[[ZMP-HH/20-1]Hamburger Beiträge zur Mathematik Nr. 819
Defects in Kitaev models and bicomodulealgebras
Vincent Koppen
Fachbereich Mathematik, Universität Hamburg, Germany [email protected]
We construct a Kitaev model, consisting of a Hamiltonian which is the sum ofcommuting local projectors, for surfaces with boundaries and defects of dimension0 and 1. More specifically, we show that one can consider cell decompositions ofsurfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeledby semisimple bicomodule algebras. We introduce an algebra whose representationslabel the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in theabsence of defects. In this way we generalize the algebraic structure underlyingthe standard Kitaev model without defects or boundaries, where all 1-cells and2-cells are labeled by a single Hopf algebra and where point defects are labeledby representations of its Drinfeld double. In the standard case, commuting localprojectors are constructed using the Haar integral for semisimple Hopf algebras. Acentral insight we gain in this paper is that in the presence of defects and boundaries,the suitable generalization of the Haar integral is given by the unique symmetricseparability idempotent for a semisimple (bi-)comodule algebra.
1. Introduction
The Kitaev model has been constructed as a simple model for topological quantum computing,using a degenerate ground-state space as the code space and a set of commuting local projectorsto correct local errors. It is also known as the quantum double model, surface code or toric code[Kit, BMCA]. The algebraic input datum for such a construction is, in the simplest situation,a finite-dimensional semisimple complex Hopf algebra [BMCA, M]; for the toric code it is thegroup algebra of the group with two elements. The ground states of this model are describedby a three-dimensional topological field theory of Turaev-Viro type [BK], which provides linksto quantum topology.On the other hand, it is interesting to consider such models not just on surfaces, but onsurfaces with additional structure. In terms of physics, we want to allow for defects andboundaries; in mathematical terms, we consider the theories on a suitable class of stratifiedmanifolds called defect surfaces in the sense of [FSS19], but see also e.g. [CMS]. (Here westudy models on oriented surfaces, whereas in [FSS19] surfaces with -framings are considered.)Defects in topological field theories are known to lead to higher-dimensional ground-state spacesand more interesting mapping class group representations of the underlying surfaces on these;see e.g. [BJQ, FS, LLW]. This is, in particular, relevant for applications to topological quantum1 a r X i v : . [ m a t h . QA ] J u l omputing, where quantum gates are implemented by mapping class group actions on the codespace [FLW]. There have been already several approaches to include defects or boundaries inKitaev models based on group algebras [BK, BMD, BSW, CCW], but our approach deals withthe more general case of semisimple Hopf algebras.The main result of this paper is the construction of a Kitaev-type model, consisting ofa commuting-projector Hamiltonian, for surfaces with general defects and boundaries, usinggeneral Hopf-algebraic and representation-theoretic input data.For our construction it is necessary to realize the data labeling the defects, which are knownfor Turaev-Viro theory in a category-theoretic language, concretely in Hopf-algebraic and re-presentation-theoretic terms. Specifically, topological field theories of Turaev-Viro type areparameterized by spherical fusion categories [BW2]. The data for defects separating two suchtheories are semisimple bimodule categories [KK, FSV, FSS19]. The idea for obtaining the datafor a Kitaev construction is to invoke Tannaka-Krein duality [D]. It states that a semisimpleHopf algebra is equivalent to specifying a fusion category (the representation category of theHopf algebra, admitting a canonical spherical structure) together with a monoidal fibre functorvalued in finite-dimensional vector spaces (the forgetful functor assigning to a representationits underlying vector space). This recovers semisimple Hopf algebras as the input datum forthe Kitaev models without defects, which we think of here as the labels for the two-dimensionalstrata of the defect surface.We extend this idea and employ, for the bimodule categories labelling line defects on thesurface in Turaev-Viro theory, the appropriate bimodule versions of fibre functors. By a bi-module version of Tannaka-Krein duality, which we explain in Subsection 2.1, this realizes thesecategories as the representation categories of bicomodule algebras over Hopf algebras. We thusidentify bicomodule algebras as the labels for line defects and, as a special case, comodulealgebras for boundaries.Having established the algebraic data for line defects of the surface, we turn our attention tovertices where such line defects can join. They are labeled by objects in a category which servesas possible labels for generalized Wilson lines in a corresponding three-dimensional topologicalfield theory, including boundary Wilson lines and Wilson lines at the intersection of surfacedefects. This category has been determined as a suitable generalization [FSS14, FSS19] ofthe Drinfeld center for a spherical fusion category, which labels bulk Wilson lines. Here, inSubsection 2.3, this category is realized as a representation category as follows: For a vertexat which line defects meet, the bicomodule algebras of the line defects and the algebras dualto the Hopf algebras attached to the adjacent two-dimensional strata naturally assemble intoan algebra, defined in Definition 5. This algebra, which in this paper we call vertex algebra ,generalizes the Drinfeld double of the Hopf algebra, whose representations label point-likeexcitations in the Kitaev model without defects. The category of possible labels for sucha vertex is then the category of modules over this algebra. Theorem 8, which we prove inAppendix A, states that this category is equivalent to the category of generalized Wilson linesat the intersection of surface defects in a corresponding three-dimensional field theory [FSS19].Furthermore, a choice of cell decomposition on the underlying surface enters the constructionof the Kitaev model. In the standard Kitaev model without defects, every -cell (or edge ) of thecell decomposition is labeled by a single Hopf algebra. In our setting this should be seen as theregular bicomodule algebra and we consider this label as the transparent defect. In our case,edges of the cell decomposition are either transparently labeled or they constitute a non-trivialdefect and are labeled by an arbitrary bicomodule algebra.Our construction proceeds in the following steps – mirroring the construction of the standardKitaev model without defects, as in e.g. [BMCA, BK]. We first define in Subsection 3 avector space with local degrees of freedom for each edge and each -cell (or vertex ) of the cell2ecomposition. Then we show in Subsection 3.1 that this vector space admits, locally withrespect to the cell decomposition, the structure of a bimodule over the algebras attached tothe vertices. This is analogous to the representations of the Drinfeld double for each site, apair of a vertex and an adjacent 2-cell (or plaquette ), in the standard Kitaev model withoutdefects. In this case one then proceeds to use the Haar integral for any semisimple Hopf algebrato define local projectors via these local representations. One of our main insights, establishedin Subsection 3.2, is that, in the presence of defects, the suitable generalization of the Haarintegral to semisimple bicomodule algebras is given by the symmetric separability idempotent,see Definition 15. The symmetric separability idempotent of a semisimple algebra is unique,which we recall in Proposition 17. Furthermore, we show in Proposition 19 that for a semisimple(bi-)comodule algebra, the symmetric separability idempotent satisfies a compatibility with the(bi-)comodule structure which generalizes a basic property of the Haar integral of a semisimpleHopf algebra. In the absence of defects, the symmetric separability idempotent reduces to theHaar integral, as we show in Example 18.Using such separability idempotents, in Subsection 3.3 we finally construct projectors foreach vertex, as usual called vertex operators , and for each plaquette, as usual called plaquetteoperators . Our main result, Theorem 25, is that all vertex operators and plaquette operatorscommute – giving rise to an exactly solvable Hamiltonian defined as a sum of commutingprojectors, which project to the ground states of the model.Concerning the ground states, our construction can be seen as a concrete representation-theoretic realization of the category-theoretic construction in [FSS19]. While in [FSS19] moregeneral categories than representation categories of Hopf algebras and bicomodule algebrasare considered, for us the additional structure of fibre functors on the categories is necessaryin order to define a larger vector space which contains the pre-block space and block space assubspaces. Moreover, while for the construction in [FSS19] no semisimplicity is required, in thispaper semisimplicity is essential for the construction of commuting local projectors, since wedefine them in terms of the symmetric separability idempotents. (See [KMS] for some progresson projectors for non-semisimple Hopf algebras.) Lastly, since semisimple Hopf algebras havean involutive antipode, they have a canonical trivial pivotal structure. Hence, we can defineour model on any surface with orientation. Acknowledgments
I would like to thank my Ph.D. advisor Christoph Schweigert for introducing to me the topicof the paper and for many helpful discussions and valuable advice and feedback. FurthermoreI am grateful to Ehud Meir, Catherine Meusburger and Thomas Voß for fruitful discussionsand to Vincentas Mulevičius for help with the figures. The author is partially supported bythe RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory” and bythe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’sExcellence Strategy – EXC 2121 “Quantum Universe” – 390833306.
2. Hopf-algebraic and representation-theoretic labels forsurfaces with cell decomposition
Following the discussion in the introduction, we will explain in this section the input data forour construction.Let Σ be a compact oriented surface together with a cell decomposition (Σ , Σ , Σ ) with non-empty sets of -cells (or vertices ), -cells (or edges ) and -cells (or plaquettes ), respectively. This3an be thought of as an embedding of a graph (Σ , Σ ) into Σ such that its complement in Σ is the disjoint union of a set Σ of disks. Furthermore, let the edges be oriented, i.e. there aresource and target maps s, t : Σ −→ Σ . If the surface Σ has a boundary, then we require thatthe 1-skeleton of the cell decomposition be contained in the boundary.For the construction of a Kitaev model one needs as a further input Hopf-algebraic and rep-resentation-theoretic data labelling the various strata of the cell decomposition. In the ordinaryKitaev model without defects as in [BMCA], all edges of the cell decomposition are labeled bya single semisimple Hopf algebra H , and wherever point-like excitations are considered [BK], avertex is labeled by a representation of the Drinfeld double D ( H ) of the Hopf algebra H . In thispaper we consider more general labels for the edges, thereby implementing arbitrary line defects(also known as domain walls in condensed matter theory) and boundaries in the Kitaev model.Accordingly we also consider more general labels for vertices, implementing point defects (alsoknown as point-like excitations ) inside defect lines or boundaries. For the remainder of thissection we will specify the three types of Hopf-algebraic and representation-theoretic data thatlabel the plaquettes, edges and vertices of a cell decomposition. We fix once and for all an algebraically closed field k of characteristic zero. For the necessarybackground on Hopf algebras and conventions regarding the notation, see [Mo, Ka, BMCA]. Definition 1. • Let H and H be Hopf algebras over k . An H - H -bicomodule algebra K is a k -algebra K together with an H - H -bicomodule structure, i.e. with co-associative co-action writtenin Sweedler notation for comodules as K −→ H ⊗ K ⊗ H ,k (cid:55)−→ k ( − ⊗ k (0) ⊗ k (1) , which is required to be a morphism of algebras. If H = k or H = k , then K is just aright H -comodule or a left H -comodule algebra, respectively.A semisimple bicomodule algebra is a bicomodule algebra whose underlying algebra issemisimple.• Let Σ be an oriented surface with a cell decomposition with oriented edges. A label H p for a plaquette p ∈ Σ is a semisimple Hopf algebra H p over k .For any edge e ∈ Σ let p ∈ Σ and p ∈ Σ be the labelled plaquettes on the left and onthe right of e , respectively, with respect to the orientation of e relative to the orientation of Σ . Then a label K e for the edge e is a finite-dimensional semisimple H p - H p -bicomodulealgebra K e over k . 4 p p H p : Hopf algebra H p : Hopf algebra K e : H p - H p -bicomodule algebraFigure 1: An edge e and the adjacent plaquettes p and p with their algebraic data. The twoarrows denote the orientations of the edge and, respectively, of the surface Σ intowhich the edge is embedded.If the edge e lies in the boundary of Σ and hence only has a plaquette p on one side (leftor right), then K e is just a left or right H p -comodule algebra, respectively. Examples .
