arXiv:1709.00522v1 [math.QA] 2 Sep 2017

Kitaev model for mirror bicrossproduct Hopf algebras and its tensor network representation

Florian Girelli,a Prince K. Osei,b Abdulmajid Osumanua a Department

of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

b Perimeter

E-mail: [email protected], [email protected], [email protected] Abstract: Kitaev’s model [1, 2] is usually defined in terms of the Drinfeld double. We propose a new version defined in terms of the mirror bicrossproduct quantum group [3]. By some aspects, even though the bicrossproduct quantum group is more complicated than the Drinfeld double, the construction of the new model is actually simpler and relies on the use of a covariant system. We also provide the definition of the ground state of the new Hamiltonian in terms of a tensor network representation.

Contents 1 Introduction

1

2 Kitaev model and covariant system 2.1 Some notations 2.2 Definition of the Kitaev lattice model in terms of a covariant system

3 3 4

3 Bicrossproduct quantum groups 3.1 Bicrossproduct Hopf algebras from semidualization 3.2 Mirror bicrossproduct Hopf algebra

7 8 8

4 Kitaev model for mirror bicrossproduct Hopf algebras

10

5 Tensor network representations for Bicrossproduct models 5.1 Diagrammatic scheme for tensor network states 5.2 Tensor network representation for the bicrossproduct model

16 16 19

6 Outlook

23

A Some relevant features of Hopf Algebras A.1 Star structure and dual pairing A.2 Semisimplicity and Haar integrals

24 24 25

B Face and vertex operators for some simple examples of graph

25

1

Introduction

Topological quantum field theories (TQFT) despite being simpler than local quantum field theories contain rich structures interesting both for the mathematical and physical standpoints. Defined in terms of (quasitriangular) Hopf algebras or ‘quantum groups’, TQFT’s in 3d provide models for different frameworks such as quantum gravity (QG) [4] and topological quantum information (TQI) [1]. Each framework comes with its own motivations but they share similar mathematical concepts. For example, in both cases, one deals with a (ribbon) graph decorated by Hopf algebra elements. Furthermore, in the TQI framework, we deal with a Hamiltonian defined in terms of operators acting on the nodes of the graph and the faces. The vacuum state can be interpreted from the quantum gravity perspective as the pure gravity case, whereas the excitations of the

–1–

TQI Hamiltonian, used to perform quantum computations, are interpreted as particles with mass or spin depending on their location. In the loop quantum gravity case, on the nodes we have torsion excitations, i.e. spin, whereas on the faces, we have curvature excitations, i.e. mass. The most relevant algebraic structure, for example to deal with representations which classify particles for example and indicate their braiding, is not only the Hopf algebra H but the associated Drinfeld double D(H). Once again, this structure was identified using different arguments in each of the different frameworks. In the TQI case, we deal with the Drinfeld’s quantum double of finite dimensional semisimple Hopf algebras (e.g. built from finite groups) [1, 2, 5] whereas in the quantum gravity case, we deal with infinite dimensional Hopf algebras as built from Lie groups or their quantum deformation [6–15]. This illustrates how the same structures appear in these differently physically motivated frameworks. One can find more rigorous discussions on how to bridge TQI models with QG models for example in [16] or [17]. Due to these common roots, we can expect fruitful interactions by confronting results obtained in one framework to the other framework. For example, recently, a dual picture was introduced in the quantum gravity setting where the excitations have been swapped [18]. Even though this was discovered independently, this result could have been guessed in light of the notion of electro-magnetic duality well known in TQI [19]. The Drinfeld double is in a sense the most commonly used quantum group, however it is not the only one. The bicrossproduct quantum group has been proposed by Majid [3] using motivations from QG. This quantum group is different in nature than the quantum double, that is its representation theory is different. Roughly speaking we can obtain a bicrossproduct quantum group by dualizing half of the Drinfeld double, through a semi-dualization map [20]. For a long time, the universal R-matrix for this quantum group was missing (when it was expected to exist) and was only very recently identified [21], allowing now for a full use of this quantum group. This new R-matrix is obtained by a Drinfled twist between the quantum double and the bicrossproduct quantum group. Particles corresponding to the irreducible representations of the bicrossproduct quantum group can then be braided using this quasitriangular structure. The bicrossproduct quantum group has been advocated to be relevant to construct non-commutative space-time in any dimensions in particular in the context of quantum gravity phenomenology [22]. In the study of quantum gravity models formulated in terms of the combinatorial quantization of the Chern-Simons action, these bicrossproduct quantum group also appeared as relevant [23–25]. From this perspective, the bicrossproduct quantum group is well-motivated from the gravity/Chern-Simons framework. Hence it is natural to ask how it can be useful in the TQI framework. For this we first need to define the analogue of Kitaev’s model, in terms of this quantum group instead of the Drinfeld double. This is what we intend to do in the following.

