Poisson-geometric analogues of Kitaev models

We define Poisson-geometric analogues of Kitaev's lattice models. They are obtained from a Kitaev model on an embedded graph $\Gamma$ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $\mathcal H(G)$. Each vertex (face) of $\Gamma$ defines a Poisson action of $G$ (of $G^*$) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group $D(G)$. We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of $D(G)$. We construct an isomorphism of Poisson $D(G)$-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly's Poisson structure for the graph $\Gamma$ and the Poisson-Lie group $D(G)$. This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the Poisson structure on the moduli space of flat $D(G)$-bundles on a surface with boundary constructed from $\Gamma$.


Introduction
Kitaev models. Besides their role as models for topological quantum computing, Kitaev's lattice models have gained popularity in condensed matter physics and are also of interest due to their connection to 3d topological quantum field theory. In Kitaev's original definition [Kit03], the edges of an embedded graph were decorated with elements of the group algebra of Z 2 . This was generalized first by Bombin and Martin-Delgado [BMD08] to group algebras of finite groups, and finally to finite-dimensional semi-simple Hopf * -algebras by Buerschaper et al. [BMCA13]. Balsam and Kirillov have shown in [BK12] that the ground state of Kitaev models for a Hopf algebra H, the protected space [Kit03], is isomorphic to the vector space that topological quantum field theory of Turaev-Viro type [TV92,BW96] (for the category H-Mod) assigns to a surface. A Kitaev model on a surface is defined via an embedded graph Γ with edge set E. A copy of a finite-dimensional semi-simple Hopf * -algebra H over C is assigned to each edge to obtain the extended space H ⊗E . Meusburger showed that its endomorphism algebra End C (H ⊗E ) is isomorphic to H(H) ⊗E , where H(H) is the Heisenberg double of H [Meu17]. To the vertices and faces of Γ one associates vertex and face operators. The operators for a vertex and adjacent face can be combined to obtain a representation of the Drinfeld double D(H) on H ⊗E [Kit03,BMCA13] and a D(H)module algebra structure on the endomorphism algebra End C (H ⊗E ) [Meu17]. The protected space is the subspace of H ⊗E on which all vertex and face operators act trivially. Its endomorphism algebra is the subalgebra of invariants of End C (H ⊗E ) with respect to the D(H)-module algebra structures.
with the Poisson subalgebra of invariant functions. We prove that a pair of actions for a vertex and adjacent face can be combined into a Poisson D(G)-action. In Section 3.4 we discuss the Poisson analogues of vertex and face operators and show that they are related to the Poisson-Lie group actions associated with the respective vertices and faces. In Sections 3.5 and 3.6 we prove our main results. The isomorphism between Poisson-Kitaev models and Fock and Rosly's Poisson structure is shown in Section 3.5. The relation between Poisson-Kitaev models and moduli spaces of flat D(G)-bundles is proven in Section 3.6.

Embedded graphs
In this section, we introduce the notion of ribbon graphs (also called fat graphs, embedded graphs or maps). For more background on ribbon graphs see, for instance, [EMM13] or [LZ04]. First we introduce some general terms for directed graphs.
Definition 2.1. Let Γ be a directed graph, V its set of vertices and E its set of edges.
(i) For an edge e ∈ E we write s(e), t(e) ∈ V for the source and target vertices of e, respectively. We set s(e ∓ ) = t(e ± ) for the edge e −1 with reversed orientation. We say that e is incoming at ( An edge e of f is oriented clockwise (oriented counterclockwise) with regard to f if there is a face path p = e εn n • . . . • e ε 1 1 of f with e = e j and ε j = 1 (ε j = −1) for some j and p traverses e exactly once. Notation 2.3. In the following Γ always denotes a ribbon graph. We write V, E and F for its sets of vertices, edges and faces, respectively. When we depict a piece of Γ, we will assume that the edge ends at vertices are ordered counterclockwise. This implies that face paths are traversed in a clockwise manner.
Remark 2.4. Ribbon graphs are in correspondence with graphs embedded into compact oriented surfaces (with boundary). To obtain the associated surface, glue an annulus to Γ for each face f ∈ F along an associated face path.
Conversely, consider a finite directed graph Γ that is embedded into a surface S. Then the orientation of the surface S naturally induces a cyclic ordering of the edge ends at every vertex v ∈ V , turning Γ into a ribbon graph.
The name ribbon graph is motivated by the fact that ribbon graphs can be thickened, thus turning each edge into a ribbon: The dual Γ * of a ribbon graph Γ is also a ribbon graph in a natural way: the edge ends of Γ * correspond to the edge sides of Γ D and the cyclic ordering at a vertex of Γ * is obtained from a face path of the corresponding face in Γ. The thickening of Γ * coincides with that of Γ (up to edge orientation) if we switch the roles of edge ends and sides. The dual of a ciliated ribbon graph Γ, however, does not inherit a ciliated ribbon graph structure. For this reason, we consider ciliated ribbon graphs with the following additional structure.
Definition 2.6. A doubly ciliated ribbon graph is a ciliated ribbon graph Γ together with a choice of a face path for every face.
The chosen face path for a face equips it with a linear ordering of the adjacent edge sides. Graphically, the orderings of a doubly ciliated ribbon graph Γ can be expressed by adding a cilium to each vertex and to each face as shown in

Poisson-Lie groups and Poisson G-spaces
In this section we introduce the data for Fock and Rosly's Poisson structure [FR99] and for Poisson-Kitaev models: Poisson-Lie groups and manifolds with Poisson actions of Poisson-Lie groups. We mostly follow the presentation in [CP94].

i) A Lie group G that is also a Poisson manifold is called a Poisson-Lie group if the multiplication map µ : G × G → G is Poisson with respect to the product Poisson structure on G × G. (ii) A homomorphism Φ : G → H of Poisson-Lie groups is a homomorphism of Lie groups that is also a Poisson map. (iii) A Poisson-Lie subgroup H of G is a Lie subgroup that is also a Poisson submanifold:
H is a Poisson-Lie group together with an injective immersion ι : H → G that is a group homomorphism and a Poisson map.
(Poisson-)Lie groups and smooth manifolds are always assumed finite dimensional. We do not require submanifolds to be embedded. The Poisson structure of a Poisson-Lie group G equips its Lie algebra L(G) with the structure of a Lie bialgebra.
(i) A Lie algebra (g, [ , ]) together with a skew symmetric linear map δ : g → g ⊗ g, called the cocommutator of g, is called a Lie bialgebra if δ * : g * ⊗ g * → g * defines a Lie bracket on g * and δ is a 1-cocycle of g with values in g ⊗ g: (ii) A homomorphism of Lie bialgebras ϕ : g → h is a homomorphism of Lie algebras such that ϕ * : h * → g * is also a homomorphism of Lie algebras.
Denote by L g : G → G the left multiplication h → g · h with an element g ∈ G and by R g : G → G the right multiplication. We write T L g and T R g for the associated tangent maps.
where we use the following notation for r, s ∈ g ⊗ g: [r 12 , s 13 ] :

A solution of the CYBE is called a classical r-matrix. (ii) A Poisson-Lie group G is called coboundary if its Poisson bivector is of the form
for an element r ∈ g ⊗ g. It is called quasi-triangular if r is a classical r-matrix, and triangular if in addition r is antisymmetric.
Lemma 2.14. [Dri83] Any Poisson-Lie group G together with the multiplication map µ : G×G → G is a Poisson G-space.
If G is quasi-triangular, then there is another Poisson structure (G, w H(G) ) on G so that (G, w H(G) ) becomes a Poisson G-space: Theorem 2.16. [STS85,STS92] (i) The following bivector defines a Poisson structure on G: (ii) The Poisson manifold (G, w H(G) ) becomes a Poisson G-space when equipped with any of the following Poisson actions:

Double Poisson-Lie groups
Similarly to the Drinfeld double of a Hopf algebra, to any Poisson-Lie group G we can associate a quasi-triangular Poisson-Lie group D(G), the double of G. We define the double of a Lie bialgebra as in [CP94]. Let (g, [, ], δ) be a Lie bialgebra with dual Lie bialgebra (g * , [, ] g * , δ g * ), where [, ] g * = δ * and δ g * = [, ] * . Denote by g * cop the Lie bialgebra g * with opposite cocommutator. Consider the symmetric bilinear form ( , ) : (g ⊕ g * ) ⊗2 → R given by Proposition 2.19. [Dri83] There is a unique Lie bialgebra structure on the vector space g ⊕ g * such that the inclusions g → g ⊕ g * , g * cop → g ⊕ g * are homomorphisms of Lie bialgebras and ( , ) is invariant under the adjoint representation. The Lie bracket is given by This Lie bialgebra structure on g ⊕ g * is quasi-triangular with classical r-matrix r = id g , where we view id g as an element of g ⊗ g * ⊆ (g ⊕ g * ) ⊗2 . For any quasi-triangular Poisson-Lie group G the classical r-matrix defines two linear maps These maps are used to relate the Poisson-Lie group G with its dual G * if G is factorizable, that is, if the symmetric component r s of its r-matrix is non-degenerate. (See, for instance, [RSTS88] or [WX92].) This is the case if G is a double Poisson-Lie group. The CYBE implies that the maps σ ± are homomorphisms of Lie algebras, where g * is equipped with the commutator δ * . Thus, they can be integrated to homomorphisms of Lie groups S + , S − : G * → G.
If G is double, the images σ + (g * ), σ − (g * ) coincide with the Lie subbialgebras g + and g − , respectively. The map t : so that S is a local diffeomorphism in a neighbourhood of 1 ∈ G * . Computing the pullback of the Poisson bivector of G * along the local inverse of S allows us to (locally) express the Poisson structure of G * on G.
Lemma 2.22. The bivector defines a Poisson structure on G. The smooth map S : G * → (G, w G * ) is Poisson and a local diffeomorphism in a neighbourhood of the unit 1 ∈ G * .
The CYBE also implies that σ ± are anti-Lie coalgebra homomorphisms. Therefore, the images G + := S + (G * ), G − := S − (G * ) are Poisson-Lie subgroups. In a neighbourhood U ⊆ G of the unit we obtain from S two local projections π ± : U → G ± given by As shown by Lu and Weinstein [LW90,Theorem 3.14], the induced (local) right action of G + on G − defined by (g − , g + ) → π − (g −1 + g − ) coincides with the right dressing action of the Poisson-Lie group (G + , −w G + ) with negative Poisson bivector on G − . Therefore, the projection π − : U → G − is Poisson and the same can be shown for π + : U → G + . In terms of the Poisson bivector w G of G this means T π ⊗2 ± w G | U = w G •π ± . A direct computation shows that T π ⊗2 ± w H(G) | U and T π ⊗2 ± w G * | U coincide up to a sign with T π ⊗2 ± w G | U for the Poisson bivectors w H(G) from (3) and w G * from (5). The following lemma summarizes these results.
Lemma 2.23 (Poisson properties of π ± ). There is an open neighbourhood U ⊆ G of the unit such that (i) for all g ∈ U there exist unique g − ∈ G − , g + ∈ G + with g = g − g + , Example 2.25. [LW90,Theorem 3.7] Let G be a connected and simply connected double Poisson-Lie group and G + , G − the connected Poisson-Lie subgroups with tangent Lie bialgebra g + and g − , respectively. If G − is compact and G + is closed in G, then the map µ : G + × G − → G from Definition 2.24 is a diffeomorphism.
We conclude this section by deriving certain identities for the projections π ± that will be frequently used in the following.
Lemma 2.26 (Computation rules for π ± ). Let G be a global double Poisson-Lie group. Then: Proof. This follows directly from the unique decomposition g = g − g + for all elements g ∈ G.
Lemma 2.27 (Relations of π ± with the bivector w H(G) and the r-matrix).
(i) For all d, g ∈ G the following equations hold: (ii) For all x ∈ G − , α ∈ G + one has: Proof. Both statements can be shown using the Ad-invariance of r s = 1 2 (r + r 21 ) and the identities that follow from r ∈ g − ⊗ g + .

Fock and Rosly's Poisson structure
In [AB83], Atiyah and Bott showed that the moduli space M = Hom(π 1 (S), G)/G of flat Gbundles on a compact oriented surface S has a natural symplectic structure if the Lie group G is equipped with a non-degenerate symmetric Ad-invariant bilinear form. A combinatorial description of this symplectic structure in terms of intersection points of curves has been given by Goldman [Gol84,Gol86]. Fock and Rosly have shown in [FR99] that on a compact oriented surface S with boundary the symplectic structure on M can be represented as a graph gauge theory on a ciliated ribbon graph Γ embedded into S. In this section we briefly present these results. Let G be a quasi-triangular Poisson-Lie group with tangent Lie bialgebra g and Γ a ciliated ribbon graph. Consider the product manifold FR := G ×E obtained by associating a copy of G to every edge e ∈ E. Denote by π e : FR → G the projection associated with the edge e ∈ E. Write I v for the ordered set of edge ends at a vertex v ∈ V and denote the corresponding edges by e i for i ∈ I v . Define for v ∈ V and i, j ∈ I v the element s ij v ∈ {0, ±1} by For i ∈ I v let V i : g → Γ(T FR) be the map that associates to the Lie algebra element X ∈ g the vector field V i (X) defined by if e = e i and i is outgoing from v.
Choose for every vertex v ∈ V a classical r-matrix of the form r(v) = r a (v) + r s with fixed Adinvariant symmetric component r s ∈ g ⊗ g. Define the bivector To a vertex v ∈ V we associate the action otherwise. (11) Let G v be the quasi-triangular Poisson-Lie group with the Poisson bivector from (2) for the classical r-matrix r(v). Consider the compact oriented manifold S that is obtained by gluing annuli to the faces of Γ (or, equivalently, by thickening Γ as in Definition 2.5) and let where π ′ e : FR ′ → G is the edge projection map associated to e ∈ E ′ . (ii) Gluing two edges along a bivalent vertex. 4 Let h 1 , h 2 be distinct edges incident at the bivalent vertex v := t(h 1 ) = s(h 2 ). In the transformed graph Γ ′ , replace these edges by the edge h ′ : s(h 1 ) → t(h 2 ) and erase v. The associated map φ : FR → FR ′ is defined by: (iii) Erasing an edge. Choose an edge h ∈ E and let Γ ′ be the graph obtained by removing h from Γ. Define the associated map φ : Denote the vertex set of the graph Γ ′ obtained by a graph transformation by V ′ .
Proposition 2.31. [FR99,Proposition 4] For all v ∈ V ′ , the maps associated to these graph transformations are morphisms of Poisson G v -spaces with respect to the actions There is a natural notion of flatness for Fock and Rosly's graph gauge theory: an element γ ∈ FR is flat at a face f if the holonomy around f is trivial. We express this by a more general functor on graph groupoid G(Γ D ) (Definition 2.1 (ii)) of the thickening Γ D (Definition 2.5). Consider the set of smooth maps C ∞ (FR, G) as a groupoid with a single object * so that Hom( * , * ) = C ∞ (FR, G) and composition is given by pointwise multiplication.
This determines the functor Hol FR uniquely as G(Γ D ) is freely generated by the edges of Γ D .

Kitaev models
We summarize the construction and some important properties of Kitaev models, mostly following the presentation in [Meu17]. First we introduce some Hopf-algebraic notions.
This Hopf algebra is denoted by D (H) Kitaev models are constructed from a doubly ciliated ribbon graph Γ and a semi-simple 5 finitedimensional Hopf algebra H over C. To the graph Γ one associates the extended space H ⊗E by assigning a copy of H to every edge of Γ. For k ∈ H, α ∈ H * consider the linear maps We then define for each edge e ∈ E the triangle operators L k e± , T α e± : H ⊗E → H ⊗E for k ∈ H, α ∈ H * by extending L k ± , T α ± : H → H to linear maps H ⊗E → H ⊗E in the obvious way. For the remainder of this section assume that the graph Γ is paired (Definition 2.7 (iv)). Consider a vertex v ∈ V with incident edges e 1 < · · · < e n numbered according to the ordering of their edge ends at v (counterclockwise). Let ε 1 , . . . , ε n ∈ {±1} such that e ε i i is incoming at v. The vertex operator A k v : H ⊗E → H ⊗E for k ∈ H is the linear map where we set S 0 = id H . Similarly, let e 1 < · · · < e n be the edges adjacent to a face f ∈ F , numbered according to the ordering of their edge sides at f (clockwise). Let ε i ∈ {±1} such that e ε i i is oriented clockwise for all i. The face operator B α f : H ⊗E → H ⊗E for α ∈ H * is defined as Vertex and face operators are subject to the following commutation relations:

Lemma 2.36 (Commutation relations of vertex and face operators). [Kit03, BMCA13] (i) The vertex operators for distinct vertices
the associated operators commute: Due to the semi-simplicity of H, one can project on a subspace of H ⊗E that is invariant under the action of vertex and face operators. Denote by η ∈ H * , l ∈ H the normalized Haar integrals of H * and H, respectively. Let S be the surface obtained by gluing a disk to every face of the graph Γ.
Proposition 2.37. [Kit03,BMCA13] The set where the summation is over Y i ∈ Irr(D(H)) and Y * i denotes the dual representation for all i. The vector space M(Y 1 , . . . , Y n ) is the protected space for the Kitaev model with excitations [Kit03,BK12]. Let S ′ be the surface obtained by gluing annuli to the faces f 1 , . . . , f n of Γ and disks to all other faces. [KJB10] for the category H-Mod assigns to S ′ , where the boundary component of S ′ corresponding to f i is labeled with Y i for i = 1, . . . , n.
In particular, the space M(Y 1 , . . . , Y n ) only depends on the homeomorphism class of the surface S ′ and the irreducible representations Y 1 , . . . , Y n of D(H). Excited states γ ∈ L L can be interpreted as quasi-particles, or anyons, located at the sites (v 1 , f 1 ), . . . , (v n , f n ) [Kit03]. In [Meu17], Meusburger characterizes the protected space H pr by its endomorphism algebra, a subalgebra of the algebra End C (H ⊗E ) of endomorphisms of the extended space. The latter is described by the Heisenberg double H(H): equips H(H) op⊗E with the structure of a D(H)-right module algebra.
As the protected space H pr is a topological invariant, so is the algebra End C (H pr ) of its endomorphisms. It is a subalgebra of the algebra which we identify with a subalgebra of End C (H ⊗E ) using Lemma 2.39. More specifically, for each using the Haar integrals l ∈ H, η ∈ H * . Then one has: inv is an algebra homomorphism with image End C (H pr ).