1. Let H be a Hopf algebra. The regular H -bicomodule algebra is the algebra underlyingthe Hopf algebra H together with left and right co-action given by the co-multiplicationof H . Note that the regular H -bicomodule algebra is semisimple if and only if the Hopfalgebra H is semisimple, since both are defined by the semisimplicity of the underlyingalgebra.2. Let G be a finite group and k G its group algebra, which has a basis ( b g ) g ∈ G parametrizedby G and multiplication induced by the group multiplication. k G is a semisimple Hopfalgebra with comultiplication given by the diagonal map b g (cid:55)→ b g ⊗ b g for all g ∈ G .Further, let U ⊆ G be a subgroup and ζ ∈ Z ( U, k × ) a group -cocycle. Then thecocycle-twisted group algebra k U ζ with multiplication b u · b v := ζ ( u, v ) b uv for all u, v ∈ U is a k G -comodule algebra with co-action given by the diagonal map b u (cid:55)→ b u ⊗ b u . Let us explain the emergence of bicomodule algebras from the point of view of Tannaka-Kreinduality, as outlined in the Introduction. We thereby relate the algebraic input data for ourconstruction, as defined in Definition 1, to the category-theoretic data for the state-sum con-struction of a modular functor in [FSS19]. For the relevant category-theoretic notions andbackground, see e.g. [EGNO].First of all, for a finite-dimensional Hopf algebra H over k , it is well known that the category H –mod of finite-dimensional left H -modules is a finite k -linear tensor category. This tensorcategory comes equipped with a forgetful functor H –mod −→ vect( k ) into the tensor categoryof finite-dimensional vector spaces. The forgetful functor is monoidal, exact and faithful.In fact, it is known [EGNO] that the datum of a finite-dimensional Hopf algebra H over k isequivalent to the datum of a finite k -linear tensor category A together with a monoidal fiberfunctor ω : A −→ vect( k ) , i.e. an exact and faithful k -linear tensor functor to the categoryof finite-dimensional vector spaces. More precisely, the Hopf algebra H can be reconstructedas the algebra of natural endo-transformations of the fiber functor ω and the tensor structureon the fiber functor ω induces the additional coalgebra structure on the algebra H , such that A ∼ = H –mod as tensor categories. 5e extend this idea to bimodule categories as follows. For a finite-dimensional H - H -bicomodule algebra K for Hopf algebras H and H , the category K –mod has the structure ofan ( H –mod) - ( H –mod) -bimodule category in a natural way. Indeed, if X is an H -module, X is an H -module and M is a K -module, then X (cid:46) M (cid:47) X := X ⊗ k M ⊗ k X becomes a K -module by pulling back the natural ( H ⊗ K ⊗ H ) -action on it along the co-action map K −→ H ⊗ K ⊗ H that belongs to K .On the other hand, let ( A , ω : A −→ vect( k )) and ( A , ω : A −→ vect( k )) be finite k -linear tensor categories together with monoidal fiber functors. Consider vect( k ) as an A - A -bimodule category via the monoidal functors ω and ω . Let M be a finite k -linear A - A -bimodule category. Then we define a bimodule fiber functor ω : M −→ vect( k ) for M tobe an exact and faithful A - A -bimodule functor from M to the category of finite-dimensionalvector spaces. Let H and H be the corresponding finite-dimensional Hopf algebras over k corresponding to ( A , ω ) and ( A , ω ) . Then, by the same argument as for tensor categories mutatis mutandis , the bimodule structure on the fiber functor ω induces the structure of an H - H -bicomodule algebra K on the algebra of natural endo-transformations of ω , such that ω induces an equivalence of bimodule categories M ∼ = K –mod .Hence, we conclude that bicomodule algebras emerge naturally as the algebraic input datafor Kitaev models, if one follows the following idea in order to obtain concrete Hopf-algebraicdata: Take the category-theoretic data underlying the corresponding topological field theoriesor modular functors, which are tensor categories and bimodule categories [FSS19, KK], andequip them with fiber functors of the appropriate type. It remains to determine the possible labels for the vertices of the cell decomposition. This isthe content of Subsection 2.3. Before that, in this Subsection 2.2, we first introduce suitablenotation and terminology in order to extract and conveniently speak about the combinatorialinformation contained in the cell decomposition.Fix a vertex v ∈ Σ . Then let Σ . v be the set of half-edges incident to v . This is the set ofincidences of an edge with the given vertex v ∈ Σ . (A loop at v yields two half-edges incidentto v .) Note that we have a map Σ . v −→ Σ , assigning to any half-edge its underlying edge,which is in general not injective due to the possible existence of loops. We will denote by Σ v its image in Σ , that is the set of edges starting or ending at the given vertex v .We will say that e ∈ Σ . v is directed away from v ∈ Σ if v = s ( e ) and, that e ∈ Σ . v is directed towards v ∈ Σ if v = t ( e ) . Then for any half-edge e ∈ Σ . v incident to the vertex v ∈ Σ , let the sign ε ( e ) ∈ { +1 , − } be positive if the half-edge e ∈ Σ . v is directed away fromthe vertex v : ve Figure 2: A half-edge e ∈ Σ . v incident to v with sign ε ( e ) := +1 and negative if e ∈ Σ . v is directed towards v : ve Figure 3: A half-edge e ∈ Σ . v incident to v with sign ε ( e ) := − Let p ∈ Σ be the plaquette on the left of the half-edge e ∈ Σ . v , as seen from the vertex v ∈ Σ , and let p (cid:48) ∈ Σ be the plaquette on the right, as in Figure 4.6 ep p (cid:48) Figure 4: A half-edge e at v with neighboring plaquettes p and p (cid:48) What we have not represented in the figure is that the half-edge e comes with an orientation,expressed by the sign ε := ε ( e ) . By our assignment of labels, if the half-edge e is directed awayfrom the vertex v , i.e. ε = +1 , then it is labeled with an H p - H p (cid:48) -bicomodule algebra K e , withco-action written in Sweedler notation for comodules: K e −→ H p ⊗ K e ⊗ H p (cid:48) k (cid:55)−→ k ( − ⊗ k (0) ⊗ k (1) (cid:27) if ε ( e ) = +1 . If, on the other hand, the half-edge e points towards v , that is ε = − , then K e is an H p (cid:48) - H p -bicomodule algebra: K e −→ H p (cid:48) ⊗ K e ⊗ H p k (cid:55)−→ k ( − ⊗ k (0) ⊗ k (1) (cid:27) if ε ( e ) = − . We shall introduce notation that allows us to treat both cases ε = +1 and ε = − at once. Let K +1 e := K e K − e := K op e , where K op e is the algebra with opposite multiplication. Moreover, let H +1 p := H p ,H − p := H opcop p , where H opcop p is the Hopf algebra with opposite multiplication and opposite comultiplication.If K e is a left (or right, respectively) H p -comodule algebra, then K − e is canonically a left (orright, respectively) H − p -comodule algebra.Hence, in both above cases we can write that K εe is an H εp - H εp (cid:48) -bicomodule algebra, withco-action in Sweedler notation: K εe −→ H εp ⊗ K εe ⊗ H εp (cid:48) ,k (cid:55)−→ k ( − ε ) ⊗ k (0) ⊗ k ( ε ) . Denote by Σ sit v the set of sites incident to v . These are incidences of a plaquette p ∈ Σ withthe given vertex v ∈ Σ . (Note that a plaquette p ∈ Σ can have two separate incidences withthe vertex v . This happens when an edge in its boundary is a loop.) Dually, for a plaquette p ∈ Σ denote by Σ sit p the set of sites incident to p . These are incidences of a vertex v ∈ Σ with the given plaquette p . It is justified to use the name site for both notions: To any site p ∈ Σ sit v at a vertex v ∈ Σ corresponds a unique site (cid:101) v ∈ Σ sit p with underlying vertex v at theplaquette that underlies the site p ∈ Σ sit v .Now let p ∈ Σ sit v be such a site at the vertex v ∈ Σ . There is a half-edge e (cid:48) p ∈ Σ . v bounding p on the left as seen from the vertex v and there is a half-edge e p ∈ Σ . v bounding p on theright. For an example consider Figure 5. 7 p e p e (cid:48) p Figure 5: A site p ∈ Σ sit v with neighboring half-edges e (cid:48) p and e p .Then, in consideration of the respective signs ε := ε ( e p ) and ε (cid:48) := ε ( e (cid:48) p ) of the half-edges e p and e (cid:48) p , we have by our assignment of labels that K ε (cid:48) e (cid:48) p is a right H ε (cid:48) p -comodule algebra and that K εe p is a left H εp -comodule algebra. In other words, we have a left (( H ε (cid:48) p ) cop ⊗ H εp ) -comodulestructure on the algebra K { e p ,e (cid:48) p } := (cid:78) e ∈{ e p ,e (cid:48) p }⊆ Σ . v K ε ( e ) e = (cid:40) K ε (cid:48) e (cid:48) p ⊗ K εe p , e p (cid:54) = e (cid:48) p ∈ Σ . v K εe p , e p = e (cid:48) p ∈ Σ . v . (1)Next we introduce, for a fixed site p ∈ Σ sit v , a canonical left (( H ε (cid:48) p ) cop ⊗ H εp ) -module algebra,which we think of as associated to the site p : Definition 3.
Let v ∈ Σ be a vertex and p ∈ Σ sit v a site at v with neighboring half-edges e p , e (cid:48) p ∈ Σ . v with signs ε, ε (cid:48) ∈ { +1 , − } as before.The ε (cid:48) - ε -balancing algebra H ∗ p , or more explicitly ( H p ) ∗ ( ε (cid:48) ,ε ) , is the left (( H ε (cid:48) p ) cop ⊗ H εp ) -modulealgebra, whose underlying k -algebra is the dual algebra of the Hopf algebra H p , with thefollowing action. (( H ε (cid:48) p ) cop ⊗ H εp ) ⊗ H ∗ p −→ H ∗ p ,a (cid:48) ⊗ a ⊗ f (cid:55)−→ f ( a (cid:48)(cid:104)− ε (cid:48) (cid:105) · ? · a (cid:104) ε (cid:105) ) , where a (cid:104) ε (cid:105) := (cid:26) a, ε = +1 S ( a ) , ε = − (cid:27) for all a ∈ H p and where S : H p −→ H p denotes the antipode.Together, the (( H ε (cid:48) p ) cop ⊗ H εp ) -comodule algebra K { e p ,e (cid:48) p } , associated to the half-edges e p ∈ Σ . v and e (cid:48) p ∈ Σ . v , and the (( H ε (cid:48) p ) cop ⊗ H εp ) -module algebra H ∗ p , associated to the site p ∈ Σ sit v situatedbetween the edges e p and e (cid:48) p , can be coupled into a single k -algebra, denoted by H ∗ p (cid:61) K { e p ,e (cid:48) p } (2)which has underlying vector space H ∗ p ⊗ K { e p ,e (cid:48) p } and which is an instance of the following generalconstruction. For related constructions see [Mo]. Definition 4.
Let H be a Hopf algebra over k , let A be a left H -module algebra and let K be a left H -comodule algebra. Then the crossed product algebra A (cid:61) K is the k -algebra withunderlying vector space A ⊗ K and multiplication ( a ⊗ k ) · ( a (cid:48) ⊗ k (cid:48) ) := a ( k ( − .a (cid:48) ) ⊗ k (0) k (cid:48) for ( a ⊗ k ) , ( a (cid:48) ⊗ k (cid:48) ) ∈ A ⊗ K.
8n particular, the algebra H ∗ p (cid:61) K { e p ,e (cid:48) p } contains H ∗ p and K { e p ,e (cid:48) p } as subalgebras and thecommutation relation between these is k · f = f ( k (cid:104)− ε (cid:48) (cid:105) ( ε (cid:48) ) · ? · k (cid:104) ε (cid:105) ( − ε ) ) · k (0) ∀ f ∈ H ∗ p , k ∈ K { e p ,e (cid:48) p } , (3)the so-called straightening formula . This generalizes the straightening formula of the Drinfelddouble of a Hopf algebra, see Example 6. In this subsection we introduce, for each vertex v ∈ Σ , an algebra over k , which is constructedfrom the algebraic labelling in the neighbourhood of the vertex v . The representations of thisalgebra will serve as possible labels for the vertex v . In a corresponding three-dimensionaltopological field theory these are the possible labels for generalized Wilson lines.Let us collect the algebras K ε ( e ) e of all half-edges e ∈ Σ . v incident to the vertex v ∈ Σ intoa tensor product K Σ . v := (cid:79) e ∈ Σ . v K ε ( e ) e . With the notation of the previous subsection, for each site p ∈ Σ sit v with neighboring half-edges e p and e (cid:48) p as in Figure 5, the algebra K { e (cid:48) p ,e p } is a left comodule over (cid:0) H ε ( e (cid:48) p ) p (cid:1) cop ⊗ H ε ( e p ) p . This trivially extends to an (( H ε ( e (cid:48) p ) p ) cop ⊗ H ε ( e p ) p ) -comodule structure on the tensor product K Σ . v of K { e,e (cid:48) } with the algebras attached to the remaining half-edges in Σ . v . The co-actionson K Σ . v for different sites commute with each other, because they come from the bicomodulestructures of the tensor factors ( K e ) e ∈ Σ . v , making K Σ . v a left comodule algebra over the tensorproduct of Hopf algebras (cid:78) p ∈ Σ sit v (cid:0) H ε ( e (cid:48) p ) p (cid:1) cop ⊗ H ε ( e p ) p . (4)For each site p ∈ Σ sit v we want to couple the balancing algebra H ∗ p to K Σ . v , similarly as in (2).For this we collect the balancing algebras of the sites around the vertex v into a tensor product H ∗ Σ sit v := (cid:79) p ∈ Σ sit v H ∗ p . This is a left module algebra over the tensor product of Hopf algebras as in (4). Now we haveall the ingredients to introduce:
Definition 5.
