actions of G assigned to the vertices of Γ. The group action v : G × G ×E → G ×E for a vertex v ∈ V acts only on those copies of G that correspond to edges incident at v. The action is by left multiplication for edges that are incoming at v, by right multiplication with the inverse for edges that are outgoing at v and by conjugation for loops at v: As the group actions v and w for distinct vertices v, w ∈ V commute, combining them yields a group action of G ×V on G ×E . One thus defines the group of graph gauge transformations as G(Γ) = G ×V and obtains a group action of G(Γ) on the set A(Γ) of graph connections. As this group action sends flat connections to flat connections, it restricts to an action : G(Γ) × F (Γ) → F (Γ). The moduli space of flat G-connections on Σ is then given as the orbit space of this group action Hom(π 1 (Σ), G)/G = F (Γ)/G(Γ).
In [21][22][23], Fock and Rosly showed that if G has a quasitriangular Poisson-Lie group structure, the symplectic structure on Hom(π 1 (Σ), G)/G can be described in terms of products of Poisson G-spaces. More specifically, this requires one or more classical rmatrices r ∈ g⊗g whose symmetric components are dual to a fixed non-degenerate Adinvariant symmetric bilinear form on g. They define a Poisson G ×V -space structure on G ×E that induces Goldman's and Atiyah-Bott's symplectic structure on Hom(π 1 (Σ), G)/G. This Poisson structure on G ×E is obtained by assigning classical r-matrices r(v) to the vertices of Γ whose symmetric components are dual to a fixed non-degenerate Ad-invariant symmetric bilinear form on g. This amounts to assigning a quasitriangular Poisson-Lie group to each vertex of Γ. Each edge of Γ is assigned a generalisation of the Heisenberg double Poisson structure from Proposition 3, which is given by the r-matrices at its starting and target vertex and an example of the mixed product Poisson structures in [25].
Taking the product of the Poisson G-spaces from Theorem 7 at each vertex of Γ then yields Fock and Rosly's Poisson structure on G ×E . To describe it explicitly, we choose a basis {T a } of g. We denote by T Le a and T Re a the right and left invariant vector fields on the copy of G for each edge e ∈ E, whose action on functions F ∈ C ∞ (G ×E ) is given by T Le a .F(g 1 , ..., g n ) = d dt | t=0 F(g 1 , . . . , e −tT a · g e , . . . , g n ) T Re a .F(g 1 , ..., g n ) = d dt | t=0 F(g 1 , . . . , g e · e tT a , . . . , g n ).
Fock and Rosly's Poisson structure can then be summarised as follows.
(48) as the sum of a bivector B a that depends only on their antisymmetric and a bivector B s that depends only on their symmetric components. One finds that the former is given by and independent of the choice of the cilia, which differ by cyclic permutations of the orderings at each vertex. 3. For each function F ∈ C ∞ (G) G and any face f , the function F • µ f ∈ C ∞ (G ×E ) is G ×V -invariant and hence its Poisson bracket with any function in C ∞ (G ×E ) is independent of the choice of the cilia. One may thus orient the cilia at the vertices in such a way that the associated path f does not traverse any cilia. This implies that there are no edges that are between consecutive edges in f with respect to the ordering at their common vertex. A direct computation then shows that the Poisson bracket of F • µ f with any function in C ∞ (G ×E ) vanishes.
The description in Theorem 8 involves the graph gauge transformations acting on the graph connections but not the moment maps µ f : G ×E → G from (46) that encode the flatness conditions. From the viewpoint of constrained mechanical systems, the conditions µ f (g 1 , ..., g n )= 1 for the moment maps µ f : G ×E → G from (46) can be viewed as group valued first-class constraints that generate the graph gauge transformations. Reducing this gauge freedom corresponds to a Poisson reduction of the Poisson G ×V -space structure from Theorem 8. It was shown by Fock and Rosly [21][22][23] that this yields the canonical symplectic structure on the moduli space of flat G connections.
Theorem 9 ([21-23]). Poisson reduction of the Poisson structure from Theorem 8 with respect to the moment maps µ f : G ×E → G for the faces f ∈ F and the group action : G ×V × G ×E → G ×E induces Goldman's and Atiyah-Bott's symplectic structure on Hom(π 1 (Σ), G)/G for the nondegenerate symmetric bilinear form defined by the symmetric component of r.

Remark 6.
This description of the moduli space of flat G-connections is closely related to its description via quasi Hamiltonian systems and Lie group value moment maps [30], for an accessible review see also the lecture notes [31]. In fact, it was shown in [24] that the monoidal category of Poisson G-spaces over a quasitriangular Poisson-Lie group G is monoidally equivalent to the monoidal category of quasi Hamiltonian G-spaces. The monoidal structures are the product of Poisson G-spaces from Theorem 7 and the fusion product of Poisson G-spaces, respectively.

Remark 7.
In the case where the quasitriangular Poisson-Lie group G in Theorem 8 is a classical double D(H) of a Poisson-Lie group H, the description in Theorem 8 can be factorised such that group elements of H are associated with Γ and elements of the dual Poisson-Lie group H * with the dual graph of Γ. This yields a factorisation of the Poisson structure from Theorem 8 as a product of Heisenberg double Poisson structures on D(G) associated with each edge of Γ. For a graph with one vertex, this was first shown in the symplectic framework in [32]. For a general graph, this was established in [33], where it was shown that the resulting description of the moduli space can be viewed as a Poisson analogue of Kitaev lattice models [34] considered in condensed matter physics and topological quantum computing.