–2–

In order to do so we will work with the concept of covariant system (K, A), which consists essentially into a Hopf algebra K and an algebra A which is carries a covariant action of the Hopf algebra K. It is covariant in the sense that A is a module algebra, that is the action is consistent with the product of A. Interestingly the standard Kitaev model in terms of the Drinfeld double is not an example of such covariant system (except for very special graphs), so that we really have a new model. Kitaev model/Hamiltonian is defined in terms of two types of operator, the vertex operator and the face operator, which provide a representation of K. As such, it is usually assumed that K will be constructed in terms of two sub-Hopf algebras, H1 and H2 . Each of the operators will then consist in the action of a H1 or H2 on A such that they define a representation of K. We will then define a tensor network representation which will allow to express explicitly the ground state of the Hamiltonian. The tensor network states are build from a diagrammatic framework similar to that introduced in [2]. The scheme of the article goes as follows. In Section I, we provide the general recipe for constructing the Kitaev model in terms of a covariant system. In Section II, we review the definition of the bicrossproduct quantum group and the specific example given in terms of the mirror bicrossproduct. Essentially it is obtained by semi-dualizing the Drinfeld double. This will be the key example of covariant system which we will focus on to build the new Kitaev model. In Section III, we construct explicitly the Kitaev Hamiltonian for such mirror bicrossproduct quantum group. In Section IV, we define the tensor representation of the model, providing in particular the realization of the ground state in this setting.

2

Kitaev model and covariant system

We present here a general scheme to describe Kitaev lattice models in terms of a covariant system. Let us begin by fixing some notations. 2.1

Some notations

We follow conventions for Hopf algebras in [3]. A Hopf algebra or ‘quantum group’ H is an algebra and a coalgebra, with a linear coproduct ∆ : H → H ⊗ H which is an algebra homomorphism and satisfies (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆. We use Sweedler notation for the coproduct so that for all h ∈ H, ∆(h) = h(1) ⊗ h(2) = h(1) ⊗ h(2) . There is also a counit  : H → C and an antipode S : H → H defined by (Sh(1) )h(2) = h(1) Sh(2) = (h) for all h ∈ H. We say that an algebra A is an H-module if H acts on A from the left. An algebra A is an H-module algebra if A is a left H-module (i.e. H acts on A from the left) and this action is covariant, i.e. h.(ab) = (h(1) .a)(h(2) .b),

h.1 = (h), a ∈ A, h ∈ H,

–3–

(2.1)

where . is a left action. We refer to the pair (H, A)L (resp. (H, A)R ) as a left (resp. right)covariant system if A is a module algebra under the left (resp. right) action of H. An important fact to be seen later on covariant systems, is that if H acts covariantly on A from the right then one can turn this to a left action of H on Aop , according to the relation (K, A)R → (K, Aop )L ,

h . a = a / S −1 h.

(2.2)

By default, when no index L, R is specified, we will deal with a left covariant system. A coalgebra C is a left H-module coalgebra if ∆(h.c) = h(1) .c(1) ⊗ h(2) .c(2) ,

(h.c) = (h)(c), c ∈ C, h ∈ H.