Let v ∈ Σ . The k -algebra C v associated to the vertex v , or vertex algebra , isdefined as follows. For any site p ∈ Σ sit v denote by e (cid:48) p and e p ∈ Σ . v the half-edges bounding p on the left and on the right, respectively, from the perspective of the vertex v , as illustrated inFigure 5. Then let C v := H ∗ Σ sit v (cid:61) K Σ . v = (cid:18) (cid:78) p ∈ Σ sit v H ∗ p (cid:19) (cid:61) (cid:18) (cid:78) e ∈ Σ . v K ε ( e ) e (cid:19) be the crossed product algebra, as introduced in Definition 4, for the left module algebra H ∗ Σ sit v and the left comodule algebra K Σ . v over the tensor product (4) of Hopf algebras.9n particular, the algebra contains H ∗ Σ sit v = ⊗ p ∈ Σ sit v H ∗ p and K Σ . v = ⊗ e ∈ Σ . v K ε ( e ) e as subalgebrasand, for each site p (cid:48) ∈ Σ sit v , we have the commutation relation (3); so in other words, H ∗ p (cid:48) (cid:61) K { e p (cid:48) ,e (cid:48) p (cid:48) } ⊆ (cid:18) (cid:78) p ∈ Σ sit v H ∗ p (cid:19) (cid:61) (cid:18) (cid:78) e ∈ Σ . v K ε ( e ) e (cid:19) = C v (5)is a subalgebra of C v . Example . Let us consider the situation where the vertex v ∈ Σ has precisely one half-edge e , which is directed away from the vertex and which is labeled by the regular H -bicomodulealgebra H , the transparent label. v H Figure 6: A vertex v with a single half-edge trans-parently labeled by H ;the associated algebra C v is the Drinfelddouble D ( H ) Then for the algebra C v at the vertex v we have H ∗ Σ sit v (cid:61) K Σ . v = H ∗ (cid:61) H and the commutationrelation (3) gives h · f = f ( S ( h (3) ) · ? · h (1) ) · h (2) . (6)This is precisely the so-called straightening formula of the Drinfeld double D ( H ) of a semisimpleHopf algebra H [Ka]. In the Kitaev model without defects as in [BMCA, BK], representationsof the Drinfeld double D ( H ) label point-like excitations.Up to this point we have explained how, for a given vertex v ∈ Σ , the algebraic labelling ofthe edges and plaquettes and the combinatorial structure of the cell decomposition around thatvertex gives rise to the k -algebra C v = H ∗ Σ sit v (cid:61) K Σ . v . Definition 7.
We declare the category of possible labels for a vertex v ∈ Σ for the Kitaevconstruction to be the k -linear category C v –mod of finite-dimensional left modules over the k -algebra C v .Indeed, in [FSS19], the category-theoretic data assigned to a vertex v ∈ Σ is as follows. In thelanguage of [FSS19], a vertex v corresponds to a boundary circle L v with marked points on whichdefect lines end. A -cell p ∈ Σ is labelled by a finite tensor category; in our context this is therepresentation category H p –mod of a finite-dimensional Hopf algebra H p . An edge e ∈ Σ islabelled by a finite bimodule category; in our context this is the representation category K e –mod of a bicomodule algebra K e . Then according to [FSS19, Definitions 3.4 and 3.9] the category ofpossible labels of a vertex v ∈ Σ is given by the category T ( L v ) of so-called balancings on theDeligne tensor product (cid:2) e ∈ Σ . v ( K ε ( e ) e –mod) of the bimodule categories labelling the half-edgesaround the vertex v . Theorem 8.
Let v ∈ Σ . There is a canonical equivalence of k -linear categories T ( L v ) ∼ = C v –mod between the category assigned by the modular functor T , constructed in [FSS19], to the circle L v with marked points corresponding to the half-edges incident to v and the representation categoryof the algebra C v . roof. The proof requires the introduction of significant additional notation and is thereforerelegated to the Appendix A, see Theorem 33.Furthermore, in the case that the edges incident to the vertex v are labeled transparently bya single Hopf algebra H seen as the regular H -bicomodule algebra, then the category C v –mod is equivalent to the Drinfeld center Z ( H –mod) [FSS19, Remarks 3.5 (iii) and 5.23], which isequivalent to the category of representations of the Drinfeld double D ( H ) . These are also thepossible labels for point-like excitations in the Kitaev model without defects, cf. [BK].
3. Construction of a Kitaev model with defects
Having specified in the preceding subsections the algebraic input data for the Kitaev modeland, in particular, having identified the possible labels for vertices, we are now in a positionto construct, for any oriented surface Σ with labeled cell decomposition, the vector space andlocal projectors of the model.We recall that we have for each plaquette p ∈ Σ a semisimple Hopf algebra H p , for eachedge e ∈ Σ a semisimple algebra K e with a compatible bicomodule structure over the Hopfalgebras of the incident plaquettes, and for each vertex v ∈ Σ a left module Z v over the algebra C v = H ∗ Σ sit v (cid:61) K Σ . v , introduced in Definition 5. We abbreviate K Σ := (cid:78) e ∈ Σ K e ,Z Σ := (cid:78) v ∈ Σ Z v , for the tensor products as vector spaces over k . More precisely, K Σ enters our construction ofthe local projectors and the Hamiltonian of the model not only as a vector space, but togetherwith its structure as the regular ( (cid:78) e ∈ Σ K e ) -bimodule and its various co-actions with respectto the Hopf algebras labeling the plaquettes. Similarly, we will regard Z Σ together with its C v -module structure for every vertex v ∈ Σ .The first thing we construct is the vector space, on which subsequently the commuting localprojectors and the Hamiltonian will be defined. Definition 9.
The state space assigned to an oriented surface Σ with labeled cell decompositionas above is the vector space H := Hom k ( K Σ , Z Σ ) = ( (cid:78) e ∈ Σ K ∗ e ) ⊗ ( (cid:78) v ∈ Σ Z v ) . (7)We refer to a tensor factor associated to an edge e or to a vertex v as a local degree of freedom associated to e or v , respectively. Remarks .
1. In the standard Kitaev construction without defects, the vector space is a tensor productof copies of a single Hopf algebra H for every edge, which we interpret in our context asthe regular bicomodule algebra over H (the transparent labeling), and for every vertex thedual vector space of a module over D ( H ) [BMCA, BK]. In our construction, we insteadconsider a module over the algebra C v for every vertex v ∈ Σ and the vector space dualsof the bicomodule algebras for the edges. This dual version will make it easier to compareour ground-state spaces with the block spaces of [FSS19].11. In order to define the state space H we are implicitly using that we do not only have thecategories ( K e –mod) e ∈ Σ and ( H p –mod) p ∈ Σ as algebraic input data, but we also have thealgebras ( K e ) e ∈ Σ and ( H p ) p ∈ Σ , of which they are the representation categories. In otherwords, we need fibre functors on the categories ( K e –mod) e ∈ Σ and ( H p –mod) p ∈ Σ to thecategory of vector spaces in order to define H as a space of k -linear homomorphisms.3. Note that we are only defining a vector space over k , and not a Hilbert space, i.e. wedo not consider a scalar product here. Accordingly, when we speak of projectors on thisvector space we always mean idempotent endomorphisms. By a Hamiltonian we mean adiagonalizable endomorphism.
Next, we exhibit on the vector space H a natural C v -bimodule structure for each vertex v ∈ Σ ,that is local in the sense that it acts non-trivially only on the local degrees of freedom in aneighborhood of the vertex v ∈ Σ . This is analogous to the existence of local actions of theDrinfeld double D ( H ) on the state space in the ordinary Kitaev model without defects for asemisimple Hopf algebra H [BMCA, BK]. In our construction, however, the algebras C v arein general not Hopf algebras and we only obtain bi module structures on H . (A C v -bimodulestructure is equivalent to a left ( C v ⊗ C op v ) -action, where C op v has the opposite multiplicationof C v . Whenever C v is a Hopf algebra, such as D ( H ) , any C v -bimodule structure can bepulled back to a left C v -action via the algebra map (id ⊗ S ) ◦ ∆ : C v → C v ⊗ C op v , using theco-multiplication ∆ and the antipode S of the Hopf algebra.)Let v ∈ Σ be any vertex. Recall from Subsection 2.3 that the algebra C v = H ∗ Σ sit v (cid:61) K Σ . v is a crossed product of H ∗ Σ sit v and K Σ . v and contains these as subalgebras, and that H ∗ Σ sit v = (cid:78) p ∈ Σ sit v H ∗ p is the tensor product of the algebras H ∗ p for each site p ∈ Σ sit v . A C v -bimodule structure on H is therefore fully determined by a K Σ . v -bimodule structure and H ∗ p -bimodule structures foreach site p ∈ Σ sit v , provided that for each p ∈ Σ sit v the left and right actions of K Σ . v and H ∗ p each satisfy the straightening formula (3) of the crossed product algebra H ∗ p (cid:61) K Σ . v , which weprove in Theorem 13.We start by exhibiting a K Σ . v -bimodule structure on the vector space H . This is the anal-ogon of the action of the Hopf algebra H for every vertex in the ordinary Kitaev model for asemisimple Hopf algebra H . Definition 11.
Let v ∈ Σ . The K Σ . v -bimodule structure on H (cid:101) A v : K Σ . v ⊗ K opΣ . v ⊗ H −→ H , is defined on the vector space of linear maps H = Hom k ( K Σ , Z Σ ) in the standard way bypre-composing with the left action on K Σ and post-composing with the left action on Z Σ ,which are defined as follows:• Firstly, the vector space K Σ becomes a left K Σ . v -module as follows. Restrict the regular K Σ -bimodule structure of K Σ , seen as a left ( K Σ ⊗ K opΣ ) -action, to the subalgebra K Σ . v ⊆ K Σ ⊗ K opΣ . 12 Secondly, the vector space Z Σ becomes a left K Σ . v -module as follows. Restrict the given C v -module structure on Z v to the subalgebra K Σ . v ⊆ (cid:78) v ∈ Σ ( H ∗ Σ sit v (cid:61) K Σ . v ) = C v andextend the action trivially to the vector space Z Σ = Z v ⊗ (cid:78) w ∈ Σ \{ v } Z w .Next we will exhibit, for any site p ∈ Σ sit v incident to a vertex v ∈ Σ , an H ∗ p -bimodulestructure on H .Recall that Σ sit p denotes the set of incidences of a vertex with a given plaquette p (which wealso call sites ) and denote by Σ . p the set of incidences of an edge with the given plaquette p (which we call plaquette edges ). We consider their union Σ sit p ∪ Σ p together with a cyclic orderon it, given by the clockwise direction along the boundary of p with respect to the orientationof Σ , as illustrated in Figure 7 p Figure 7: Cyclic order on the set Σ sit p ∪ Σ p of sites andplaquette edges of a plaquette p Furthermore, for any plaquette edge e ∈ Σ p at the plaquette p , let the sign ε p ( e ) ∈ { +1 , − } be positive if the plaquette edge e ∈ Σ p is clockwise directed around the plaquette p : p e Figure 8: A plaquette edge e with sign ε p ( e ) := +1 and negative if e ∈ Σ p is directed counter-clockwise around p : p e Figure 9: A plaquette edge e with sign ε p ( e ) := − Recall that, attached to each plaquette p ∈ Σ , there is a Hopf algebra H p . Now, dependingon choice of a site v ∈ Σ sit p at p , we define an H ∗ p -bimodule structure on the vector space H .This is the analogon of the action of the dual Hopf algebra H ∗ for every site in the ordinaryKitaev model for a semisimple Hopf algebra H . Definition 12.