Poisson analogues of Kitaev models
In this chapter we define Poisson analogues of Kitaev models by exchanging the data from the representation theory of a Hopf algebra with Poisson-Lie counterparts, that is, certain Poisson G-spaces. Our goals are: (i) to give a Poisson analogue of Kitaev models that has close structural similarities to (quantum) Kitaev models and (ii) to show that this Poisson analogue is related to the moduli space of flat G-bundles for a surface with boundary.

Kitaev models with Poisson-geometric data
To construct an analogue of Kitaev models, we replace the Hopf algebraic data in the Kitaev model by Poisson geometrical counterparts as in Table 1 We will now define counterparts of the triangle operators and the vertex and face operators as functions on the Poisson manifold K with values in G. The triangle operators L k ± and T α ± are replaced by the projections π ± : G → G ± ⊂ G from Lemma 2.23. For this we introduce a holonomy functor on the graph groupoid G(Γ D ) (Definition 2.1 (ii)) of the thickened graph Γ D (Definition 2.5). It is defined similarly to the functor Hol FR from (14). Consider the set of smooth maps C ∞ (K, G) as a groupoid with a single object with composition given by pointwise multiplication.
Definition 3.2. The holonomy functor Hol : for all edges e ∈ E , where π e : K → G stands for the projection on the component associated with the edge e, and η : G → G is the inversion map.

Because of the factorization
= π e also holds. In fact, π e can be computed from any holonomy along an edge end and adjacent edge side of e. The functor Hol is an analogue of the Hopf algebra valued holonomy functor in [Meu17]. The latter is a functor G(Γ  [Kit03,BMD08,BMCA13]. It assigns to the edge ends of an edge e the triangle operators L k e± from Equation (15), indexed by k ∈ H. To its edge sides it assigns the operators T α e± with α ∈ H * . In analogy to the quantum case, the holonomy functor of a Poisson-Kitaev model assigns elements of G + to paths along edge sides and elements of the dual Poisson-Lie group G − to paths along edge ends. The edge side r(e) connects vertices of Γ, whereas the edge end f (e) connects the faces left and right of e or, equivalently, the corresponding vertices of the dual graph Γ * . We can thus view r(e) as an edge of Γ, that is decorated with an element of G + , and f (e) as the corresponding edge of the dual graph, which is labeled with an element of the dual Poisson-Lie group G − . The graphs Γ and Γ * are combined in the thickened graph Γ D . The interaction of an edge and its dual edge is described by the Heisenberg double. and I e (p) = p for all other edge sides and edge ends. Let η * e : where η e : K ′ → K is the involution that inverts the group element at the edge e ′ and leaves the elements at all other edges invariant. It satisfies Hol ′ •I e = η * e • Hol, where Hol ′ is the holonomy functor for (K ′ , Γ ′ ). By Theorem 2.16, the map η e : K ′ → K is Poisson and intertwines the two different actions ⊲, ⊲ ′ : G × H(G) → H(G) on the copy of H(G) at e ′ .
The functor I e describes a rotation of the edge e by 180 degrees. Hence, reversing an edge in Γ is the same as rotating an edge by 180°in Γ D and corresponds to inverting the associated element of G. As edge reversal is compatible with the Poisson structures of K and K ′ and with the Poisson actions ⊲, ⊲ ′ , the orientation of edges is irrelevant. All constructions introduced in the following are compatible with edge reversal. The compatibility will not be mentioned explicitly unless it is not obvious. Paths around vertices and faces play an important role in the following. We define such paths and the associated holonomies.

Definition 3.4.
(i) Consider a vertex v ∈ V and let i 1 < · · · < i n be the linearly ordered edge ends incident at v. For k = 0, . . . , n we define the partial vertex path where  Under the holonomy functor from [Meu17] the vertex and face operators A k v , B α f from Equations (16) and (17) correspond to paths around the vertex v and the face f , respectively. This allows us to define Poisson counterparts by applying our holonomy functor (21) instead.
Definition 3.5. A function a ∈ C ∞ (K, R) of the form a = g • Hol v for some v ∈ V and g ∈ C ∞ (G, R) is called a vertex operator. Likewise, a function of the form b = g • Hol f for an f ∈ F is called a face operator.
Note that both vertex and face paths turn clockwise around the associated vertex or face. Vertex and face operators of quantum Kitaev models (Equations (16) and (17)) are also defined in a clockwise order. We define the notion of flatness of a vertex or face by requiring the associated holonomies to be trivial.
. Denote the subsets of elements flat at v (respectively f ) by and the set of flat elements on a subset L ⊆ V∪F by An element γ ∈ K L for L = ∅ is called flat at L.
Note that flatness at a vertex or face only depends on the respective cyclic ordering because vertex and face paths for different linear orderings are related by cyclic permutations. otherwise.

Definition 3.7. (i) The vertex action associated with the vertex v is the map
(ii) The face action for the face f is given by Remark 3.8 (Relation to dressing actions). If the incident edges e 1 < · · · < e m at a vertex v are all incoming, the action ⊲ v can be written more simply as Likewise, for a face f where all adjacent edges e 1 < · · · < e n are ordered clockwise one has: The Theorem 3.14]. Therefore, the factor π − (d k (γ)x −1 ) in the first line of (30) is given by the dressing action of the partial face holonomy d k (γ) on x −1 . Likewise, Equation (29) involves the dressing action of G − on G + .
Proof. Direct computation using the computation rules for π ± : G → G ± from Lemma 2.26.
That ⊲ v , ⊲ f are also Poisson maps will be shown in Proposition 3.18.  (i) Let v ∈ V, f ∈ F and L ⊂ V∪F . We set where K L is the set of elements flat at L from (24). For L = ∅ we simply write, respectively, We can identify elements of A(Γ, L) with functions on the space K L , which we interpret as a Poisson-Kitaev model with excitations at the non-flat vertices and faces. The set A(Γ, L) of invariant functions on K L takes the role of the algebra of endomorphisms on the subspace L L ⊆ H ⊗E from (19) that is associated to a quantum Kitaev model with excitations. The protected space H pr from Equation (18) satisfies H pr = L L ′ for L ′ = V∪F . By Proposition 2.41, the endomorphism algebra of L L ′ is obtained from the algebra of invariants H(H) ⊗E inv by applying the projector Q f lat . Analogously, the set A(Γ, L) is obtained from C ∞ (K, R) inv L by restriction to the subset K L of elements flat at L. As counterpart of the endomorphism algebra of L L it should form a Poisson algebra. We will show this in Proposition 3.21 and prove in Theorem 3.30 that it is isomorphic to the moduli space of flat G-bundles for the surface obtained by gluing disks to the faces in L and annuli to all other faces.