(2.3)

We denote by H ∗ the dual Hopf algebra with dual pairing given by the non-degenerate bilinear map h , i and H cop , H op denote taking the opposite coproduct or opposite product. 2.2

Definition of the Kitaev lattice model in terms of a covariant system

Lattice definition: We consider Σ a 2d compact oriented manifold without boundary for simplicity. We will note Γ a discretization of Σ which can be the 1-skeleton of a polytope decomposition noted ∆Σ , or of the dual of the polytope decomposition ∆∗Σ . We will note V, E, F respectively the set of vertices, edges, faces of ∆Σ or ∆∗Σ . The vertex will need to be equipped with a cilium which specifies an order for the edges. By convention, we take the inverse trigonometric orientation. In the following, we will consider a site (v, p) of ∆Σ or ∆∗Σ which consists in a choice of face p and adjacent vertex v. This choice naturally specifies a cilium. Hilbert space: The states of the theory live on the edges e of Γ. They are given in terms of a finite-dimensional semi-simple Hopf ?-algebra A which we will note H when Γ is the 1-skeleton of ∆Σ , or by H ∗ , the dual of H, when Γ is the 1-skeleton of ∆∗Σ . This duality choice allows to implement consistently the Poincar´e duality between the two possible decorated graphs. Equipping A with an inner product, thanks to the ?-structure and a Haar integral, the Hopf algebra is also an Hilbert space. We refer to the Appendix A for some Hopf algebra notations and relevant properties. The total Hilbert space is then given by O HΓ = A. e∈Γ

–4–

Covariant system choice: The fundamental object for the construction is given by a (left) covariant system (K, A), with A = H or H ∗ . By definition, K is a Hopf algebra1 acting on A and A is then a module algebra. One can view A as the canonical representation of K. We require K to be built from two Hopf algebras H1 and H2 . This is motivated by the fact that we will define geometric operators which encode each the action of these sub-Hopf algebras, while providing a representation of K. As a consequence there is a covariant action of H1 on A and of H2 on A. Furthermore we demand also that (i) H2 is equipped with a Haar integral and it is dually paired with A, (ii) A is equipped with a Haar integral and it is an H1 -module coalgebra. From our assumption, we can split the algebra K into K = H2 ./ H1 (or H1 ./ H2 ), where some actions might be trivial. This means that we can use the algebra product · of K to define some natural actions of H1 on H2 and vice versa. k1 · k2 = (1 ⊗ h) · (a ⊗ 1) = (h0  a0 ) ⊗ (h00
a00 ),

a, a0 , a00 ∈ H2 , h, h0 , h00 ∈ H1 . (2.4)

The elements h0 , h00 (resp. a0 , a00 ) are depending on h (resp. a), their exact value given by the specific shape of the product, coproduct and antipode structures of K. We do not specify them further since we do not restrict further the Hopf structure of K in terms of H1 and H2 . Triangle operators: We use the actions of H1 and H2 on A to define the triangle operators L± , T± which are the basic building blocks of the Kitaev lattice model. The left action of H1 on A and the left action of H2 on A are noted respectively L+ and T+ . h ∈ H1 , a ∈ H2 , φ ∈ A. (2.5) Lh+ (φ) := h.φ, T+a (φ) := a.φ, We will also need the corresponding right actions of H1 and H2 on A, constructed thanks to the antipodes of A and Hi . S(h)

Lh− (φ) = (S ◦ L+

◦ S)(φ),

S(a)

T−a (φ) = (S ◦ T+

◦ S)(φ).

(2.6)

Given an oriented edge e, we associate geometrically the action of • L− to the source of the edge e, and L+ to the target of the edge e, • T− to the face on the left of the edge e, and T+ to the face on the right of the edge e.

–5–

T L

H1

•

H2

A

•

H

L+ 1

H

T+ 2

Figure Figure 1. 1. An An oriented oriented edge edge showing showing the the di↵erent different operators operators on on A. A. We We have have the the left/right left/right actions of H on A associated to the vertices of e and the left/right actions of H 1 2 actions of H1 on A associated to the vertices of e and the left/right actions of H2 on on AA on on the faces delimited by e. the faces delimited by e.