Let p ∈ Σ . We define, for each site v ∈ Σ sit p , the H ∗ p -bimodule structure on H ,or left action of the enveloping algebra H ∗ p ⊗ ( H ∗ p ) op , (cid:101) B ( p,v ) : H ∗ p ⊗ ( H ∗ p ) op ⊗ H −→ H , by the following left and right H ∗ p -actions on H .• We start by declaring that H ∗ p acts from the left on H = ( (cid:78) e ∈ Σ K ∗ e ) ⊗ ( (cid:78) w ∈ Σ Z w ) bythe action of H ∗ p ⊆ H ∗ Σ sit v (cid:61) K Σ . v on the ( H ∗ Σ sit v (cid:61) K Σ . v ) -module Z v and by acting as theidentity on the remaining tensor factors of H .13 For the right action of H ∗ p on H , we use the total order on the set (Σ sit p ∪ Σ . p ) \ { v } starting right after v ∈ Σ sit p in Σ sit p ∪ Σ . p with respect to the cyclic order declared above,given by the clockwise direction around the plaquette p . We first exhibit individual right H ∗ p -actions on the tensor factors of ( (cid:78) e ∈ Σ p K ∗ e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z w ) : – For any e ∈ Σ . p , the vector space K ∗ e becomes a right H ∗ p -module as follows. K e isa right H ε p ( e ) p -comodule and, hence, a left ( H ∗ p ) ε p ( e ) -module. Thus the vector spacedual K ∗ e becomes a right ( H ∗ p ) ε p ( e ) -module, and finally, by pulling back along thealgebra isomorphism ? (cid:104) ε p ( e ) (cid:105) : H ∗ p → H ∗ p ε p ( e ) , a right H ∗ p -module.Recall that ? (cid:104) +1 (cid:105) def = id H ∗ p and ? (cid:104)− (cid:105) def = S , the antipode of H ∗ p . Explicitly, this right H ∗ p -action is given by K ∗ e ⊗ H ∗ p −→ K ∗ e ,ϕ ⊗ f (cid:55)−→ (cid:16) k (cid:55)→ ϕ (cid:16) k (0) f (cid:16) k (cid:104) ε p ( e ) (cid:105) ( ε p ( e )) (cid:17)(cid:17)(cid:17) . – For any w ∈ Σ sit p \ { v } , the vector space Z w becomes a right H ∗ p -module as follows.The ( H ∗ Σ w (cid:61) K Σ w ) -module Z w comes with a left H ∗ p -action since H ∗ p ⊆ H ∗ Σ w (cid:61) K Σ w is a subalgebra. We let H ∗ p act on Z w from the right by pulling back this left actionalong the antipode ? (cid:104)− (cid:105) = S : H ∗ p → H ∗ p .Then we declare H ∗ p to act from the right on the tensor product ( (cid:78) e ∈ Σ p K ∗ e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z w ) by applying the co-multiplication on H ∗ p suitably many times and then acting individuallyon the tensor factors in the sequence given by the image of the clockwise linear order thatwe have prescribed on the set (Σ sit p ∪ Σ . p ) \ { v } under the map (Σ sit p ∪ Σ . p ) \ { v } → (Σ p ∪ Σ p ) \ { v } that assigns to a site its underlying vertex and to a plaquette edge itsunderlying edge. Finally, this gives a right H ∗ p -action on H = ( (cid:78) e ∈ Σ K ∗ e ) ⊗ ( (cid:78) w ∈ Σ Z w ) by acting with the identity on all remaining tensor factors.So far we have defined, in Definitions 11 and 12, on the vector space H an K Σ . v -bimodulestructure (cid:101) A v for each vertex v ∈ Σ and an H ∗ p -bimodule structure (cid:101) B ( p,v ) for each site p ∈ Σ sit v .These are analogous to the actions of the Hopf algebra H and the dual Hopf algebra H ∗ defined for each site in the ordinary Kitaev model without defects. Just as the latter are shownto interact with each other non-trivially, giving a representation of the Drinfeld double D ( H ) at each site [BMCA], we will now proceed to study how the bimodule structures (cid:101) A v and (cid:101) B ( p,v (cid:48) ) of K Σ . v and H ∗ p for various v and ( p, v (cid:48) ) interact with each other.In order to simplify the proof we will make a certain regularity assumption on the celldecomposition of the surface Σ : We call a cell decomposition regular if it has no looping edges,i.e. there is no edge which has the same source vertex as target vertex and if the Poincaré-dualcell decomposition also has no looping edges, i.e. in the original cell decomposition there is noplaquette that has two incidences with one and the same edge (on its two sides). Theorem 13.
Let H be the vector space defined in Definition 9 for an oriented surface Σ witha labelled cell decomposition. Recall from Definitions 11 and 12 the K Σ . v -bimodule structure (cid:101) A v on H for every vertex v ∈ Σ , and the H ∗ p -bimodule structure (cid:101) B ( p,v ) on H for every plaquette p ∈ Σ together with incident site v (cid:48) ∈ Σ sit p . Then• For any pair of vertices v (cid:54) = v ∈ Σ , the actions (cid:101) A v and (cid:101) A v commute with each other.• For any pair of sites ( p ∈ Σ , v ∈ Σ sit p ) and ( p ∈ Σ , v ∈ Σ sit p ) such that p (cid:54) = p , theactions (cid:101) B ( p ,v ) and (cid:101) B ( p ,v ) commute with each other. Assume that the cell decomposition of Σ is regular. For any site ( p ∈ Σ , v ∈ Σ sit p ) , theactions (cid:101) A v and (cid:101) B ( p,v ) compose to give on H a bimodule structure over the crossed productalgebra H ∗ ( p,v ) (cid:61) K Σ . v , (cid:101) B ( p,v ) (cid:101) A v : H ∗ p ⊗ K Σ . v ⊗ ( H ∗ p ⊗ K Σ . v ) op ⊗ H −→ H ,f ⊗ k ⊗ f (cid:48) ⊗ k (cid:48) ⊗ x (cid:55)−→ (cid:101) B f ⊗ f (cid:48) ( p,v ) (cid:101) A k ⊗ k (cid:48) v ( x ) . Proof. • The left K Σ . v - and K Σ . v -actions act as the identity on all tensor factors of H except on Z v and Z v , respectively. It is thus clear that they commute for v (cid:54) = v .The right K Σ . v - and K Σ . v -actions only have a common tensor factor on which they donot act by the identity for every edge e ∈ Σ that joins the vertices v and v . Such anedge is directed away from one of the vertices and directed towards the other. Hence, theaction for one of the vertices comes from left multiplication of K e and the other one fromright multiplication, so they commute.• The left H ∗ p - and H ∗ p -actions act as the identity on all tensor factors of H except on Z v and Z v , respectively. It is thus clear that they commute for v (cid:54) = v . In the remainingcase v = v =: v , H ∗ p and H ∗ p are commuting subalgebras in C v . Since their actions on Z v are by Definition 12 the restrictions of the C v -action that Z v comes with, they musttherefore commute.The right H ∗ p - and H ∗ p -actions only have a common tensor factor on which they donot act by the identity for every vertex v ∈ Σ and for every edge e ∈ Σ that lies inthe boundaries of both plaquettes p and p . For any such vertex v , the two actionscome from the ( H ∗ Σ sit v (cid:61) K Σ . v ) -action on Z v restricted to the two subalgebras H ∗ p and H ∗ p , respectively. These subalgebras commute inside H ∗ Σ sit v (cid:61) K Σ . v , therefore showing theclaim.• The left K Σ . v - and H ∗ p -actions on H are simply the restrictions of the left C v -action on Z v to K Σ . v and H ∗ p , respectively, and the identity on all other tensor factors of H . Hence,by construction they satisfy the commutation relations of the crossed product algebra H ∗ p (cid:61) K Σ . v ⊆ C v , see also (5).The right K Σ . v - and H ∗ p -actions on H are non-trivial only on the tensor factors (cid:78) e ∈ Σ v K ∗ e and ( (cid:78) e ∈ Σ p K ∗ e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z w ) , respectively. We can therefore restrict our attentionto the vector space ( (cid:78) e ∈ Σ v ∪ Σ p K ∗ e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z w ) , on which K Σ . v and H ∗ p act fromthe right.For convenience, for the remainder of the proof we now switch to the dual vector space ( (cid:78) e ∈ Σ v ∪ Σ p K e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z ∗ w ) , with the corresponding left actions of K Σ . v and H ∗ p .With the notation of Subsection 2.2, let e p , e (cid:48) p ∈ Σ . v be the half-edges at v on thetwo sides of the site p ∈ Σ sit v , with signs ε := ε ( e p ) and ε (cid:48) := ε ( e (cid:48) p ) . The K Σ . v - and H ∗ p -actions only overlap on the tensor factors ( K e ) e ∈ Σ v ∩ Σ p corresponding to the edgesunderlying the half-edges e p , e (cid:48) p ∈ Σ . v . Due to our regularity assumption on the celldecomposition, the half-edges e p and e (cid:48) p have distinct underlying edges. Then the actionof K Σ . v = ( K εe p ⊗ K ε (cid:48) e (cid:48) p ) ⊗ (cid:78) e ∈ Σ . v \{ e p ,e (cid:48) p } K ε ( e ) e on (cid:78) e ∈ Σ v K e , which is a tensor productof algebras, decomposes into a tensor product of the action of K εe p ⊗ K ε (cid:48) e (cid:48) p on K e p ⊗ K e (cid:48) p and the action of (cid:78) e ∈ Σ . v \{ e p ,e (cid:48) p } K ε ( e ) e on (cid:78) e ∈ Σ v \{ e p ,e (cid:48) p } K e . On the latter vector space, H ∗ p does not act non-trivially by our regularity assumption on the cell decomposition. Hence,15t remains to consider the interactions of the left actions of K εe p ⊗ K ε (cid:48) e (cid:48) p and H ∗ p on thevector space K e p ⊗ K e (cid:48) p ⊗ ( (cid:78) e ∈ Σ p \{ e p ,e (cid:48) p } K e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z ∗ w ) . We abbreviate by V :=( (cid:78) e ∈ Σ p \{ e p ,e (cid:48) p } K e ) ⊗ ( (cid:78) w ∈ Σ p \{ v } Z ∗ w ) the tensor factor on which only H ∗ p acts non-trivially.Furthermore, without loss of generality, we write the left H ∗ p -action on V in terms of theSweedler notation for the corresponding right H p -coaction, V → V ⊗ H p , v (cid:55)→ v (0) ⊗ v (1) : H ∗ p ⊗ V −→ V, v (cid:55)−→ f.v =: f ( v (1) ) v (0) . Finally, it is left to analyze the interaction between the H ∗ p -action H ∗ p ⊗ K e p ⊗ K e (cid:48) p ⊗ V −→ K e p ⊗ K e (cid:48) p ⊗ V,f ⊗ x ⊗ x (cid:48) ⊗ v (cid:55)−→ f (3) .x ⊗ f (1) .x (cid:48) ⊗ f (2) .v = f (cid:16) x (cid:48)(cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) v (1) x (cid:104)− ε (cid:105) ( − ε ) (cid:17) x (0) ⊗ x (cid:48) (0) ⊗ v (0) , and the ( K εe p ⊗ K ε (cid:48) e (cid:48) p ) -action ( K εe p ⊗ K ε (cid:48) e (cid:48) p ) ⊗ K e p ⊗ K e (cid:48) p ⊗ V −→ K e p ⊗ K e (cid:48) p ⊗ V,a ⊗ a (cid:48) ⊗ x ⊗ x (cid:48) ⊗ v (cid:55)−→ a.x ⊗ a (cid:48) .x (cid:48) ⊗ v ( a · ε x ) ⊗ ( a (cid:48) · ε (cid:48) x (cid:48) ) ⊗ v, where · ε and · ε (cid:48) denote the multiplication in K εe p and K ε (cid:48) e (cid:48) p , respectively, that is a · ε x := (cid:40) ax, ε = +1 ,xa, ε = − . It remains to show that that these actions satisfy the straightening formula f ( a (cid:48)(cid:104)− ε (cid:48) (cid:105) ( ε (cid:48) ) · ? · a (cid:104) ε (cid:105) ( − ε ) ) . ( a (0) ⊗ a (cid:48) (0) ) . ( x ⊗ x (cid:48) ⊗ v ) = ( a ⊗ a (cid:48) ) .f. ( x ⊗ x (cid:48) ⊗ v ) , for all f ∈ H ∗ p , a ⊗ a (cid:48) ∈ K εe p ⊗ K ε (cid:48) e (cid:48) p and x ⊗ x (cid:48) ⊗ v ∈ K e p ⊗ K e (cid:48) p ⊗ V . Indeed, the followingcalculation, which is analogous to the calculation in the proof of [BMCA, Theorem 1] butmore general and at the same time shorter, verifies this. f (cid:16) a (cid:48)(cid:104)− ε (cid:48) (cid:105) ( ε (cid:48) ) · ? · a (cid:104) ε (cid:105) ( − ε ) (cid:17) . ( a (0) ⊗ a (cid:48) (0) ) . ( x ⊗ x (cid:48) ⊗ v )= f (cid:16) a (cid:48)(cid:104)− ε (cid:48) (cid:105) ( ε (cid:48) ) · ? · a (cid:104) ε (cid:105) ( − ε ) (cid:17) . (( a (0) · ε x ) ⊗ ( a (cid:48) (0) · ε (cid:48) x (cid:48) ) ⊗ v )= f (cid:16) a (cid:48)(cid:104)− ε (cid:48) (cid:105) (2 ε (cid:48) ) · ( a (cid:48) (0) · ε (cid:48) x (cid:48) ) (cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) · v (1) · ( a (0) · ε x ) (cid:104)− ε (cid:105) ( − ε ) · a (cid:104) ε (cid:105) ( − ε ) (cid:17) (( a (0) · ε x ) (0) ⊗ ( a (cid:48) (0) · ε (cid:48) x (cid:48) ) (0) ⊗ v (0) )= f (cid:16) a (cid:48)(cid:104)− ε (cid:48) (cid:105) (2 ε (cid:48) ) · a (cid:48)(cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) · x (cid:48)(cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) · v (1) · x (cid:104)− ε (cid:105) ( − ε ) · a (cid:104)− ε (cid:105) ( − ε ) · a (cid:104) ε (cid:105) ( − ε ) (cid:17) (( a (0) · ε x (0) ) ⊗ ( a (cid:48) (0) · ε (cid:48) x (cid:48) (0) ) ⊗ v (0) )= f (cid:16) x (cid:48)(cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) · v (1) · x (cid:104)− ε (cid:105) ( − ε ) (cid:17) (( a · ε x (0) ) ⊗ ( a (cid:48) · ε (cid:48) x (cid:48) (0) ) ⊗ v (0) )= ( a ⊗ a (cid:48) ) . (cid:18) f (cid:16) x (cid:48)(cid:104) ε (cid:48) (cid:105) ( ε (cid:48) ) · v (1) · x (cid:104)− ε (cid:105) ( − ε ) (cid:17) ( x (0) ⊗ x (cid:48) (0) ⊗ v (0) ) (cid:19) = ( a ⊗ a (cid:48) ) . (cid:0) f. ( x ⊗ x (cid:48) ⊗ v ) (cid:1) . This proves that H ∗ p and K εe p ⊗ K ε (cid:48) e (cid:48) p together give a representation of the crossed productalgebra H ∗ p (cid:61) ( K εe p ⊗ K ε (cid:48) e (cid:48) p ) , as claimed. 16 emark . Taking all sites p ∈ Σ sit v around a given vertex v ∈ Σ together, we thus get, dueto Theorem 13, on H a bimodule structure over the vertex algebra C v . It is remarkable thatthis makes the crossed product algebra structure on C v show up naturally – analogous to theappearance of the algebra structure of the Drinfeld double in the commutation relation of thevertex and plaqette actions in the standard Kitaev model without defects. Before we proceed to use the bimodule structures on the state space H defined in Subsection3.1 to define commuting local projectors on the vector space H , we need to invoke anotheralgebraic ingredient.The standard Kitaev construction for a semisimple Hopf algebra H makes use of the Haarintegrals of H and of H ∗ , in order to define commuting local projectors on the state space viathe actions of H and H ∗ . The Haar integral of a semisimple Hopf algebra H over k is theunique element (cid:96) ∈ H satisfying x(cid:96) = ε ( x ) (cid:96) = (cid:96)x for all x ∈ H and ε ( (cid:96) ) = 1 . This means that (cid:96) is the central idempotent which projects to the H -invariants: for any H -module M , we have (cid:96).M = M H := { m ∈ M | h.m = ε ( h ) m ∀ h ∈ H } . Furthermore, (cid:96) ∈ H is cocommutative, i.e. (cid:96) (1) ⊗ (cid:96) (2) = (cid:96) (2) ⊗ (cid:96) (1) in Sweedler notation. The idempotence, centrality and cocommutativityof the Haar integral are crucial in showing that the Haar integral gives rise to commuting localprojectors in the standard Kitaev construction [BMCA].In our setting, instead of a semisimple Hopf algebra acting on the state space, we have, foreach vertex v ∈ Σ , a bimodule structure on the state space over a semisimple (bi-)comodulealgebra K Σ . v . Hence, we need a notion replacing the Haar integral, that works in this setting.Our main insight is that the suitable generalization of the Haar integral to our setting is theunique symmetric separability idempotent, which exists for any semisimple algebra over analgebraically closed field k with characteristic zero. Definition 15.