Graph transformations
In this section we present certain transformations between ribbon graphs and associate to them Poisson maps that relate the Poisson-Kitaev model (K, Γ) with the model (K ′ , Γ ′ ) on the transformed graph Γ ′ . These transformations do not change the homeomorphism class of the surface with boundary associated to Γ. The associated Poisson maps induce bijections between the sets A(Γ, L) and A(Γ ′ , L ′ ).
First we relate graphs that only differ by a cyclic permutation of the linear ordering at a vertex or face. It turns out that the sets of invariant functions C ∞ (K, R) v L with respect to different orderings at the vertex v coincide if v ∈ L, and an analogous statement holds for faces. Consider a vertex v ∈ V with ordered edge ends i 1 < · · · < i m . Denote by ⊲ v : G + × K → K the vertex action at v from Equation (27) and denote by ⊲ ′ v : G + × K → K the vertex action for the cyclically permuted linear ordering i 2 < · · · < i m < i 1 . Similarly, consider the face action

Lemma 3.11 (Vertex and face actions for cyclically permuted orderings).
(i) Let p 1 (v) be the partial vertex path for the ordering i 1 < · · · < i m . One has: (ii) Similarly, let p 1 (f ) be the partial face path for the ordering i ′ 1 < · · · < i ′ n . We obtain:  (22) in the transformed ordering. The proof for Statement (ii) is analogous and (iii) is a direct consequence of (i) and (ii).
In the following, denote by E ′ , V ′ , F ′ the sets of edges, vertices and faces, respectively, of the graph Γ ′ obtained from a graph transformation. Let K ′ = H(G) ×E ′ be the associated product Poisson manifold and Hol ′ the holonomy functor for the Poisson-Kitaev model (K ′ , Γ ′ ). Next, we transform Γ by gluing two distinct edges e 1 , e 2 along a bivalent vertex v m . We require that there are no faces f ∈ F with v(f ) = v m . Orient e 1 , e 2 so that t(e 1 ) = v m = s(e 2 ) and suppose that the ordering at v m is given by f (e 1 ) < b(e 2 ), as shown in Figure 3.3. In the transformed graph Γ ′ replace the edges e 1 , e 2 by an edge e ′ with s(e ′ ) = s(e 1 ) and t(e ′ ) = t(e 2 ). The ordering of the transformed faces left and right of e ′ is obtained by replacing the consecutive edge sides r(e 1 ) < r(e 2 ) by r(e ′ ) and l(e 2 ) < l(e 1 ) by l(e ′ ).
e ′ e 2 e 1 There is an analogous gluing transformation for faces. Let f m ∈ F be a face with exactly two distinct adjacent edges e 1 , e 2 . We require that there is no vertex Choose the cilium of f m such that the associated face path is given by p(f m ) = l(e 2 ) −1 • r(e 1 ). This implies that t(e 1 ) = t(e 2 ), s(e 1 ) = s(e 2 ) and that the edge sides are ordered r(e 1 ) < l(e 2 ) as in Figure 3.4. In the transformed graph Γ ′ replace the edges e 1 , e 2 by a single edge e ′ : s(e 1 ) → t(e 1 ). Choose the natural ordering at the faces and vertices adjacent to e ′ by replacing edge ends and sides of e 1 , e 2 by the edge ends and sides of e ′ . The associated map ψ fm : K → K ′ assigns to e ′ the holonomy along f (e 1 ) • f (e 2 ) • r(e 2 ): (π e • ψ fm )(γ) := π − (π e 1 (γ)) π e 2 (γ) for e = e ′ π e (γ) otherwise.
As a face with just two edge sides corresponds to a bivalent vertex in the dual graph Γ * , this transformation can be viewed as gluing two edges along a bivalent vertex in Γ * (up to exchanging the roles of G − and G + ).
(iii) For γ ∈ K l the map ψ l intertwines vertex and face actions: (iv) For γ ∈ K l the map ψ l intertwines vertex and face holonomies: (v) The map ψ l preserves flatness: one has ψ l (K L ) = K ′ L ′ for all L ′ ⊆ V ′∪ F ′ and L := L ′ ∪ {l}.
Proof. Statement (i) follows from Lemma 2.23 and Theorem 2.16 (ii). Statement (ii) is a direct computation using Lemma 2.26. We prove (iii) and (iv) for the map ψ vm as the proof for ψ fm is completely analogous. Note that the vertex holonomy at t(e 2 ) = t(e ′ ) and the face holonomy to the right of e ′ are invariant under ψ vm per construction. To show (iv), it remains to show that the holonomy functor Hol ′ for Γ ′ D satisfies as this is equivalent to saying that the vertex holonomy at s(e ′ ) = s(e 1 ) and the face holonomy to the left of e 1 and e 2 are invariant under ψ vm . Direct computation using Lemma 2.26 shows that Equation (35) is equivalent to γ ∈ K vm , which proves (iv). Equation (35) in particular implies that for γ ∈ K vm the partial vertex and face paths from Equations (22) and (23) are invariant under ψ vm . Therefore, so are the elements c k (γ) and d k (γ) in Equations (25) and (26), that are used to define vertex and face actions. This implies (iii). It remains to prove Statement (v). That ψ l (K L ) ⊆ K ′ L ′ follows from Statement (iv). To show that ψ l (K L ) ⊇ K ′ L ′ , we construct a right inverse φ l : K ′ → K to ψ l . First let l = v m . As illustrated in Figure 3.5, the map φ vm is defined by: otherwise.
(36) Figure 3.5: The map φ vm corresponds to creating a flat bivalent vertex.
In the proof of (iii), we first consider the case l = v m . By Lemma 3.12 (v), one has ψ vm ( To see that it is bijective, we consider the right inverse φ vm : K ′ → K of ψ vm from Equation (36). A direct computation using Lemma 2.26 shows that φ vm (K ′ L ′ ) ⊆ K L and therefore the map Set h ′ := h • φ vm . The map φ vm is no left inverse of ψ vm . However, for γ ∈ K vm the map φ vm • ψ vm can be expressed by a vertex action at v m : We therefore obtain for which is (38). It remains to show that h ′ ∈ C ∞ (K ′ , R) inv L ′ . A direct computation using Lemma 2.26 shows that Together with the fact that It is easy to see that β ⊲ v γ ∈ K vm , so that Equation (39) can be applied to obtain: One can check directly using Lemma 2.26 that the actions ⊲ v and ⊲ vm commute. This implies: A computation using the properties of π ± from Lemma 2.26 shows that γ ∈ K L implies α⊲ vm γ ∈ K L for all α ∈ G + . We use the invariance of h ∈ C ∞ (K, R) inv L under ⊲ v and ⊲ vm on K L to obtain For the gluing transformation of a face l = f m one proceeds in the same way using the right inverse φ fm from Equation (37). Again, φ fm is not a left inverse to ψ fm , but for γ ∈ K fm it satisfies (φ fm • ψ fm )(γ) = x(γ) ⊲ fm γ with x(γ) = π − (π + (π e 1 (γ)) −1 π e 2 (γ)) −1 .
Proposition 3.13 allows us to relate a Poisson-Kitaev model with one on a paired doubly ciliated ribbon graph (Definition 2.7 (iv)).
Corollary 3.14 (Transformation into a paired graph). Let (K, Γ) be a Poisson-Kitaev model and The surfaces associated to Γ and Γ ′ by gluing annuli to f 1 , . . . , f n and f ′ 1 , . . . , f ′ n , respectively, and disks to all other faces are homeomorphic.
Proof. To obtain the paired graph Γ ′ , we perform the following steps, as illustrated in Figure 3.7: 1. Split every loop into two edges by creating a bivalent vertex. The resulting graph Γ ′ has no loops because of step 1.
Step 2. ensures that the faces left and right of every edge differ. The selected sites (v 1 , f 1 ), . . . , (v n , f n ) are left intact and become the sites (v ′ 1 , f ′ 1 ), . . . , (v ′ n , f ′ n ) of the transformed graph. Steps 3. and 4. guarantee that all remaining vertices and faces can be combined into sites, so that Γ ′ is a paired graph.
The map ψ : K ′ → K is the map associated to gluing the pairs of edges created in Steps 1. through 4. back together in reverse order. It is Poisson by Lemma 3.12, which proves (i). Statement (ii) follows from Proposition 3.13 (iii). We obtain the surface associated to Γ ′ by gluing annuli to f ′ i = f i for all i and disks to all other faces, which includes the faces created in Steps 2. and 3. Doubling an edge and gluing a disk to the new face f m and an annulus or disk to an adjacent face f yields a homeomorphic topological space as only gluing an annulus or disk to f . Therefore, the surfaces obtained from Γ and Γ ′ by gluing annuli to f 1 , . . . , f n and disks to all other faces are homeomorphic, which proves (iii).

Invariance and flatness
In this section, we show how the vertex and face actions from Definition 3.7 interact with the vertex and face holonomies Hol v , Hol f from Definition 3.4. We prove commutation relations of vertex and face actions that are Poisson counterparts of the commutation relations for quantum vertex and face operators in Lemma 2.36. We then prove that the Poisson bracket on K induces a Poisson algebra structure on the set A(Γ, L) from Definition 3.10 (ii).

the associated actions and holonomies satisfy:
Hol

see Definition 2.7) we have:
Hol (iii) For each site (v, f ) we obtain: Proof. (i) and (ii): Direct computations using Lemma 2.26. For (iii), one can first assume that there are no loops at v and that every edge is traversed at most once by the face path p(f ). In this case, (iii) follows from a computation using Lemma 2.26. In the general case, we transform the graph Γ by introducing a bivalent vertex on every loop of v and doubling every edge of f that occurs twice in the face path p(f ), similar to Steps 1. and 2. in the proof of Corollary 3.14. Write Γ ′ for the resulting graph and let L ′ be the set of vertices and faces of Γ ′ that have been created by this procedure. From Lemma 3.12 we obtain a corresponding Poisson map ψ : K ′ → K that implements gluing the new pairs of edges back together. Denote by Hol ′(v,f ) the combined holonomy from Definition 3.4 (iii) for the site (v, f ) in Γ ′ . We use Lemma 3.12 (iii) and (iv) to compute for γ ′ ∈ K ′ L ′ , x ∈ G − , α ∈ G + : In these equations we used that which follows from a direct computation using Lemma 2.26. As Γ ′ has no loops at v and no edge of f occurs twice in its face path, we have already proven Statement (iii) for Γ ′ . So we can apply it to obtain: As ψ| K ′ L ′ : K ′ L ′ → K is surjective by Lemma 3.12 (v), this concludes the proof.