The The exact exact element element of of H Hi i by by which which these these operators operators act act will will depend depend on on the the number number of edges. Indeed, A is associated to one edge of . However in general we will of edges. Indeed, A is associated to one edge of Γ. However in general we will deal deal with the action of H and H on many edges. Hence we need to extend the action 1 2 with the action of H1 and H2 on many edges. Hence we need to extend the action of of H H11 and and H H22 to to many many copies copies ofof A, A, thanks thanks to to the the coproduct coproduct of of K. K. Geometric Geometric operators: operators: We We will will focus focus on on specific specific combinations combinations of of the the triangle triangle operoperators ators––noted notedrespectively respectivelyAAhh(v, (v,p) p)and andBBaa(v, (v,p) p)to toencode encodetheir theirdependence dependenceon onthe thesite site (v, p) – which contain a specific geometric information. (v, p) – which contain a specific geometric information. •• AAhh(v, (v,p) p)encodes encodesthe theaction actionofofH H11 on onthe thestates stateswhich whichedges edgesshare sharethe thevertex vertexvv of of , starting from the cilium specified by the face p. Γ, starting from the cilium specified by the face p. Y Y HΓ −→ HΓ,, (2.7) (v,p) p)== LL±± ::H AAhh(v, !H (2.7) e3v e3v

(v,p) p)encodes encodesthe theaction actionofofH H22 on onthe thestates stateswhich whichedges edgesdelimitate delimitatethe theface faceff •• BBaa(v, ∗ ⇤ starting from from the the vertex vertex v.v. of ∆⌃Σ or or ∆⌃Σ,, starting of Y Y HΓ −→ HΓ.. (2.8) (v,p) p)== BBaa(v, TT±± ::H !H (2.8) e∈∂f e2@f

The choice choice of of triangle triangle operator operator LL++ or or LL− (resp. (resp. TT++ or or TT−)) in in AAvv (resp. (resp. BBff)) The depends on the orientation of the edge of interest. Furthermore we have not encoded depends on the orientation of the edge of interest. Furthermore we have not encoded the dependence dependence in in aa or or hh in in the the triangle triangle operators operators since since itit depends depends heavily heavily on on the the the coproduct of K which we have not specified here. coproduct of K which we have not specified here. The important important feature feature of of these these operators operators isis that that they they provide provide aa representation representation of of The the algebra algebra structure structure of of K, K, as as given given in in (2.4) (2.4) when when they they act act on on the the same same site site (v, (v,p). p). the 0 (v, p)Ba00
h00 (v, p). (v,p)A p)Ah(v, (v,p) p)==AAa0 0 h0 (v, BBaa(v, p)Ba00 h00 (v, p). h a ⇤h

(2.9) (2.9)

This shows shows that that the the graph graph provides provides aa representation representation of of the the quantum quantum group group K. K. Note Note This that to to obtain obtain such such relation, relation, the the coproduct coproduct ofof K K isis again again fully fully needed. needed. The The geometric geometric that operators are also expected to satisfy the following properties. operators are also expected to satisfy the following properties. 1 1 Later

on, to talk about excitations and braiding, one might require it to be quasi-triangular. Later on, to talk about excitations and braiding, one might require it to be quasi-triangular.

–– 66 ––

(i) Ah (v, p) ◦ Ag (w, p0 ) = Ag (w, p0 ) ◦ Ah (v, p) if the two vertices v and w do not coincide. (ii) Ba (v, p) ◦ Bb (w, q) = Bb (w, q) ◦ Ba (v, p) if the two faces p and q do not coincide. (iii) Ah (v, p) ◦ Bb (v 0 , p0 ) = Bb (v 0 , p0 ) ◦ Ah (v, p) at different sites. Hamiltonian: The vertex and face operators, for each vertex and face of Γ are ”summed” over, thanks to the Haar integrals l and k of respectively H1 and H2 . We define the ”averaged” geometric operators A(v) := Al (v, p),

B(p) := Bk (v, p).

(2.10)

An important feature of these operators is that they are actually projectors. A2 (v) = Al2 (v, p) = Al (v, p) = A(v),

B 2 (p) = Bk2 (v, p) = Bk (v, p) = B(p).(2.11)

We can now define the Hamiltonian of the theory: X X (id − B(p)). (id − A(v)) + H=

(2.12)

p∈F

v∈V

By requiring the operators A(v) and B(p) to be self-adjoint, we insure the Hamiltonian to be self adjoint. The last important piece is given by the ground state or protected space of the Hamiltonian (4.12). It is given by the invariant subspace P of H: PΓ := {φ ∈ HΓ : H(φ) = 0}.

(2.13)

The protected space is also a topological invariant of the oriented surface Σ [1, 2].