Let A be an algebra over a field k . A symmetric separability idempotent for A is an element p ∈ A ⊗ A , which we write as p = p ⊗ p ∈ A ⊗ A omitting the summationsymbol, satisfying ( x · p ) ⊗ p = p ⊗ ( p · x ) ∀ x ∈ A, (8) p · p = 1 , (9) p ⊗ p = p ⊗ p , (symmetry) (10)where on both sides of equation (8) and in equation (9) we are using the multiplication in A .The properties (8) and (9) immediately imply that p ⊗ p is an idempotent when seen as anelement of the enveloping algebra A ⊗ A op . Remarks .
1. The structure of a separability idempotent, i.e. an element p ⊗ p ∈ A ⊗ A satisfying(8) and (9), is equivalent to an A -bimodule map s : A −→ A ⊗ A that is a section of themultiplication m : A ⊗ A −→ A , by defining s ( x ) := p ⊗ p x for all x ∈ A . An algebraendowed with such a structure is called separable and, in general, such a separabilitystructure might not exist or be unique. A symmetric separability structure, however, isalways unique – see the end of the proof of Proposition 17.17. Representation-theoretically, a separability idempotent p ⊗ p ∈ A ⊗ A op plays the role ofprojecting to the subspace of invariants for any A -bimodule M . Indeed, due to property(8), one has p .M.p = M A := { m ∈ M | a.m = m.a ∀ a ∈ A } ⊆ M. This is in analogy to the Haar integral (cid:96) ∈ H of a semisimple Hopf algebra H whichprojects to the invariants (cid:96).M = M H := { m ∈ M | h.m = ε ( h ) m ∀ h ∈ H } of any left H -module M .Just as every finite-dimensional semisimple Hopf algebra over a field k has a unique Haarintegral, for every finite-dimensional semisimple k -algebra there exists a unique symmetricseparability idempotent: Proposition 17 ([A]) . Let A be a finite-dimensional semisimple algebra over a field k which isalgebraically closed and of characteristic zero. Then there exists a unique symmetric separabilityidempotent p ⊗ p ∈ A ⊗ A op for A .Proof. For a more detailed proof, see [A, Thm. 3.1, Cor. 3.1.1]. Here we recall the main ideathat the unique symmetric separability idempotent can be described in terms of the trace formon A , because we will use this description in Proposition 19.Due to semisimplicity, the following symmetric bilinear pairing on A is non-degenerate: T : A ⊗ A −→ k ,a ⊗ b (cid:55)−→ t ( a · b ) := tr A ( L a · b ) , defined in terms of the trace form where L ? denotes the left multiplication of A . In fact, thisnon-degenerate bilinear pairing turns A into a symmetric special Frobenius algebra. Considerthe isomorphism T : A ∼ −→ A ∗ , a (cid:55)→ t ( a · − ) , induced by this non-degenerate bilinear pairing.This is an isomorphism of A -bimodules. It induces an isomorphism A ⊗ A ∼ −→ A ∗ ⊗ A ∼ = End k ( A ) .Consider the pre-image p ∈ A ⊗ A of the identity id A under this isomorphism. As usual, wewrite an element p ∈ A ⊗ A as p = p ⊗ p , omitting the summation symbol. In fact, if wechoose a basis ( p i ) i for A and let ( p i ) i be its dual basis of A with respect to the non-degeneratepairing T , then p ⊗ p is the sum (cid:80) i p i ⊗ p i . With this definition of p ⊗ p ∈ A ⊗ A it isstraightforward to verify the defining properties (8), (9) and (10) of a symmetric separabilityidempotent.To prove that the symmetric separability idempotent is unique, let p ⊗ p , q ⊗ q ∈ A ⊗ A op be any two symmetric separability idempotents for A . Then they are equal by the followingcomputation: p ⊗ p (9) = q q p ⊗ p (8) = q p ⊗ p q (10) = q p ⊗ p q (8) = q ⊗ p p q (10) = q ⊗ p p q (9) = q ⊗ q (10) = q ⊗ q , using the defining properties (8), (9) and (10). Example . Let H be a finite-dimensional semisimple Hopf algebra over k with Haar integral (cid:96) ∈ H . Then the symmetric separability idempotent for H is (cid:96) (1) ⊗ S ( (cid:96) (2) ) ∈ H ⊗ H op .Indeed, the invariance property of the Haar integral, x(cid:96) = ε ( x ) (cid:96) for all x ∈ H , implies thecorresponding invariance property (8) of (cid:96) (1) ⊗ S ( (cid:96) (2) ) . The normalization ε ( (cid:96) ) = 1 of the Haarintegral implies the corresponding normalization property 9 for the separability idempotent.Finally, using that the Haar integral is two-sided, which implies S ( (cid:96) ) = (cid:96) , it can be shown that (cid:96) (1) ⊗ S ( (cid:96) (2) ) is symmetric.Hence we see that, in the sense of this example, the symmetric separability idempotent of asemisimple algebra generalizes the Haar integral of a semisimple Hopf algebra.18n our construction of a Kitaev model, however, we are not only dealing with semisimplealgebras, but semisimple algebras together with a compatible bicomodule structure. On theother hand, the Haar integral (cid:96) ∈ H has the property of being cocommutative, (cid:96) (1) ⊗ (cid:96) (2) = (cid:96) (2) ⊗ (cid:96) (1) , which is crucial in showing that it gives rise to commuting projectors in [BMCA]and we have not exhibited an analogous property of the symmetric separability idempotent. Inthe following proposition we prove such a property, which holds for the symmetric separabilityidempotent of a semisimple (bi-)comodule algebra and which generalizes the cocommutativityof the Haar integral, see Example 20. Proposition 19.
Let H be a semisimple Hopf algebra over k and let K be a semisimple right H -comodule algebra with symmetric separability idempotent p ⊗ p ∈ K ⊗ K op . Consider theright H -coaction on the tensor product K ⊗ K op : K ⊗ K op −→ K ⊗ K op ⊗ H,k ⊗ k (cid:48) (cid:55)−→ k (0) ⊗ k (cid:48) (0) ⊗ k (1) k (cid:48) (1) . Then p ⊗ p ∈ K ⊗ K op is an H -coinvariant element of K ⊗ K op , i.e. p ⊗ p ⊗ p p = p ⊗ p ⊗ H ∈ K ⊗ K op ⊗ H , and this is equivalent to p ⊗ p ⊗ p = p ⊗ S ( p ) ⊗ p ∈ K ⊗ H ⊗ K op . (11) Analogously, if K is a left H -comodule algebra, then p ⊗ p − ⊗ p = p ⊗ S ( p − ) ⊗ p ∈ K ⊗ H ⊗ K op . (12) Proof.