Poisson Structures from Dynamical r-Matrices
In this section, we generalise some of the results from Section 4.1 to dynamical classical r-matrices. While a classical r-matrix is an element of g⊗g for a Lie algebra g, a dynamical r-matrix is a function with values in g⊗g. Thus, a classical r-matrix can be viewed as a special case of a dynamical r-matrix, namely a constant one.
Dynamical r-matrices were first considered in [35,36] in the context of integrable systems. They also arise in the Poisson reduction and symplectic reduction of systems with Poisson-Lie symmetries [37], for instance, in the description of moduli spaces of flat connections and the Chern-Simons gauge theory [38,39]. For an accessible introduction to classical dynamical r-matrices, see [8], for classification results [40][41][42][43].
To keep the discussion simple, we restrict attention to dynamical r-matrices associated with Cartan subalgebras. Most of the results hold in more generality but their description becomes more complicated. the unitarity condition: r (s) = 1 2 (r ab + r ba )T a ⊗T b is a constant element of g⊗g that is invariant under the adjoint action of g.
The concept of a dynamical classical r-matrix generalises the concept of a classical r-matrix. The latter is simply a dynamical classical r-matrix for the trivial Lie subalgebra h = {0} ⊂ g. However, dynamical r-matrices have some important advantages over the classical r-matrices from Definition 9. The first is that classical r-matrices whose symmetric component coincides with a fixed ad-invariant symmetric element t ∈ g⊗g for a real Lie algebra g may not exist unless one complexifies g, while the differential equation that defines dynamical classical r-matrices still has a solution. If a classical r-matrix for a fixed ad-invariant element of g⊗g exists, it often arises as a limit of a dynamical one. This is illustrated by the following examples.  Then, a dynamical classical r-matrix for the Lie subalgebra h = RJ 0 ∼ = R is given y r : R → su(2)⊗su(2), t → J 0 ⊗J 0 + J 1 ⊗J 1 + J 2 ⊗J 2 − tan(t)J 1 ∧ J 2 .
The Lie algebra su (2) has no real classical r-matrix in the sense of Definition 9.

2.
Consider the Lie algebra sl(2, R) with the basis {J 0 , J 1 , J 2 } from Example 6. Then, a dynamical classical r-matrix for h = RJ 0 is given by r : R → sl(2, R)⊗sl(2, R), t → J 0 ⊗J 0 − J 1 ⊗J 1 − J 2 ⊗J 2 − tan(t)J 1 ∧ J 2 and a dynamical classical r-matrix for h = RJ 1 by r : R → sl(2, R)⊗sl(2, R), The Lie algebra sl(2, R) has no real classical r-matrix whose antisymmetric component is a real multiple of J 1 ∧ J 2 . A classical r-matrix whose antisymmetric component is a real multiple of J 2 ∧ J 0 is given in (29). This classical r-matrix is obtained from r in the limit t → −∞.
The second major advantage of dynamical over classical r-matrices is that the latter can be related by dynamical gauge transformations. This was used in [40][41][42][43] to classify them. Dynamical gauge transformations are especially simple for the case where h ⊂ g is a Cartan subalgebra, and we restrict attention to this case. Definition 17 ([41]). Let h ⊆ g a Cartan subalgebra and H ⊂ G the associated connected subgroup. A dynamical gauge transformation for h is a holomorphic map γ : U → H with U ⊆ h * . Its action on a classical dynamical r-matrix r : U → g⊗g is given by where η : U → h⊗h is the map defined by the left invariant one-form ω = γ −1 dγ and η 21 is obtained from η by flipping the two factors in the tensor product.

Proposition 4 ([41]
). If r : U → g⊗g is a classical dynamical r-matrix for a Cartan subalgebra h ⊆ g, then for all dynamical gauge transformations γ : U → H the map γ r : U → g⊗g is a dynamical classical r-matrix for h.
This result was used in [40,42] to classify all dynamical r-matrices for complex simple Lie algebras, for a more general classification under weaker assumptions, see also [43]. These classification results uses dynamical gauge transformations to bring the dynamical r-matrix into a standard form, in which the dynamical classical Yang-Baxter equation becomes particularly simple.
Just as a classical r-matrix for a Lie algebra g defines a quasitriangular Poisson-Lie group structure on the associated Lie group G, dynamical classical r-matrices define Poisson groupoid structures. The Poisson groupoid structure defined by classical dynamical rmatrices r : U ⊂ h * → g⊗g and s : V ⊂ k * → g⊗g is a Poisson structure on U × G × V, where linear functions on U and V are identified with elements of h and k. Dynamical Poisson groupoids were introduced by Etingof and Varchenko in [42]. We summarise the more accessible formulation from [37], with some changes in sign conventions and a restriction to Cartan subalgebras not present in [37].

Proposition 5 ([37]
). Let g be a Lie algebra with basis {T a } and h, k ⊂ g Cartan subalgebras with bases {H a = h b a T b } and {K a = k b a T b }, respectively, and dual bases {H a } and {K a }. Let r = r ab T a ⊗T b : U ⊂ h * → g⊗g s = s ab T a ⊗T b : V ⊂ k * → g⊗g structure from Proposition 5. Coupling the contributions of the incident edges at a vertex as in Theorem 8, but with a dynamical Poisson structure then yields a Poisson structure on the group Π v∈V U v × G ×E that generalises the Poisson structure from Theorem 8. This is a special case of the results obtained in [44], in which this Poisson structure was considered for more general dynamical Poisson G-spaces.