3

Bicrossproduct quantum groups

We briefly review the features of the bicrossproduct construction which are required in the current application and refer the reader to the book [3] for a comprehensive construction of bicrossproduct quantum groups. Some features of semi-simple Hopf algebras are summarized in appendix A and we refer to [3] or [26] for a general background on Hopf algebras. For simplicity, we will always focus on finite dimensional Hopf algebras.

–7–

3.1

Bicrossproduct Hopf algebras from semidualization

Consider a Hopf algebra H which factorizes into two sub-Hopf algebras H1 , H2 and built on the vector space H1 ⊗ H2 . Factorization here implies an isomorphism of linear spaces given by the map H1 ⊗ H2 → H. This gives rise to the actions . : H2 ⊗ H1 → H1 and / : H2 ⊗ H1 → H2 of each Hopf algebra on the vector space of the other and define a double crossproduct Hopf algebra H1 ./H2 . The actions enter the definition of the product on H1 ⊗H2 as (1⊗a).(h⊗1) = a(1) .h(1) ⊗a(2) /h(2) . The coproduct of H1 ./H2 is given by the tensor coproduct coming from each factor. There is a covariant left action of H1 ./H2 on H2∗ as a module algebra, leading to a left covariant system (H1 ./H2 , H2∗ ) with H2∗ >/(H1 ./H2 ) as its associated left cross product algebra. The semidual of the double cross product is obtained by dualising half of the match pair data and gives a bicrossproduct Hopf algebra. More precisely, replacing H2 with H2∗ gives a bicrossproduct Hopf algebra H2∗ I/H1 , which then acts covariantly on H2 from the right as an algebra. The covariant action of the bicrossproduct gives the right covariant system (H2∗ I/H1 , H2 ). The explicit details are given in [21] as follows. The left action . : H1 ⊗ H2∗ → H2∗ of H1 on H2∗ and a right coaction ∆R : H1 → H1 ⊗ H2∗ of H2∗ on H1 are defined by φ ∈ H2∗ ,

(h.φ)(a) := φ(a/h), 0

1

h hh , ai = a.h,

h ∈ H1 ,

a ∈ H2 ,

a ∈ H2

h ∈ H1

∆R h = h ⊗ h1 ∈ H1 ⊗ H2∗ . 0

These define the bicrosspropduct H2∗ I/H1 by a left handed cross product H2∗ >/H1 as an algebra and a right handed cross coproduct H2∗ I

/ and .< put together. The bicrossproduct is I/ (or .J) and is the semidual of ./ as explained above

–8–

product by a mutual left coadjoint action of H ∗op on H and a right coadjoint action of H on H ∗op which are given respectively by φ.h = h(2) hh(1) , φ(1) ihSh(3) , φ(2) i,

φ/h = φ(2) hh(1) , φ(1) ihSh(2) , φ(3) i, h ∈ H, φ ∈ H ∗op . (3.4)

This product is given by (h ⊗ φ)(g ⊗ ψ) = hg(2) ⊗ ψφ(2) hg(1) , φ(1) ihSg(3) , φ(3) i,

h, g ∈ H, φ, ψ ∈ H ∗op .

(3.5)

The coproduct is the tensor product coproduct of the individual Hopf algebras H and H ∗op , ∆(h ⊗ φ) = h(1) ⊗ φ(2) ⊗ h(2) ⊗ φ(1) . (3.6)

Following the general construction of double crossproduct above, D(H) canonically acts on (H ∗op )∗ = H cop from the left as an algebra and we have (D(H), H cop ) as a left covariant system. The mirror bicrossproduct for a Hopf algebra H is M (H) = H cop I/H. It is easy to see from the previous section that one can obtain this by semidualising the quantum double D(H). The left action of H on H cop and the right coaction of H cop on H given respectively as h . a = h(1) aSh(2) , ∆R h = h(2) ⊗ h(1) Sh(3) . (3.7) The algebra is

(a ⊗ h)(b ⊗ g) = a(h(1) bSh(2) ) ⊗ h(3) g,

h, g ∈ H, a, b ∈ H cop ,

(3.8)

which is fully determined by hb := (1 ⊗ h)(b ⊗ 1) = (h(1) bSh(2) )h(3) ,

(3.9)

with the embeddings h → 1H cop ⊗ h and b → b ⊗ 1H being algebra morphisms. The coproduct is given by ∆(a ⊗ h) = a(2) ⊗ h(2) ⊗ a(1) h(1) Sh(3) ⊗ h(4) .