Without loss of generality we only show the case where K is a right H -comodule algebra.Recall from the proof of Proposition 17 that the symmetric separability idempotent p ⊗ p ∈ K ⊗ K op for K can be characterized in terms of the multiplication and the trace form t : K −→ k on K , namely by t ( p · x ) p = x ∀ x ∈ K , as explained in the proof of Proposition 17. Anotherway of phrasing this is that the map K ∗ −→ K defined by f (cid:55)−→ f ( p ) p is the inverse ofthe isomorphism K −→ K ∗ , k (cid:55)−→ t (? · k ) induced by the non-degenerate pairing t ◦ µ , where µ : K ⊗ K −→ K is the multiplication on K .The crucial step for the present proof is the observation that the multiplication and the traceform on K are morphisms of H -comodules if K is an H -comodule algebra. For the multiplicationthis means that x (0) y (0) ⊗ x (1) y (1) = ( xy ) (0) ⊗ ( xy ) (1) ∀ x, y ∈ K , which holds by definition ofa comodule algebra, see Definition 1. As for the H -colinearity of the trace form, note that t = ev K ◦ ( µ ⊗ id K ∗ ) ◦ (id K ⊗ coev K ) , where µ : K ⊗ K → K denotes the multiplication, and coev K : k −→ K ⊗ K ∗ and ev K : K ⊗ K ∗ −→ k are the standard coevaluation and evaluationmorphisms for vector spaces. Due the involutivity of the antipode S of H , both ev K and coev K are morphisms of right H -comodules for the H -comodule structure on the dual K ∗ given by K ∗ −→ K ∗ ⊗ H, ϕ (cid:55)−→ ϕ (0) ⊗ ϕ (1) , where ϕ (0) ( x ) ϕ (1) := ϕ ( x (0) ) S ( x (1) ) for all x ∈ K . (Weare here implicitly using the canonical trivial pivotal structure on the tensor category of right H -comodules, which exists due to the involutivity of the antipode of H .) Since therefore thetrace form t is composed only of morphisms of right H -comodules, it is itself a morphism ofright H -comodules, i.e. t ( k (0) ) k (1) = t ( k )1 H ∀ k ∈ K. (13)As a consequence, the isomorphism K −→ K ∗ , k (cid:55)−→ t (? · k ) induced by the pairing t ◦ µ is an isomorphism of H -comodules. Indeed, for all x ∈ K one has t ( xk (0) ) k (1) = t ( x (0) k (0) ) S ( x (2) ) x (1) k (1) (13) = t ( x (0) k ) S ( x (1) ) def = ( t (? · k )) (0) ( x )( t (? · k )) (1) .This immediately implies that the inverse map, K ∗ −→ K, ϕ (cid:55)−→ ϕ ( p ) p , must also be amorphism of H -comodules, which spelled out means that ϕ ( p ) p ⊗ S ( p ) def = ϕ (0) ( p ) p ⊗ ϕ (1) = ( p ) p ⊗ p for all ϕ ∈ K ∗ . This implies the equation (11) of the claim. To show that thisis equivalent to p ⊗ p ∈ K ⊗ K op being H -coinvariant, we compute p ⊗ p ⊗ p p (11) = p ⊗ p ⊗ S ( p ) p = p ⊗ p ⊗ H . Example . Let H be a semisimple Hopf algebra and consider it as the regular H -bicomodulealgebra, as in Example 2.(1). Recall that for H the symmetric separability idempotent is p ⊗ p = (cid:96) (1) ⊗ S ( (cid:96) (2) ) ∈ H ⊗ H . Let us spell out Proposition 19 for the left and right H -comodule structures on the regular bicomodule algebra H . Equation (11) boils down to theequation ( (cid:96) (1) ) (1) ⊗ ( (cid:96) (1) ) (2) ⊗ S ( (cid:96) (3) ) = (cid:96) (1) ⊗ S ( S ( (cid:96) (2) ) (2) ) ⊗ S ( (cid:96) (2) ) (1) . But due to S = id H bothsides of the equation are equal to (cid:96) (1) ⊗ (cid:96) (2) ⊗ S ( (cid:96) (3) ) . On the other hand, equation (12) boilsdown to the equation ( (cid:96) (1) ) (2) ⊗ ( (cid:96) (1) ) (1) ⊗ S ( (cid:96) (3) ) = (cid:96) (1) ⊗ S ( S ( (cid:96) (2) ) (1) ) ⊗ S ( (cid:96) (2) ) (2) , which in turndue to S = id H simplifies to (cid:96) (2) ⊗ (cid:96) (1) ⊗ S ( (cid:96) (3) ) = (cid:96) (1) ⊗ (cid:96) (3) ⊗ S ( (cid:96) (2) ) . This is equivalent to thecocommutativity property (cid:96) (1) ⊗ (cid:96) (2) = (cid:96) (2) ⊗ (cid:96) (1) .Hence we have shown that the coinvariance property of the symmetric separability idempo-tent for a bicomodule algebra, proven in Proposition 19, is the appropriate analogue of thecocommutativity of the Haar integral. In the proof of Lemma 21 we will use it in a crucial way,on the way towards proving in Theorem 25 that symmetric separability idempotents allow fordefining commuting projectors. Lemma 21.
Let H be a semisimple Hopf algebra over k and let K be a semisimple left H -comodule algebra and A a semisimple left H -module algebra. Let p ⊗ p ∈ K ⊗ K op and π ⊗ π ∈ A ⊗ A op be the symmetric separability idempotents for K and A , respectively.Then (1 A ⊗ p ) ⊗ (1 A ⊗ p ) and ( π ⊗ K ) ⊗ ( π ⊗ K ) commute in the algebra ( A (cid:61) K ) ⊗ ( A (cid:61) K ) op , where A (cid:61) K is the crossed product algebra defined in Definition 4.Proof. Due to the co-invariance of the symmetric separability idempotent of a semisimple co-module algebra over k , proven in Proposition 19, we have p − ⊗ p ⊗ p (12) = S ( p − ) ⊗ p ⊗ p and ( h.π ) ⊗ π = π ⊗ ( S ( h ) .π ) for all h ∈ H , where the latter can be derived from equation (11) by regarding A as a right H ∗ -comodule algebra, which is equivalent to a left H -module algebra [Mo]. By definition ofthe multiplication in ( A (cid:61) K ) ⊗ ( A (cid:61) K ) op we have: (1 A ⊗ p ) ⊗ (1 A ⊗ p ) · ( π ⊗ K ) ⊗ ( π ⊗ K ) = ( p − .π ⊗ p ) ⊗ ( π ⊗ p ) and ( π ⊗ K ) ⊗ ( π ⊗ K ) · (1 A ⊗ p ) ⊗ (1 A ⊗ p ) = ( π ⊗ p ) ⊗ ( p − .π ⊗ p ) But the right-hand sides of these equations are equal by the following computation: ( p − .π ⊗ p ) ⊗ ( π ⊗ p ) = ( S ( p − ) .π ⊗ p ) ⊗ ( π ⊗ p )= ( π ⊗ p ) ⊗ ( S ( p − ) .π ⊗ p )= ( π ⊗ p ) ⊗ ( p − .π ⊗ p ) . .3. Local commuting projector Hamiltonian from vertex andplaquette operators In this subsection we define on the vector space H assigned to a surface Σ with a labelled celldecomposition a set of commuting local projectors and finally, in the spirit of Kitaev latticemodels, a Hamiltonian on H as the sum of commuting projectors.Recall that in Subsection 3.1 we have defined on H a K Σ . v -bimodule structure (cid:101) A v for eachvertex v ∈ Σ and a H ∗ p -bimodule structure (cid:101) B ( p,v ) for each site ( p, v ) , p ∈ Σ , v ∈ Σ sit p . A K Σ . v -bimodule structure is equivalent to a left ( K Σ . v ⊗ K opΣ . v ) -action on H , so that specifyingan element of the so-called enveloping algebra ( K Σ . v ⊗ K opΣ . v ) determines an endomorphismof H . By assumption, all bicomodule algebras K e labelling the cell decomposition of Σ aresemisimple and, hence, the tensor product K Σ . v is semisimple and possesses a unique symmetricseparability idempotent p v ⊗ p v ∈ ( K Σ . v ⊗ K opΣ . v ) according to Proposition 17. Definition 22.
Let v ∈ Σ . The vertex operator for the vertex v is the idempotent endomor-phism of the state space H A v := (cid:101) A v ( p v ⊗ p v ) : H −→ H given by acting with the unique symmetric separability idempotent p v ⊗ p v ∈ K Σ . v ⊗ K opΣ . v via the K Σ . v -bimodule structure (cid:101) A v , defined in Definition 11.This operator is local in the sense that it acts as the identity on all tensor factors in H =( ⊗ e ∈ Σ K ∗ e ) ⊗ ( ⊗ w ∈ Σ Z w ) except for those associated to the vertex v ∈ Σ and to the edges e ∈ Σ v incident to v . Since the symmetric separability idempotent of a semisimple bicomodulealgebra generalizes the Haar integral of a semisimple Hopf algebra, as explained in Subsection3.2, we see that the vertex operator defined here provides a suitable analogon to the vertexoperators in the ordinary Kitaev model for a semisimple Hopf algebra.Next we want to define a projector on H for each plaquette p ∈ Σ in analogy to the plaquetteoperators of the ordinary Kitaev model for a semisimple Hopf algebra H , which are definedby acting with the Haar integral of the dual Hopf algebra H ∗ . In our construction, we havedefined in Definition 12 an H ∗ p -bimodule structure (cid:101) B ( p,v ) on H for every plaquette p ∈ Σ withincident site v ∈ Σ sit p and we can again use this to define a projector (cid:101) B ( p,v ) ( λ p (1) ⊗ S ( λ p (2) )) on H by acting with the symmetric separability idempotent of the semisimple algebra H ∗ p , whichis λ p (1) ⊗ S ( λ p (2) ) ∈ H ∗ p ⊗ ( H ∗ p ) op , see Example 18. However note that, as opposed to the vertexoperator here it is actually not necessary to invoke the concept of the symmetric separabilityidempotent, since H ∗ p is a Hopf algebra just as in the ordinary Kitaev model, and its symmetricseparability idempotent is given by the Haar integral.When considering the projector (cid:101) B ( p,v ) ( λ p (1) ⊗ S ( λ p (2) )) on H , it seems that a priori it dependsnot only on the plaquette p but also on the site v ∈ Σ sit p that we had to choose in Definition 12 inorder to define the bimodule structure (cid:101) B ( p,v ) . Just like the plaquette operators in the ordinaryKitaev model, we will show that due to the properties of the Haar integral the projector onlydepends on the plaquette p : Lemma 23.
Let p ∈ Σ . If λ p ∈ H ∗ p is the Haar integral of H ∗ p , then the endomorphism (cid:101) B ( p,v ) ( λ p (1) ⊗ S ( λ p (2) )) : H −→ H does not depend on the choice of the site v ∈ Σ sit p . roof. The endomorphism (cid:101) B ( p,v ) ( λ p (1) ⊗ S ( λ p (2) )) is equal to the endomorphism of H obtainedby acting with the Haar integral λ via the left H ∗ p -action B (cid:48) ( p,v ) on H that is the pullback of theleft ( H ∗ p ⊗ ( H ∗ p ) op ) -action (cid:101) B ( p,v ) along the algebra map (id H ∗ p ⊗ S ) ◦ ∆ : H ∗ p −→ H ∗ p ⊗ ( H ∗ p ) op .Next we observe that the action B (cid:48) ( p,v ) is independent of v for any cocommutative element λ of the Hopf algebra H ∗ p . Indeed, looking carefully at Definition 12, we extract from it that B (cid:48) ( p,v ) ( λ ) acts with the multiple coproduct of λ on the degrees of freedom of H in the boundaryof the plaquette p in a cyclic order starting at the vertex v . Therefore, for a different vertex v (cid:48) ∈ Σ sit p , the endomorphism B (cid:48) ( p,v (cid:48) ) ( λ ) will only differ by a cyclic shift in the multiple coproductof λ . But since λ is cocommutative, any multiple coproduct of it is invariant under such cyclicshifts of its tensor factors.Thus we have shown that the following is well-defined. Definition 24.
Let p ∈ Σ . The plaquette operator for the plaquette p is the idempotentendomorphism of the state space H B p := (cid:101) B ( p,v ) ( λ p (1) ⊗ S ( λ p (2) )) : H −→ H given by acting via the H ∗ p ⊗ ( H ∗ p ) op -action (cid:101) B ( p,v ) introduced in Definition 12 with the uniquesymmetric separability idempotent λ p (1) ⊗ S ( λ p (2) ) ∈ H ∗ p ⊗ ( H ∗ p ) op for H ∗ p . Here λ p ∈ H ∗ p is theHaar integral for H ∗ p .This operator is local in the sense that it acts as the identity on all tensor factors in H =( ⊗ e ∈ Σ K ∗ e ) ⊗ ( ⊗ v ∈ Σ Z v ) except for those associated to the edges e ∈ Σ p and the vertices v ∈ Σ p incident to the plaquette p .We have thus defined a family of projectors ( A v ) v ∈ Σ and ( B p ) p ∈ Σ on the vector space H .We now finally reach our main result that they all commute with each other. Theorem 25.
Let Σ be an oriented compact surface with a regular cell decomposition labeledby semisimple Hopf algebras, semisimple bicomodule algebras and representations of the ver-tex algebras, and let H be the associated vector space defined in Definition 9 with vertex andplaquette operators { ( A v ) v ∈ Σ , ( B p ) p ∈ Σ } defined in Definitions 22 and 24.Then any pair of vertex or plaquette operators commutes.Proof. Due to Theorem 13, the only non-trivial commutation relations between a K Σ . v -actionand an H ∗ p -action on H may occur when v and p are incident to each other. In that case, the K Σ . v -bimodule structure (cid:101) A v and the H ∗ p -bimodule structure (cid:101) B ( p,v ) together form a bimodulestructure over the crossed product algebra H ∗ p (cid:61) K Σ . v . However, due to Lemma 21 the symmet-ric separability idempotents for K Σ . v and H ∗ p commute in ( H ∗ p (cid:61) K Σ . v ) ⊗ ( H ∗ p (cid:61) K Σ . v ) op and,hence, the vertex operator A v and the plaquette operator B p commute with each other.This is completely analogous to the standard Kitaev model without defects: We have afamily of commuting projectors on the state space. Since any family of commuting projectorsis simultaneously diagonalizable, this allows for the definition of an exactly solvable Hamiltonianas the sum of commuting projectors. We thus conclude our construction of the Kitaev latticemodel with defects as follows: Definition 26.