Lemma 3.16 (Stability of K L under vertex and face actions). Let
, the subspace K L is stable under the vertex action ⊲ v (face action ⊲ f ) from Definition 3.7: Proof. We only show the statement for the vertex action ⊲ v as the proof for ⊲ f is analogous. First we consider the case v ∈ V ∩ L. We prove ) is a site, this follows from Equation (40) and the properties of the projection π − in Lemma 2.26 because Hol v = π − • Hol (v,f (v)) . Otherwise, first cyclically permute the ordering of f (v) so that (v, f (v)) becomes a site; then Equation (40) holds with respect to the new ordering.
Then v(f ′ ) = v and f (v) = f ′ , so that Lemma 3.15 (ii) applies and we have G To translate this into a statement for the graph Γ, let γ ∈ K f ′ . By Lemma 3.12 (v) there is a m , so we can apply (34) to the right hand side to obtain For the graph Γ ′ we have already shown that G + ⊲ v K ′ f ′ = K ′ f ′ , so the last term is equal to 1 G + and we obtain G We argue as before by shifting the ordering of f ′ (after potentially creating a bivalent vertex adjacent to Next, we prove that the actions ⊲ v , ⊲ f for a vertex v and face f are Poisson maps. For this we use the graph transformations from Section 3.2. To be able to do so, we first need a lemma that shows that the restriction of the Poisson bracket {g, h} to K L depends only on g| K L and h| K L if g, h ∈ C ∞ (K, R) l L for all l ∈ L. This result will also be used in Proposition 3.21 to show that A(Γ, L) is a Poisson algebra.
To ensure that K L is a submanifold, we require that Γ is connected and that there is a pair of a vertex and adjacent face that are not in L ⊆ V∪F . Under these assumptions, we have: (i) The set K L ⊆ K is a submanifold.
(ii) Let g, g ′ , h ∈ l∈L C ∞ (K, R) l L with g| K L = g ′ | K L . The Poisson bracket satisfies: Proof. As a smooth manifold one has K ∼ = G ×E − × G ×E + . We prove (i) by constructing an injective immersion φ : G Furthermore, we can implement flatness for all vertices in L ∩ V done,i+1 and faces in L ∩ F done,i+1 by parametrizing the G − -or G + -components of the copies of G associated to some edges in E done,i+1 . Because Γ is connected, we can repeat this procedure until E done,i+1 = E. We obtain a parametrization of K L in terms of G where the set E − (E + ) consists of the edges that have not been used to implement flatness at a vertex (face). This proves (i). To prove (ii), note that for γ ∈ K L the tangent space T γ K L is given by We show that for all v ∈ V ∩ L, f ∈ F ∩ L and h ∈ l∈L C ∞ (K, R) l L the following equations hold for the Poisson bivector w K of K: Assume that v ∈ V ∩ L is a vertex without loops and that the edges e 1 < · · · < e m incident at v are incoming. Set d i := π e i (γ) for i = 1, . . . , m. For γ ∈ K L the map G + → R, α → h(α ⊲ v γ) is constant. Hence, we obtain for all elements β ∈ g + , where V (β) ∈ Γ(T K) is the vector field that generates the action ⊲ v . Define for e ∈ E the inclusion map ι e (γ) : By differentiating the map (− ⊲ v γ) in Equation (29) we obtain an explicit expression for V (β) where Consider the Poisson bivector w H(G) from (3). The bivector of the product Poisson manifold K = H(G) ×E is given by w K (γ) = e∈E T (ι e (γ)) ⊗2 w H(G) (π e (γ)). The map Hol v : K → G depends only on the copies of H(G) associated with the edges e 1 , . . . , e m , so that Use Equation (7) from Lemma 2.27 (i) to obtain: In the case with loops one proceeds in the same way, with the exceptions that loops occur twice in the holonomy Hol v and one obtains two associated contributions to the vector field V (β) in (43). The proof of the statement (T Hol f ⊗dh) w K (γ) = 0 for γ ∈ K L , f ∈ F ∩ L is analogous to that for vertex holonomies.
Proposition 3.18. The maps ⊲ v : Proof. First let v ∈ V be a vertex without loops and assume that the linearly ordered edges e 1 < · · · < e n are all incoming at v. Denote the product Poisson bivector of G + × K by w π and set d i := π e i (γ). To see that ⊲ v : G + × K → K is Poisson, first note that the maps for the individual edges from Equation (29) are Poisson as a consequence of Lemma 2.23 and Theorem 2.16 (ii). This implies where w K is the Poisson bivector of the product Poisson manifold K = H(G) ×E . It remains to show that (T π e i ⊗ T π e j ) • T (⊲ v ) ⊗2 w π = 0 for i = j. Let i < j and define with the partial vertex path p k (v) from (22). This is the value of the partial vertex holonomy from the (k − 1)-th to the l-th edge end and satisfies c 1,k (γ) = c k (γ). We compute: The first term here is obtained from the contribution of G + to w π (α, d), while the other terms are associated with the edges e 1 , . . . , e i . As G + ⊆ G is a Poisson-Lie subgroup of the quasi-triangular Poisson-Lie group G, we obtain the Poisson bivector w G + of G + from Equation (2). The Adinvariance of the symmetric component r s of the classical r-matrix of G implies: Lemma 2.26 (ii) implies that T (π + • R g ) y = 0 ∀ y ∈ g − , g ∈ G. Applying this to the first term of (45) and Equation (7) from Lemma 2.27 (i) to the other terms yields: Using r 21 ∈ g + ⊗ g − and Lemma 2.26 we can simplify this expression to: Let k ∈ {1, . . . , i − 1}. The equation g π − (d k ) = π − (g π − (d k )) π + (g π − (d k )) ∀ g ∈ G implies and applying Equation (8) from Lemma 2.27 (ii) to the second term yields: Apply the map T L ⊗2 Thus, for the sum in (46) we obtain Insert this for the sum on the right hand side of (46) to obtain Now we relax the assumption that v is a vertex without loops. So let v ∈ V be a general vertex and Γ ′ the graph obtained from introducing a bivalent vertex on every loop at v. Consider the Poisson map ψ : K ′ → K obtained from Lemma 3.12 that corresponds to gluing the split loops back together. Denote the set of newly introduced vertices by where we used the Poisson property of ψ in the second equation. From Lemma 3.12 (iii) one obtains Let We can therefore apply Lemma 3.17 (ii) to substitute g i for g ′ i in (48). We obtain where we used that ⊲ v : G + × K ′ → K ′ and ψ : K ′ → K are Poisson in the third equation and Lemma 3.12 (iii) in the fourth. Because ψ| K ′ L ′ : K ′ L ′ → K is surjective by Lemma 3.12 (v), this implies that ⊲ v : G + × K → K is Poisson. The proof for ⊲ f : G − × K → K is analogous.
In particular, Proposition 3.18 shows that K is a Poisson G + -space with respect to every vertex action and a Poisson G − -space for every face action. The maps ⊲ : We show this for the vertex holonomy Hol v as the proof for Hol f is analogous. Assume that (v, f (v)) is a site. Then this follows from Equation (40) Proof. We only prove (iii) as the other statements are shown similarly. First note that the map (π + × π − ) • ∆ : G → G + × G − is Poisson, where ∆ : G → G × G is the diagonal map. This is shown by a computation using Lemma 2.26 and the fact that the maps π ± : G → G ± are Poisson (Lemma 2.23). This implies that the map We show that ⊲ (v,f ) is a group action, that is Because g i = π − (g i )π + (g i ), i = 1, 2, it suffices to show this for g i ∈ G + ∪ G − . In the cases g 1 , g 2 ∈ G − and g 1 , g 2 ∈ G + Equation (50) follows from the fact that ⊲ v , ⊲ f are group actions. For g 1 ∈ G − , g 2 ∈ G + it follows from the definition of ⊲ (v,f ) in (49). It thus remains to show If there are no loops at v and at most one edge side of any edge belongs to f , then Equation (51) follows from a direct computation using the properties of π ± from Lemma 2.26. In the general case, transform the graph Γ by adding a bivalent vertex on each loop of v and doubling every edge whose left and right face is f . Denote by (K ′ , Γ ′ ) the Poisson-Kitaev model for the transformed graph Γ ′ and by L ′ ⊆ V ′∪ F ′ the set of vertices and edges created by this procedure. From Lemma 3.12 we obtain the map ψ : K ′ → K that corresponds to gluing back together the edges that we split into two. By the surjectivity of ψ (Lemma 3.12 (v)), for γ ∈ K there is a γ ′ ∈ K ′ L ′ with γ = ψ(γ ′ ). Equation (51) holds for elements of K ′ as Γ ′ satisfies our previous assumptions. We compute: While applying Equation (33), we used G We show that the set A(Γ, L) of invariant functions on K L from Definition 3.10 (ii) inherits a Poisson bracket from C ∞ (K, R). For this we use Lemma 3.17, which requires that the graph Γ is connected and that there is a vertex and adjacent face not contained in L. We require the vertices and faces without a flatness condition to be paired into sites. Proof. Let g 1 , g 2 ∈ C ∞ (K, R) inv L and v ∈ V . For α ∈ G + , γ ∈ K L we obtain from Proposition 3.18 where we used that g i • (− ⊲ v γ) is constant in the third equation. For i = 1, 2 the identity ∈ L. Therefore, the maps g i • (α ⊲ v −) and g i coincide on all elements of the form α ⊲ v ′ γ with α ∈ G + , γ ∈ K L . Likewise, they coincide on elements of the form We can thus apply Lemma 3.17 (ii) to (52) and obtain To prove Statement (ii), it remains to show that the induced Poisson bracket on A(Γ, L) is welldefined. This follows from Lemma 3.17 (ii).