(3.10)

The Hopf algebra H cop I/H acts covariantly on H ∗op from the right according to φ / (a ⊗ h) = hah(1) , φ(1) ihSh(2) , φ(3) iφ(2) ,

(3.11)

and using (2.2), this gives rise to covariant left action on H ∗ (a ⊗ h) . φ = hSh(1) Sa, φ(1) ihh(2) , φ(3) iφ(2) .

(3.12)

We thus have the right covariant system (H cop I/H, H ∗op )R as the left covariant system (H cop I/H, H ∗ )L . We refer to [21] for details. Extracting the covariant actions of H cop on H ∗ and H on H ∗ in the covariant system (H cop I/H, H ∗ ), we get a.φ = hSa, φ(1) iφ(2) ,

h.φ = hSh(1) , φ(1) ihh(2) , φ(3) iφ(2)

–9–

(3.13)

respectively. It is interesting to note that the left action of H cop on H ∗ is a coregular action and makes H ∗ an H cop -module algebra while the left action of H on H ∗ is a coadjoint action and makes H ∗ an H-module coalgebra. It is important to note that Drinfeld’s quantum double D(H) is related to its semidual, Majid’s bicrossproduct M (H). It is shown in [21] that if H is factorisable, then D(H) = H./H ∗op acting on H is related to M (H) = H cop I/H acting on H ∗ by a Drinfeld and module algebra twist.

4

Kitaev model for mirror bicrossproduct Hopf algebras

Following Section 2 and [1, 2], our aim is to find local vertex and face operators Ah , Ba which act on HΓ and represent both copies H, H cop in the mirror product bicrossproduct M (H) so that they satisfy the product in the bicrossproduct H cop I/H as they interact nontrivially on the intersection of their support. One can find two examples of simple graphs to illustrate the construction in Appendix B. Lattice and Hilbert space: We take Γ to be the 1-skeleton of the dual of the polytope discretization ∆∗Σ . Each edge of Γ is decorated by elements of H ∗ . The vector space on which we will represent our geometric operators is therefore the C-vector space O ˜ Σ ) = HΓ = H ∗. H(Σ, ∆ e∈Γ

The Hilbert space of the lattice model is generated by assuming that H ∗ is a Hopf C ? -algebra. The non-degenerate Hermitian inner product on H ∗ is defined by [28, 29] hφ|ψiH ∗ := hl, φ? ψi,

φ, ψ ∈ H ∗ ,

(4.1)

where l is the normalized Haar integral of H. Covariant system and triangle operators: We take the covariant system (H cop I/H, H ∗ ), which is the mirror bicrossproduct acting on H ∗ . Definition 4.1. The triangle operators for an edge e ∈ Γ, h ∈ H, φ ∈ H ∗ and a ∈ H cop are the linear maps Lh± , T±a : H ∗ → H ∗ given by Lh+ (φ) := h.φ = hh, (Sφ(1) )φ(3) iφ(2) , T+a (φ) := a.φ = hSa, φ(1) iφ(2) ,

Lh− (φ) := φ/h = hh, φ(3) (Sφ(1) )iφ(2) , T−a (φ) := φ/a = ha, Sφ(2) iφ(1) . (4.2)

Here, the operators L+ and T+ are the canonical left action (3.13) while the L− and T− are right actions obtained from the relations S(h)

Lh− (φ) = (S ◦ L+

◦ S)(φ),

S(a)

T−a (φ) = (S ◦ T+

– 10 –

◦ S)(φ).

(4.3)

The inner product (4.1) makes the triangle operators L± and T± into ?-representations with adjoint maps given by The inner product (4.1) makes the triangle? operators L± and? T± into ?-representations h † h (T±a )† = T±a . with adjoint maps given by (L± ) = L± , ? a h? For example, we check this for follows: (Lh± )† T=+Las (T±a )† = T±a . ±,