The
Hamiltonian on the state space H assigned to an oriented surface Σ withlabeled cell decomposition as above is the diagonalizable endomorphism h := (cid:88) v ∈ Σ (id H − A v ) + (cid:88) p ∈ Σ (id H − B p ) : H −→ H . ground-state space is its kernel, H := ker h, i.e. the simultaneous -eigenspace for all the projectors { ( A v ) v ∈ Σ , ( B p ) p ∈ Σ } .Such a Hamiltonian is also called frustration-free , as its lowest eigenvalue is not lower thanany eigenvalue of its summands. Remark . The ground-state space H is isomorphic to the vector space that is category-theoretically realized by the modular functor constructed in [FSS19] for the defect surface Σ labeled by the corresponding representation categories of the Hopf algebras and bicomodulealgebras. We leave the detailed proof of this statement for a future update of this paper.As a consequence, the ground-state space H is invariant under fusion of defects and indepen-dent of the transparently labeled part of the cell decomposition. Moreover, due to the resultsof [FSS19], there will be a mapping class group action on H that can be explicitly computed.This allows to define quantum gates on the ground-state space in terms of the mapping classgroup action, as has been proposed before, and to address questions of universality of suchgates. We have thus constructed an explicit Hamiltonian model which offers the possibility forquantum computation, realizing a general framework for theories of the type discussed e.g. in[BJQ].A detailed investigation of the above and related questions remain for future work.23 . A category-theoretic motivation for the vertex algebras The construction in this paper takes as its input a compact oriented surface Σ , whose -cells arelabelled by Hopf algebras and whose -cells are labelled by bicomodule algebras. Furthermore,we have introduced in Definition 5, for every vertex v ∈ Σ , an algebra C v , which we call vertexalgebra. The category of possible labels for a vertex v ∈ Σ of the cell decomposition is thecategory of modules over the relevant vertex algebra C v , see Definition 7.On the other hand, in three-dimensional topological field theories and modular functorsdefined on surfaces with defects such as in [FSS19, KK], the strata are labelled by category-theoretic data: -cells by finite tensor categories and -cells by finite bimodule categories,which in our setting arise as the representation categories of the Hopf algebras and bicomodulealgebras that we use as labels for our construction.Furthermore, in [FSS19], a category is assigned to any boundary circle of a surface withdefects, which is equivalent to a Drinfeld center in the absence of defects. Such a boundarycircle can be intersected by defect lines labelled by bimodule categories, leading to markedpoints on the circle. In our construction this situation corresponds to a vertex v ∈ Σ at whicha number of edges labelled by bicomodule algebras meet. We can regard such a vertex as aboundary circle L v , cut into the surface Σ , at which defect lines end which are labelled by therepresentation categories of the corresponding bicomodule algebras.The main result of this section, Theorem 33, is that the category assigned to such a decoratedcircle with marked points L v according to the prescription of [FSS19], defined in Definition 30,is canonically isomorphic to the category of labels that we have defined in Definition 7 for sucha vertex v ∈ Σ in a labeled cell decomposition.First we must explain the category that is assigned to a boundary circle of a defect surfacein the construction of [FSS19]. For the category-theoretic background, see also [EGNO]. Weadapt the notions and notation to our setting, since it slightly differs from the one in [FSS19].Here, the tensor categories we consider are pivotal and the underlying defect surface is oriented,whereas in the reference no pivotal structures are used and instead the surfaces are framed.For a tensor category A and a sign ε ∈ { +1 , − } , write A ε := (cid:40) A , if ε = +1 , A , if ε = − , where A := A op , mop is the tensor category whose underlying linear category is the oppositecategory of A and whose tensor product is also opposite to the one of A , i.e. a ⊗ b := b ⊗ a for a, b ∈ A , where for any object a ∈ A we denote its corresponding object in the oppositecategory A by a , and likewise for morphisms. If A = H –mod for a Hopf algebra H , then A ∼ = H –mod canonically as tensor categories, where H := H op , cop is the Hopf algebra thathas the opposite multiplication as well as the opposite co-multiplication with respect to H .For X ∈ H –mod , the corresponding object X in H –mod is given by the vector space dual Hom k ( X, k ) of X with the natural induced H -action. For ε ∈ { +1 , − } , we also write H ε := H if ε = − , and H ε := H if ε = +1 .The right duality functor induces a monoidal equivalence, A −→ A , x (cid:55)−→ x ∨ . For A = H –mod for a Hopf algebra H , this equivalence takes an H -module X and turns it into an H -module by pulling back the H -action along the antipode S : H −→ H . Note that instead of theright dual functor one can also take any other odd-fold right or left dual. For our purposes thischoice does not matter, since the tensor categories which we will consider are pivotal, where allthese odd-fold duals are canonically identified. Indeed, for a semisimple Hopf algebra H , theantipode is involutive, so that all odd powers of the antipode are the same. (This is in contrast24o [FSS19] where no pivotal structures on the tensor categories are used, but instead -framingson the underlying surfaces are used to determine which multiple of the duality functor to usein a given moment in the construction.)If A and A are two tensor categories and M is an A - A -bimodule category, then theopposite linear category M := M op canonically becomes an A - A -bimodule category bydefining a (cid:46) m (cid:47) a := a (cid:46) m (cid:47) a for a ∈ A , m ∈ M , a ∈ A and likewise for morphisms.For ε ∈ { +1 , − } , we write M ε := (cid:40) M as an A - A -bimodule category , if ε = +1 , M as an A - A -bimodule category , if ε = − . If M = K –mod for an H - H -bicomodule algebra K , then M ∼ = K –mod canonically as ( H –mod) - ( H –mod) -bimodule categories, where K := K op is the opposite algebra with re-spect to K considered as an H - H -bicomodule algebra. For M ∈ K –mod , the correspondingobject M in K –mod is given by the vector space dual Hom k ( M, k ) of M with the naturalinduced K -action. For ε ∈ { +1 , − } , we also write K ε := K if ε = − , and K ε := K if ε = +1 .A boundary circle of an oriented surface with defect lines labeled by bimodule categories givesrise to the following data. Consider an oriented circle with n marked points ( e i ) i ∈ Z n that areeach labelled with a sign ε i ∈ { +1 , − } , so that we call these points oriented . Label eachsegment between two marked points e i and e i +1 by a finite pivotal tensor category A i,i +1 andlabel each marked point e i with a finite bimodule category M i , which is an A i − ,i - A i,i +1 -bimodule category if ε i = +1 , and an A i +1 ,i - A i,i − -bimodule category if ε i = − . In otherwords, then M ε i i is an A ε i i − ,i - A ε i i,i +1 -bimodule category, using the notation we have introducedabove for opposite tensor categories and opposite bimodule categories. The set ( M ε i i ) i ∈ Z n iscalled a string of cyclically composable bimodule categories , according to [FSS19].To this decorated circle with marked points, by the prescription of [FSS19], one associatesa linear category, which we will explain now, see Definition 30. First we consider the Deligneproduct M ε (cid:2) · · · (cid:2) M ε n n of the categories ( M ε i i ) i ∈ Z n . Following the above notation, corre-sponding to each segment between two marked points e i and e i +1 in the circle there is thestructure of an A ε i +1 i,i +1 - A ε i i,i +1 -bimodule category on this Deligne product. These n bimodulecategory structures on the Deligne product commute with each other (up to canonical coherentisomorphisms), since they act either on different Deligne factors or on two different sides of oneof the bimodule categories.For each of these bimodule category structures on the Deligne product we can considerso-called balancings ; e.g. for a (cid:2) -factorized object ( m ε (cid:2) · · · (cid:2) m nε n ) these are natural isomor-phisms ( m ε (cid:2) · · · (cid:2) m iε i (cid:2) ( a ε i +1 ε i +1 (cid:46) m i +1 ε i +1 ) (cid:2) · · · (cid:2) m nε n −→ m ε (cid:2) · · · (cid:2) ( m iε i (cid:47) a ε i ε i ) (cid:2) m i +1 ε i +1 (cid:2) · · · (cid:2) m nε n ) a ∈A i,i +1 Here, for any category X and ε ∈ { +1 , − } , we use the notation x ε := (cid:40) x ∈ X , if ε = +1 ,x ∈ X , if ε = − . for the object in X ε that corresponds to the object x ∈ X , and for a pivotal tensor category A we use the notation a ε := (cid:40) a, if ε = +1 ,a ∨ , if ε = − . (While this notation would make sense for any tensor category that is not necessarily pivotal,it would be unnatural as it would arguably favor the right dual functor over all other odd-foldduals. Therefore we assume that A is pivotal, which is the case of our interest anyway.)Let us recall the general definition of such balancings for bimodule categories.25 efinition 28. Let A be a pivotal tensor category, let ε, ε (cid:48) ∈ { +1 , − } and let M be an A ε - A ε (cid:48) -bimodule category.Then the category Z ε,ε (cid:48) ( M ) of balancings in M has as objects pairs ( m, β ) , where m is anobject of M and the balancing ( β a : a εε (cid:46) m ∼ −−→ m (cid:47) a ε (cid:48) ε (cid:48) ) a ∈A is a natural isomorphism satisfying ( a ⊗ b ) εε (cid:46) m ∼ = a εε (cid:46) b εε (cid:46) m a εε (cid:46) m (cid:47) b ε (cid:48) ε (cid:48) m (cid:47) ( a ⊗ b ) ε (cid:48) ε (cid:48) ∼ = m (cid:47) a ε (cid:48) ε (cid:48) (cid:47) b ε (cid:48) ε (cid:48) id a εε (cid:46)β b β a ⊗ b β a (cid:47) id b ε (cid:48) ε (cid:48) I εε (cid:46) m mm (cid:47) I ε (cid:48) ε (cid:48) ∼ = β I ∼ = or, in formulas, β a ⊗ b = ( β a (cid:47) id b ε (cid:48) ε (cid:48) ) ◦ (id a εε (cid:46)β b ) ∀ a, b ∈ A , (14) β I = id m , (15)where we have omitted the bimodule constraint isomorphisms.The morphisms in the category of balancings are defined to be the morphisms in M that arecompatible with the balancings. Remark . While this definition does not require any pivotal structure on the tensor category– one can consider every dual to be the right dual, for example – we will consider it only for apivotal tensor category, since otherwise it would not coincide with the definition of the categoryof κ -balancings from [FSS19] for an integer κ ∈ Z . In the construction in [FSS19] this integercomes from a framing of the underlying surface and determines which of the various multiplesof the double-dual functor, which are trivialised by a pivotal structure, we would need to insertin the above definition.The category that one finally assigns to the decorated circle with marked points, accordingto the prescription of [FSS19] is as follows: Definition 30 (c.f. Definition 3.4 in [FSS19]) . Let L be an oriented circle with marked orientedpoints { e i } i ∈ Z n labelled by bimodule categories – giving rise to a string ( M ε i i ) i ∈ Z n of cyclicallycomposable bimodule categories. The category T ( L ) assigned to the circle L is the categoryof balancings on the Deligne product ( (cid:2) i ∈ Z n M ε i i ) with respect to the A ε i +1 i,i +1 - A ε i i,i +1 -bimodulecategory structures for all i ∈ Z n . In formulas,T ( L ) := Z ε ,ε n ( · · · Z ε ,ε ( (cid:2) i ∈ Z n M ε i i )) . (16) Remarks . • This category is well-defined because the bimodule category structures on the Deligneproduct, with respect to which the balancings are considered, all commute with eachother (up to canonical coherent natural isomorphisms). In [FSS19] it is explained thatthe category of balancings is monadic and that the monads for the balancings for thedifferent bimodule category structures on the Deligne product satisfy a distributivitylaw, which also shows that (16) does not depend on the order in which we consider thebalancings. 26 The category assigned to a decorated circle with marked points reduces to the well-knownDrinfeld center Z ( A ) , as shown in [FSS19], if all bimodule categories M i are given by asingle tensor category A .In Theorem 33 we want to give a realization of such a category assigned to a decoratedcircle with marked points, in terms of representations of a k -algebra, namely the vertex algebra C v , if the bimodule categories ( M i ) i are the representation categories of bicomodule algebras ( K e ) e ∈ Σ . v .To this end, we first show generally that the category of balancings, as in Definition 28, canbe realized in such a representation-theoretic way. For this, let H be a finite-dimensional Hopfalgebra over k , let ε, ε (cid:48) ∈ { +1 , − } and let K be an H ε - H ε (cid:48) -bicomodule algebra. Recall fromSubsubsection 2.1.1 that the category K –mod is an H ε - H ε (cid:48) -bimodule category, so that we canconsider the category of balancings Z ε,ε (cid:48) ( K –mod) as defined in Definition 28. On the otherhand, recall from Definition 3 the so-called balancing algebra H ∗ ε,ε (cid:48) , which is an (( H ε (cid:48) ) cop ⊗ H ε ) -module algebra, and recall from Definition 4 the crossed product algebra H ∗ ε,ε (cid:48) (cid:61) K , for which weconsider K as an (( H ε (cid:48) ) cop ⊗ H ε ) -comodule algebra. This k -algebra H ∗ ε,ε (cid:48) (cid:61) K with underlyingvector space H ∗ ⊗ K is characterized by having H ∗ and K as subalgebras, and by the followinginstance of the straightening formula for the multiplication of an element f ∈ H ∗ with anelement k ∈ K : k · f = f ( k (cid:104)− ε (cid:48) (cid:105) (1) · ? · k (cid:104) ε (cid:105) ( − ) · k (0) (17)The following proposition proves that the category of balancings on K –mod is isomorphicto the representation category of the k -algebra H ∗ ε,ε (cid:48) (cid:61) K . This justifies the name “balancingalgebra” for H ∗ ε,ε (cid:48) and will be used in Theorem 33 to establish a connection between the vertexalgebras defined in this paper and the categories assigned to circles in [FSS19]. Proposition 32.