Vertex and face operators
We show an analogue of the commutation relations for vertex and face operators from Lemma 2.36 for their Poisson counterparts from Definition 3.5. Then we prove that the derivation {g, −} for a vertex or face operator g is related to the vector fields that generate the respective vertex or face action. First we need two technical lemmas. Consider two different edges e 1 , e 2 incident at a vertex v such that t(e 1 ) = v = s(e 2 ) and i s (e 2 ) comes directly after i t (e 1 ) in the ordering of the edge ends at v.
(If e 1 , e 2 are no loops, this simply means that e 1 is incoming, e 2 outgoing and that e 2 is the next edge after e 1 .) Consider the paths p 1 := r e 2 • r e 1 , p 2 := f e 1 • b −1 e 2 . These are segments of a face path and the vertex path p(v) (see Definition 3.4) that meet at v. Lemma 3.22. For these paths and all f 1 , f 2 ∈ C ∞ (G) the following equation holds: Proof. Set d i := π e i (γ), i = 1, 2. The holonomies along p 1 and p 2 are given by (see Equation (21)): Using the inclusion map ι e : G → K from Equation (42), the Poisson bivector of the product Poisson manifold K = H(G) ×E can be written as w K (γ) = e∈E T ι ⊗2 e w H(G) (π e (γ)) with the Heisenberg double bivector w H(G) from (3). In the Poisson bracket (53) only the terms associated with e 1 and e 2 contribute. We compute these terms. Let η : G → G be the inversion map. We obtain where the first and second term is obtained from the contribution to w K associated with e 1 and e 2 , respectively. Apply Equation (6) from Lemma 2.27 (i) to obtain: The computation rules for π ± from Lemma 2.26 together with the fact r ∈ g − ⊗ g + imply: 2 ) ) r 21 2 ) ) r 21 = 0 .
Lemma 3.23. For any edge e ∈ E and f 1 , f 2 ∈ C ∞ (G, R) the holonomies along opposite edge ends or edge sides commute: Proof. This follows from T π − ⊗ T (π − • η) w H(G) = T π + ⊗ T (π + • η) w H(G) = 0, which can be shown by a direct computation using Lemma 2.26 and the fact that r ∈ g − ⊗ g + . Now we are ready to prove an analogue of Lemma 2.36 for Poisson-Kitaev models: Proposition 3.24 (Commutation relations for vertex and face operators). Let g 1 , g 2 ∈ C ∞ (G, R).
(i) Vertex (face) operators for distinct vertices v 1 = v 2 (faces f 1 = f 2 ) Poisson commute: (ii) For a vertex v and face f with v(f ) = v and f (v) = f the associated operators commute:

. The combined face and vertex holonomy from Definition 3.4 (iii) induces an injective homomorphism of Poisson algebras
with regard to the Poisson bivector w G * on G from Equation (5).
Proof. We prove the statement for vertex operators in (i). Note that only those copies of G associated with edges that connect v 1 with v 2 contribute to the Poisson bracket For any such edge e the map g 1 • Hol v 1 depends on Hol(f (e)) and g 2 • Hol v 2 on Hol(b(e)) or vice versa. Lemma 3.23 implies that the contribution to {g 1 • Hol v 1 , g 2 • Hol v 2 } associated with e vanishes. The proof for face operators is analogous. To see (ii), first assume that there are no loops at v and that every edge is traversed at most once by the face path p(f ). Then edges with non-vanishing contribution to the Poisson bracket can be grouped into pairs e 1 = e 2 ∈ E of consecutive edges on the face path p(f ). With appropriate choice of orientation we have t(e 1 ) = s(e 2 ) = v. Then one argument of the Poisson bracket is a function of Hol(r e 2 • r e 1 ) and the other of Hol(f e 1 • b −1 e 2 ). By Lemma 3.22 any such contribution vanishes. In the general case, we transform the graph Γ by adding a bivalent vertex on every loop at v and doubling any edge that occurs twice in the face path p(f ). Denote by (K ′ , Γ ′ ) the Poisson-Kitaev model for the transformed graph and the set of new vertices and faces by L ′ ⊆ V ′∪ F ′ . From Lemma 3.12 we obtain a Poisson map ψ : K ′ → K that corresponds to gluing the split edges back together. The map ψ is invariant under actions at l ∈ L ′ and satisfies by Equation (34): By Lemma 3.16, the subspace K ′ L ′ is stable under the actions ⊲ l ′ for l ′ ∈ L ′ which implies that Hol ′v , Hol ′f ∈ l ′ ∈L ′ C ∞ (K, R) l ′ L ′ . Use the Poisson property of ψ and Lemma 3.17 (ii) to compute where we used Statement (ii) for the transformed graph Γ ′ . The map ψ| K ′ L ′ is surjective by Lemma 3.12 (v), so this concludes the proof of (ii). Now we prove (iii). The map Hol (v,f ) is surjective, which implies the injectivity of Hol (v,f ) * . It remains to show that Hol (v,f ) : K → (G, w G * ) is a Poisson map. Let (v, f ) be a site and consider first the simpler case where there are no loops at v and no edges whose left and right sides belong both to f . Applying (T Hol (v,f ) ) ⊗2 to w K yields: We can assume that the only two edges that occur both in the vertex path p(v) and the face path p(f ) are the first and last edge in the ordering of v (which coincide with the first and last edge of f ). To see this, recall that all other such edges form pairs (e 1 , e 2 ) where e 2 comes directly after e 1 on both p(f ) and p(v). By Lemma 3.22, the pair (e 1 , e 2 ) does not contribute to (55) and (56). Denote by f 1 < · · · < f m the edges of f and the edges incident at v by v 1 < · · · < v n . We can assume without restriction of generality that all v 1 , . . . , v n are incoming at v and that f 2 , . . . , f m−1 are oriented clockwise. This implies that f 1 = v 1 is oriented counterclockwise and f m = v n clockwise. Denote the elements associated to the edges by d v i := π v i (γ) and d f j := π f j (γ). Then is Poisson. Consider the n-fold product H(G) n equipped with the product Poisson structure. We can write H = h • ∆ 13 with the maps ∆ 13 : where for i = 1, 2, 3 the map ι i : G → G 3 is the inclusion of G into the i-th copy of G 3 at the element (d 1 , d 2 , d 1 ). One has (T ι ⊗2 Denote the inversion map on G by η. We have Lemma 2.27 (i), the computation rules for π ± from Lemma 2.26, and r ∈ g − ⊗ g + imply: Inserting this into (58) and comparing with w G * from Equation (5) proves the claim: (H(d 1 , d 2 )) .
In the general case, we again transform Γ by adding bivalent vertices to loops at v and doubling edges where both sides belong to f . From Lemma 3.12 we obtain the map ψ : K ′ → K that is associated with gluing the split edges back together. Then we argue as in the proof of (ii). By Equation (20), the D(H)-right module algebra structure at a site (v, f ) is obtained from the vertex and face operators for v and f , respectively. We show an analogue of this relation by expressing the derivation {g, −} for a vertex or face operator g in terms of the vector fields obtained from the respective vertex or face action. For β ∈ g + , y ∈ g − the vector fields generating the vertex action at v and face action at f are given by Proposition 3.26 (Vector fields for operators and actions.). Let v ∈ V, f ∈ F and g ∈ C ∞ (G, R). The vector fields generated by the vertex operator g • Hol v and face operator g • Hol f satisfy where r ∈ g − ⊗ g + is the classical r-matrix of the double Poisson-Lie group G.
Proof. Equation (59) has been shown in Equation (44) If G is connected, then Equation (61) for a function h ∈ C ∞ (K, R) in turn implies invariance under all vertex and face actions.
Proof. Equations (61) follow from Equations (59) and (60). To prove the converse statement, note that the r-matrix of G induces linear isomorphisms g * − → g + : ϕ → (ϕ ⊗ id) r and g * The argument for faces f ∈ F is similar.

Poisson-Kitaev models and Fock-Rosly spaces
In this section we show that a Poisson-Kitaev model on the graph Γ is Poisson-isomorphic to a Fock-Rosly space on Γ if all vertices and faces can be paired into sites. This isomorphism will be used in Section 3.6 to prove that the Poisson algebra A(Γ, L) is isomorphic to the canonical Poisson algebra of functions on the moduli space of flat G-bundles. In this section we assume that Γ is a paired doubly ciliated ribbon graph (see Definition 2.7 (iv)).
We consider the Poisson-Kitaev model K = H(G) ×E for the graph Γ and the Fock-Rosly space FR = (G ×E , w FR ) from Definition 2.29. The latter is equipped with the Poisson bivector w FR from Equation (10) where we set r(v) = r for all v ∈ V and r is the classical r-matrix of G. We construct a Poisson isomorphism Φ : K → FR. It is obtained from the holonomies of certain paths associated to the edges of Γ. More specifically, we associate to an edge e : v s → v t the path p(e) in the thickening Γ D (see Definition 2.5) that starts at the cilium of v s , turns counterclockwise around v s , then along e and clockwise along v t until it meets the cilium of v t . This path is depicted in Figure 3.8. Suppose that b(e) is the j s -th edge end in the linear ordering at v s and f (e) the j t -th edge end at v t . The path p(e) is given by where p js−1 (v s ) and p jt−1 (v t ) are partial vertex paths as defined by Equation (22 This map can be viewed as the Poisson-Lie counterpart of the isomorphism of D(H)-module algebras in [Meu17] that was used to relate the endomorphism algebra End C (H ⊗E ) of a Kitaev model to the quantum moduli algebra. We show that for each site (v, f ) the map Φ : K → FR is an isomorphism of Poisson G-spaces with respect to the Poisson actions ⊲ (v,f ) : G × K → K from (49) and ⊲ FR v : G × FR → FR from (11). We also prove that it intertwines the holonomies along the combined vertex and face path p(v) • p(f ) with respect to the holonomy functors Hol from (21) and Hol FR from (14).
That Ψ is the inverse to Φ can be checked directly using the properties of the projections π ± in Lemma 2.26. It remains to show that Φ : K → FR is Poisson. First we prove this in the case that for any two vertices v, v ′ there is at most one edge connecting v and v ′ . To simplify presentation, we prove the equation for functions of the form f i =f i • π e i with edges e 1 , e 2 ∈ E andf 1 ,f 2 ∈ C ∞ (G, R). It implies T π e 1 ⊗ T π e 2 (T Φ ⊗2 w K ) = T π e 1 ⊗ T π e 2 (w FR • Φ) for all edges e 1 , e 2 and hence that Φ is Poisson. Note that a change of the orientation of an edge e amounts to an inversion of the element π e (γ) in both the Poisson-Kitaev model K and the Fock-Rosly space FR. As these are Poisson automorphisms, we can choose any edge orientation. Thus, we have to consider the following cases: (a) e 1 = e 2 . (b) The edges e 1 and e 2 have no vertex in common.

Case (b):
If the edges e 1 , e 2 have no common vertex, then {f 1 , f 2 } F R = 0 by Equation (10). The only edges e that potentially contribute to {f 1 • Φ, f 2 • Φ} K connect the edges e 1 and e 2 . Therefore, e 1 occurs in p b (e) and e 2 in p f (e) or vice versa. Then f 1 is a function of Hol(b(e)) and f 2 of Hol(f (e)) or the other way around. Lemma 3.23 implies

Case (c):
Let v := s(e 1 ) = s(e 2 ). This situation is pictured in Figure 3.10 along with the paths First we consider the case when e 1 is the first edge at v and e 2 comes directly after e 1 with respect to the linear ordering, so that p b (e 1 ) = 1 s(b(e 1 )) is the identity morphism of s(b(e 1 )) and p b (e 2 ) = b(e 1 ). Additionally, we require that p f (e 1 ) = 1 t(f (e 1 )) and p f (e 2 ) = 1 t(f (e 2 )) . Then one has (π e 1 • Φ)(γ) = π e 1 (γ) (π e 2 • Φ)(γ) = π e 2 (γ) π − (π e 1 (γ) −1 ) −1 . We can assume that there are no edges in Γ that connect t(e 1 ) with t(e 2 ) since these contribute neither to {f 1 • Φ, f 2 • Φ} K nor to {f 1 , f 2 } FR by the same arguments as in case (b). Thus, we have to prove that the following map is Poisson Here, H(G) 2 is equipped with the bivector w π of the product Poisson structure and w ′ FR is the Poisson bivector from Equation (10) for the case when the graph consists only of the two edges e 1 : v → t(e 1 ) and e 2 : v → t(e 2 ) with e 1 < e 2 at v and t(e 1 ) = t(e 2 ). Denote by π 1 , π 2 : G 2 → G the projections on the components that correspond to e 1 and e 2 . Then w ′ FR is explicitly given by and the bivector w π by We decompose h into Poisson maps. For this we write h as where ∆ : Let π ′ i : G 3 → G, i = 1, 2, 3 be the projections on the components of G 3 and ι ′ i : G → G 3 , i = 1, 2, 3 the inclusion maps at the element (∆ × id G )(d 1 , d 2 ) = (d 1 , d 1 , d 2 ). They satisfy: Denote by w ′ π be the Poisson bivector of the product Poisson manifold H(G) 3 . Then one has This implies for the map h from Equation (70): By Theorem 2.16 (ii) and Lemma 2.23 (ii) the following map is Poisson and applying this fact to the first term on the right hand side of (72) yields: Apply the projection T π ⊗2 i and use (69) to obtain: 1 , d 2 )) .
By the anti-symmetry of the Poisson bivectors w π and w ′ FR it only remains to show the following equation to prove that h is Poisson: 1 , d 2 )) .
Applying the projection T π 1 ⊗ T π 2 to Equation (73) yields where we used the identity f = η •π − •η and replaced id G by η •η in the second step. The inversion η : H(G) → H(G) is Poisson by Theorem 2.16 (ii), so that Use r 21 ∈ g + ⊗ g − and the computation rules from Lemma 2.26 to obtain This shows that h : H(G) 2 → (G 2 , w ′ FR ) is Poisson. Now consider the general case as pictured in Figure 3.10. This means that the paths p b (e 1 ), p f (e 1 ) and p f (e 2 ) are not necessarily identity morphisms and there may be edges between e 1 and e 2 in the linear ordering of v = s(e 1 ) = s(e 2 ). We can still assume that there are no edges that connect t(e 1 ) with t(e 2 ) as these do not contribute to {f 1 • Φ, f 2 • Φ} K or {f 1 , f 2 } FR by the same arguments as in case (b). We have to show that the following map is Poisson where w ′ FR is again the Fock-Rosly bivector for a graph that consists only of the edges e 1 : v → t(e 1 ) and e 2 : v → t(e 2 ) with e 1 < e 2 at v and t(e 1 ) = t(e 2 ). For this we decompose Let e s,1 < · · · < e s,m < e 1 be the edges at v = s(e 1 ) = s(e 2 ) that come before e 1 in the linear ordering at v. Denote the edges at t(e 1 ) that come before e 1 with regard to the ordering at t(e 1 ) by e t,1 < · · · < e t,n < e 1 . To the edges e s,1 , . . . , e s,m , e 1 , e 2 , e t,1 , . . . , e t,n we associate the product Poisson manifold H(G) m × H(G) 2 × H(G) n . Define the path This path is the same as p(e 2 ) from Equation (62) for the shifted linear ordering at v that corresponds to a cilium at t(b(e 1 )). The path p(e 2 ) ′ and the edges e s,1 , . . . , e s,m and e t,1 , . . . , e t,n are illustrated in Figure 3.11. The map φ 1 : K → H(G) m × H(G) 2 × H(G) n assigns the holonomy along p(e 2 ) ′ to the copy of H(G) that corresponds to e 2 : It is Poisson by Lemma 2.23 (ii) and Theorem 2.16 (ii). The map φ 2 : with h : H(G) 2 → (G 2 , w ′ FR ) from Equation (68). We have shown that h is Poisson in Equations (74) and (75). This implies that φ 2 is a Poisson map. Denote by π s : K → H(G) m the projection on the components associated with the edges e s,1 , . . . , e s,m incident at v and by π t : K → H(G) n the projection associated with the edges e t,1 , . . . , e t,n incident at t(e 1 ). Let hol b : H(G) m → G − be the map that is uniquely defined by hol b • π s = Hol(p b (e 1 )) with the path p b (e 1 ) from (62). Similarly, let hol f : H(G) n → G − be the unique map with hol f • π t = Hol(p f (e 1 )). The map φ 3 : where η G − : G − → G − is the inversion map. By Lemma 2.23 (ii) and Theorem 2.16 (ii), the maps Recall that (G 2 , w ′ FR ) is the Fock-Rosly space for the graph that has only the edges e 1 : v → t(e 1 ) and e 2 : v → t(e 2 ) with t(e 1 ) = t(e 2 ) and e 1 < e 2 in the ordering at v. Using the associated vertex actions ⊲ FR v , ⊲ FR t(e 1 ) : G × (G 2 , w ′ FR ) at v and t(e 1 ) from Equation (11), we can write d 1 , d 2 )) .
The actions ⊲ FR v , ⊲ t(e 1 ) are Poisson by Proposition 2.28, and therefore so is φ 4 . One can check easily that Φ ′ = φ 4 • φ 3 • φ 2 • φ 1 by inserting the definitions of φ i , i = 1, . . . , 4 from Equations (76), (77), (78) and (79). Hence, the map Φ ′ is Poisson. Now consider the general case, where two vertices of Γ may be connected by more than one edge. We transform the graph Γ by introducing a bivalent vertex v m on every edge e. This splits e into the two edges e 1 : s(e) → v m and e 2 : v m → t(e). Choose the linear ordering at v m such that e 2 > e 1 . Denote the transformed Poisson-Kitaev model by (K ′ , Γ ′ ). From Lemma 3.12 we obtain a Poisson map ψ : K ′ → K that corresponds to gluing the split edges back together. The resulting graph Γ ′ is not paired anymore. However, the map Φ from Equation (63) can be defined for any doubly ciliated ribbon graph. So far our proof for its Poisson property did not require a paired graph, either. Denote by FR ′ the Fock-Rosly space associated to Γ ′ . As any two vertices of Γ ′ are connected at most by one edge, we thus obtain the Poisson map Φ ′ : K ′ → FR ′ defined by (63). There is a map ψ FR : FR ′ → FR that implements gluing the split edges of Γ ′ back together given by Equation (12). It is Poisson by Proposition 2.31. A direct computation shows that Φ•ψ = ψ FR •Φ ′ . Because the maps ψ, ψ FR and Φ ′ are Poisson and ψ is surjective by Lemma 3.12 (v), this proves that Φ : K → FR is Poisson.