Let H be a semisimple finite-dimensional Hopf algebra over k , let ε, ε (cid:48) ∈{ +1 , − } and let K be an H ε - H ε (cid:48) -bicomodule algebra. Then there is a canonical equivalence of k -linear categories Z ε,ε (cid:48) ( K –mod) ∼ = ( H ∗ ε,ε (cid:48) (cid:61) K )–mod . Proof.
Let ( M, β = ( β X : X εε (cid:46) M ∼ −−→ M (cid:47) X ε (cid:48) ε (cid:48) ) X ∈ H –mod ) be an object in Z ε,ε (cid:48) ( K –mod) .Recall that the vector spaces underlying the modules X εε ∈ H ε –mod and X ε (cid:48) ε (cid:48) ∈ H ε (cid:48) –mod arethe same as X ∈ H –mod . In this proof, to simplify notation, we will often write β X as a map X ⊗ M −→ M ⊗ X , keeping implicit the module structures on the respective vector spaces.We define, using β , a left H ∗ -module structure on M as follows. We denote by H reg ∈ H –mod the left regular H -module with underlying vector space H , whose H -action is defined by leftmultiplication. ρ : H ∗ ⊗ M −→ M, (18) f ⊗ m (cid:55)−→ (id M ⊗ f ) β H reg (1 H ⊗ m ) We show that this indeed satisfies the axioms of a left H ∗ -module: On the one hand we have,for f, g ∈ H ∗ and m ∈ M , ρ ( f ⊗ ρ ( g ⊗ m )) def = (id M ⊗ f ) β H reg (1 H ⊗ (id M ⊗ g ) β H reg (1 H ⊗ m ))= (id M ⊗ f ⊗ g )( β H reg ⊗ id H )(id H ⊗ β H reg )(1 H ⊗ H ⊗ m ) . On the other hand, we have ρ (( f · g ) ⊗ m ) = (id M ⊗ ( f · g )) β H reg (1 H ⊗ m )= (id M ⊗ f ⊗ g )(id M ⊗ ∆) β H reg (1 H ⊗ m ) natural = (id M ⊗ f ⊗ g ) β H reg ⊗ H reg (∆(1 H ) ⊗ m )= (id M ⊗ f ⊗ g ) β H reg ⊗ H reg (1 H ⊗ H ⊗ m ) (14) = (id M ⊗ f ⊗ g )( β H reg ⊗ id H )(id H ⊗ β H reg )(1 H ⊗ H ⊗ m ) , where we use in the third line that the coproduct of H is an H -module morphism ∆ : H reg −→ H reg ⊗ H reg . This shows one of the two axioms of an H ∗ -module. For the other axiom, let again m ∈ M . Then, indeed, we have ρ (1 H ∗ ⊗ m ) = ρ ( ε ⊗ m ) def = (id M ⊗ ε ) β H reg (1 H ⊗ m ) β natural = β k ( ε (1 H ) ⊗ m )= m, where we use in the third line that the co-unit of H is an H -module morphism ε : H reg −→ k .Hence, we have shown that ρ endows M with the structure of an H ∗ -module.To prove that ( M, ρ ) is an object of ( H ∗ ε,ε (cid:48) (cid:61) K )–mod we have to show that the just defined H ∗ -action ρ and the given K -action on M , which we simply denote by K ⊗ M → M, k ⊗ m (cid:55)→ k.m , satisfy the straightening formula (17). That is, we have to show that, for all f ∈ H ∗ , k ∈ K and m ∈ M , k. ((id M ⊗ f ) β H reg (1 H ⊗ m )) = (id M ⊗ f ( k (cid:104)− ε (cid:48) (cid:105) (1) · ? · k (cid:104) ε (cid:105) ( − )) β H reg (1 H ⊗ k (0) .m ) (19)We start with the right-hand side: (id M ⊗ f ( k (cid:104)− ε (cid:48) (cid:105) (1) · ? · k (cid:104) ε (cid:105) ( − )) β H reg (1 H ⊗ k (0) .m ) β natural = (id M ⊗ f ( k (cid:104)− ε (cid:48) (cid:105) (1) · ?)) β H reg ( k (cid:104) ε (cid:105) ( − ⊗ k (0) .m ) β H reg K -linear = (( k (0) . ?) ⊗ f ( k (cid:104)− ε (cid:48) (cid:105) (2) k (cid:104) ε (cid:48) (cid:105) (1) · ?)) β H reg (1 H ⊗ m )= k. ((id M ⊗ f ) β H reg (1 H ⊗ m )) . Here we use in the first line that right multiplication by any element h ∈ H is an H -modulemorphism (? · h ) : H reg −→ H reg for the left regular H -module H reg , and in the last linewe use the defining property of the antipode of H . This concludes the proof that ( M, ρ ) ∈ ( H ∗ ε,ε (cid:48) (cid:61) K )–mod .Conversely, assume that M ∈ ( H ∗ ε,ε (cid:48) (cid:61) K )–mod and let us define on M a balancing β X : X εε (cid:46) M −→ M (cid:47) X ε (cid:48) ε (cid:48) for all X ∈ H –mod . Denoting by ( e i ∈ H ∗ ) i and ( e i ∈ H ) i a pair ofdual bases, we define β X : X ⊗ M −→ M ⊗ X,x ⊗ m (cid:55)−→ (cid:88) i e i .m ⊗ e i .x, where e i .x refers to X as an H -module, not X ε (cid:48) ε (cid:48) as an H ε (cid:48) -module, even though we will showthat β X is a K -module morphism X εε (cid:46) M −→ M (cid:47) X ε (cid:48) ε (cid:48) . Indeed, for k ∈ K, x ∈ X, m ∈ M ,we calculate k. ( β X ( x ⊗ m ) def = (cid:88) i ( k (0) .e i .m ) ⊗ ( k (cid:104) ε (cid:48) (cid:105) (1) .e i .x ) = (cid:88) i ( e i ( k (cid:104)− ε (cid:48) (cid:105) (1) · ? · k (cid:104) ε (cid:105) ( − ) .k (0) .m ) ⊗ ( k (cid:104) ε (cid:48) (cid:105) (2) .e i .x )= (cid:88) i ( e i .k (0) .m ) ⊗ ( k (cid:104) ε (cid:48) (cid:105) (2) .k (cid:104)− ε (cid:48) (cid:105) (1) .e i .k (cid:104) ε (cid:105) ( − .x )= (cid:88) i ( e i .k (0) .m ) ⊗ ( e i .k (cid:104) ε (cid:105) ( − .x ) def = β X ( k. ( x ⊗ m )) Furthermore, it can be seen directly that ( β X ) X ∈ H –mod is a natural family. Indeed, for any H -module morphism f : X −→ Y and x ∈ X, m ∈ M , we have β Y ( f ( x ) ⊗ m ) def = (cid:80) i e i .m ⊗ e i . ( f ( x )) = (cid:80) i e i .m ⊗ f ( e i .x ) def = (id M ⊗ f ) β X ( x ⊗ m ) .It remains to show that ( β X ) X ∈ H –mod satisfies axioms (14) and (15), i.e. β X ⊗ Y = ( β X ⊗ id Y )(id X ⊗ β Y ) for all X, Y ∈ H –mod and β k = id M .For the first identity, let x ∈ X , y ∈ Y and m ∈ M . Then on the one hand we have β X ⊗ Y ( x ⊗ y ⊗ m ) def = (cid:80) i e i .m ⊗ e i . ( x ⊗ y ) = (cid:80) i e i .m ⊗ ( e i (1) .x ) ⊗ ( e i (2) .y ) . On the other hand, ( β X ⊗ id Y )(id X ⊗ β Y )( x ⊗ y ⊗ m ) def = (cid:80) i,j e j .e i .m ⊗ e j .x ⊗ e i .y = (cid:80) i e i .m ⊗ ( e i (1) .x ) ⊗ ( e i (2) .y ) , wherethe last identity uses that the multiplication of H ∗ is defined as the dual of the co-multiplicationof H .In order to show (15), we use that the unit of H ∗ is the co-unit ε : H → k of H . For λ ∈ k and m ∈ M we thus have β k ( m ⊗ λ ) def = (cid:80) i e i .m ⊗ ε ( e i ) λ = 1 H ∗ .m = m .So far in this proof, we have shown that on M ∈ K –mod one can construct out of a bal-ancing on M an H ∗ -action such that M becomes an ( H ∗ ε,ε (cid:48) (cid:61) K ) -module, and that converselyout of an ( H ∗ ε,ε (cid:48) (cid:61) K ) -module structure one can construct a balancing on M ∈ K –mod . Toconclude the proof of the proposition we have to show that these two assignments are inverseto each other.First, assume that ( M, β ) ∈ Z ε,ε (cid:48) ( K –mod) . Consider the balancing β (cid:48) on M that is con-structed from the H ∗ -action on M which in turn is constructed from β , as shown above. For X ∈ H –mod , x ∈ X and m ∈ M we have β (cid:48) X ( x ⊗ m ) def = (cid:88) i (id M ⊗ e i ) β H reg (1 H ⊗ m ) ⊗ e i .x = ( β H reg (1 H ⊗ m )) ( M ) ⊗ ( β H reg (1 H ⊗ m )) ( X ) .x β natural = β X ( x ⊗ m ) , where use the notation ( β H reg (1 H ⊗ m )) ( M ) ⊗ ( β H reg (1 H ⊗ m )) ( X ) := β H reg (1 H ⊗ m ) ∈ M ⊗ X ,and in the third line we use that (? .x ) : H reg −→ X is an H -module morphism for any x ∈ X .Finally, assume that M ∈ ( H ∗ ε,ε (cid:48) (cid:61) K )–mod with H ∗ -action ρ : H ∗ ⊗ M −→ M . Consider the H ∗ -action ρ (cid:48) on M that is constructed from the balancing on M which in turn is constructedfrom ρ , as shown above. For f ∈ H ∗ and m ∈ M we then have ρ (cid:48) ( f ⊗ m ) def = (cid:88) i (id M ⊗ f )( ρ ( e i ⊗ m ) ⊗ e i . H ) = (cid:88) i ρ ( e i ⊗ m ) f ( e i ) = ρ ( f ⊗ m ) , which concludes the proof of the proposition.Now, finally, we can prove the main result of this appendix. Most of the work for this hasalready been done in the proof of Proposition 32. Let v ∈ Σ be a vertex of a labeled celldecomposition of Σ so that ( K e ) e ∈ Σ . v are bicomodule algebras labelling the incident edgesat v . Let L v be the corresponding circle with marked points which are labeled by cyclicallycomposable bimodule categories ( K e –mod) e ∈ Σ . v .29 heorem 33. Let v ∈ Σ be a vertex in a labelled (as defined in Definition 1) cell decompositionof a compact oriented surface Σ . There is a canonical equivalence of k -linear categories T ( L v ) ∼ = C v –mod between the category assigned by the modular functor T , constructed in [FSS19], to the circle L v with marked points corresponding to the half-edges incident to a vertex v ∈ Σ and therepresentation category of the algebra C v .Proof. Consider the bicomodule algebra ( (cid:78) e ∈ Σ . v K ε ( e ) e ) , which realizes the Deligne product (cid:2) e ∈ Σ . v ( K e –mod) ε ( e ) = ( (cid:78) e ∈ Σ . v K ε ( e ) e )–mod as a representation category. For each incidentsite p ∈ Σ sit v , which corresponds to a segment between two marked points of the correspondingdecorated circle L v and is labeled by a Hopf algebra H p , it has an H ε ( e p ) p - H ε ( e (cid:48) p ) p -bicomodulestructure, where e p and e (cid:48) p are half-edges incident to v in the boundary of the plaquette p , cf.Figure 5. Denote the sites in Σ sit v in clockwise order around v by ( p , , . . . , p n, ) and abbreviate ε ( e p i,i +1 ) =: ε i +1 and ε ( e (cid:48) p i,i +1 ) =: ε i . We then repeatedly apply Proposition 32 for each of these H ε i +1 p i,i +1 - H ε i p i,i +1 -bicomodule structures. This is well-defined and does not depend on the order,since for different p ∈ Σ sit v the bicomodule structures commute with each other. 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