Statement (ii): Equation (64) is equivalent to the equations
As Γ is paired, there are no loops at v and we can assume that all edges are incoming. Then ⊲ v is given by Formula (29) and Equation (80) can be checked directly using Lemma 2.26. To see (81), use the fact that there are no edges that occur twice in the face path p(f ) to assume that all edges of f are oriented clockwise. Then ⊲ f is given by (30) and (81) follows from the properties of π ± in Lemma 2.26.

Statement (iii):
This follows from a direct computation using Lemma 2.26. Most of our constructions for Poisson-Kitaev models are based on the properties of the projections π ± from Lemma 2.23. Therefore, one could generalize Poisson-Kitaev models to (general) double Poisson-Lie groups G, but many definitions, such as the holonomy functor or vertex and face actions, would have to be replaced by local versions.

Relation with moduli spaces of flat G-bundles for surfaces with boundary
In Section 3.2, we introduced graph transformations that allow us to transform the graph Γ of a Poisson-Kitaev model (K, Γ) into a paired doubly-ciliated ribbon graph Γ ′ . By Theorem 3.28, the Poisson manifold K ′ associated with the paired graph Γ ′ is isomorphic to the Fock-Rosly space FR ′ on Γ ′ . In [FR99], Fock and Rosly have shown that the Poisson structure on FR ′ induces the canonical Poisson structure on the moduli space of flat G-bundles Hom(π 1 (S ′ ), G)/G for the surface S ′ obtained by gluing annuli to the faces of Γ ′ . We use this to relate the Poisson algebra A(Γ, L) from Definition 3.10 (ii) to moduli spaces of flat G-bundles. Recall from Proposition 2.30 that the subalgebra of invariant functions on FR ′ is isomorphic to the canonical Poisson algebra of functions on Hom(π 1 (S ′ ), G)/G defined by the non-degenerate symmetric component r s of the classical r-matrix of G. In our situation, r is the classical r-matrix of the double Lie bialgebra g from Proposition 2.19. Let Γ be a connected doubly-ciliated ribbon graph and choose a set of sites (v 1 , f 1 ) Consider the set of functions invariant on FR ′ L ′ : Similarly to A(Γ ′ , L ′ ) from Definition 3.10 (ii), we define the quotient with respect to the relation h 1 ∼ h 2 ⇔ h 1 | FR ′ L ′ = h 2 | FR ′ L ′ . Note that Hol FR (p(f )) = Hol FR (p(v) • p(f )) for each site (v, f ) (see Equation (14)). By Equation (65), one has Φ(K ′ L ′ ) = FR ′ L ′ for the set K ′ L ′ from Definition 3.6. At each site (v, f ) the map Φ intertwines the G-actions ⊲ (v,f ) and ⊲ FR v (Equation (64)). Therefore, Φ induces a bijection Φ * : One can equip A FR (Γ ′ , L ′ ) with a Poisson bracket such that the bijection Φ * /∼ becomes an isomorphism of Poisson algebras. Because Φ : K ′ → FR ′ is a Poisson isomorphism, this Poisson bracket on A FR (Γ ′ , L ′ ) is the one induced by the Poisson bracket on FR ′ . We have shown that the Poisson algebra A(Γ, L) is isomorphic to A FR (Γ ′ , L ′ ). It remains to show that A FR (Γ ′ , L ′ ) is isomorphic to the canonical Poisson algebra of functions on Hom(π 1 (S), G)/G. For this we show that A FR (Γ ′ , L ′ ) is isomorphic to the Poisson subalgebra C ∞ (FR ′′ , R) inv of invariant functions on the Fock-Rosly space FR ′′ for a suitable graph Γ ′′ , and that the surface S ′′ obtained by gluing annuli to the faces of Γ ′′ is homeomorphic to S. By Proposition 2.30, the Poisson algebra C ∞ (FR ′′ , R) inv is isomorphic to the Poisson algebra of functions on the moduli space Hom(π 1 (S ′′ ), G)/G for the bilinear form dual to r s , which then proves Theorem 3.30. We construct the graph Γ ′′ by merging the faces in F ∩ L into adjacent faces until only the selected faces f 1 , . . . , f n are left. More specifically, we transform Γ ′ as follows.
Choose a flat face f f lat,1 ∈ F ′ ∩ L ′ and choose an edge e 1 ∈ E adjacent to f f lat,1 such that the faces to the left and right of e 1 differ. Such an edge e 1 exists, since otherwise Γ ′ would have only one face or several connected components in contradiction to our assumptions. Remove the edge e 1 , thus merging the face f f lat,1 into a different, adjacent face f . Denote the transformed graph by Γ ′ 1 . The surfaces obtained by gluing a disc to Γ ′ along f f lat,1 and an annulus (disc) along f , and by gluing an annulus (disc) to Γ ′ 1 along f are homeomorphic. Thus, the surface obtained from Γ ′ By composing the right inverses ψ i we obtain a right inverse ψ : FR ′′ → FR ′ of φ. Equations (82) and (84) imply that ψ(FR ′′ ) ⊆ FR ′ L ′ . Together with Equation (85) this implies that the map ψ induces a map ψ * /∼ : A direct computation shows that the map φ i : FR ′ i−1 → FR ′ i associated with erasing the edge e i satisfies for all γ ∈ FR ′ i−1 with Hol FR (f f lat,i ) = 1 G : lat,1 , . . . , f f lat,i } .
Together with Equation (83) this implies for all i and γ ∈ FR ′ This implies ψ•φ| FR ′ L ′ = id FR ′ L ′ and the map ψ * /∼ is an inverse to φ * /∼ . This concludes the proof.
(i) Theorem 3.30 implies in particular that the Poisson algebra A(Γ, L) does not depend on the graph Γ, but only on the surface S obtained by gluing annuli to f 1 , . . . , f n and disks to all other faces. This Poisson algebra is an analogue of the endomorphism algebra of the subspace L L ⊆ H ⊗E from (19) that is associated to a quantum Kitaev model with topological excitations at the sites (v 1 , f 1 ), . . . , (v n , f n ). (ii) Alekseev and Malkin have introduced a decoupling transformation for the moduli space of flat E-bundles for a semi-simple quasi-triangular Poisson-Lie group E in [AM95]. Their construction is based on a set of generators of π 1 (S). They showed that the Poisson structure on the moduli space Hom(π 1 (S), E)/E can be obtained from the product Poisson manifold