a ? a h |T T+follows: ( )i = hl, ? (2) ihSa, (1) i H⇤ = + ( )i For example, we check this forhl,T+a as = hl, ? ihl, (2) ihSa, (1) i = hl, ? ihSa, (1) ihl, (2) i hφ|T+a (ψ)iH ∗ = hl, φ? T+a?(ψ)i = hl, φ? ψ(2) ihSa, ψ(1) i? = hl, ih(Sa)l, i = hl ⌦(Sa)l, ⌦ i ? = hl, φ? ihl, ψ(2) ihSa, ψ i = hl, φ ihSa, ψ ihl, ψ(2) i = h(Sa)l ⌦ l, ? ⌦ (1) i = h(Sa)l,? ? ihl, (1)i ? = hl, φ ih(Sa)l,?ψi = ?hl ⊗(Sa)l, φ ⊗ ψi? = hSa ⌦ l, ?(1) ⌦ (2) ihl, i = hSa, ihl, ?(2) ihl, i (1) ? = h(Sa)l ⊗ l, φ ⊗ ψi = h(Sa)l, φ ihl, ψi = hl, ?(2) ? ihSa,? ?(1) i = hl, h((Sa)??)? , ?(1) ?i ?(2) i = hSa ⊗ l, ?φ(1) ⊗ φ(2) ihl, ψi = hSa, φ(1) ihl, φ(2) ihl, ψi ? = hl,? T+a ( ) i?= hT+a ( a?)| iH ⇤ . = hl, φ(2) ψihSa, φ(1) i = hT+ (φ)|ψiH ∗ . Similar calculations are done for T and L± . Similar calculations are done for T− and L± .

(4.4)

Geometric operators: Thanks to the triangle operators, we can define the geoGeometric operators: Thanks operators, we can the geo-the metric operators. We recall that to thethe sitetriangle (v, p) defines a cilium anddefine we consider metric operators. We recall that the site (v, p) defines a cilium and we consider the clockwise orientation. clockwise orientation. Definition 4.2. ˜ Σ with all edges ingoing, and Definition 4.2. (1) Let us consider the site (v, p) of ∆ ˜ (1) Let us consider p) vertex of ⌃ operator with all edges ingoing, and the elements the elements h ∈ H, the φ ∈site H ∗ .(v, The h 2 H, 2 H ⇤ . The vertex operator Ah (v, p) : HΓ −→ HΓ Ah (v, p) : H ! H acting on a vertex v is given by acting on a vertex v is given by n

p

p

h(n) .

1

•v

2

n

7 !

•v

3

h(1) .

1

h(2) .

2

h(3) .

3 n

p

(2)

= hh, (S

1

) (1)

1

....(S (3)

n

) (1)

n

i (3)

1 (2)

•v

2 (2)

3 (2)

– 11 – – 11 –

or

or

X h(1) h ∗ ⊗ |E| h(1) Ah (v, p) = LX · · · ⊗ L+(n) :hH → H ∗ ⊗ |E| . + ⊗ Ah (v, p)h= L+ ⌦ · · · ⌦ L+(n) : H ⇤ ⌦ |E| ! H ⇤ ⌦ |E| . h

The antipode is used to change orientation away from the vertex v. In this case The antipode is used to change theindicated vertex v.inIn(4.3). this case the antipode maps the left action L+ orientation to the right away actionfrom L− as the antipode maps the left action L+ to the right action L as indicated in (4.3). ˜ Σ and the element a ∈ H cop , the face operator (2) Given a site (v, p) of ∆ (2) Given a site (v, p) of ˜ ⌃ and the element a 2 H cop , the face operator Ba (v, p) : HΓ −→ HΓ Ba (v, p) : H ! H is defined as: is defined as: v 1 v • n a(1) . n • a(n) . 1 ·· ··

2

4

3

·· ··

7 ! a(n

3)

.

a(n 4

a(n n (2)

= hSa,

or or

1

... (1)

n

i (1)

2)

1)

. v •

2

. 3 1

(2)

·· ··

2 (2)

4

3

(2)

(2)

X a a Ba (v, p) = a T+(n) ⌦ · · ·a⌦ T+(1) : H ⇤ ⌦ |E| ! H ⇤ ⌦ |E| X Ba (v, p) = T+a(n) ⊗ · · · ⊗ T+(1) : H ∗ ⊗ |E| → H ∗ ⊗ |E| a

Note that thanks to the Hermitian inner product, the operators Ah and Ba are Note that thanks to the Hermitian inner ofproduct, the operators Ah and Hermitian since they are tensor products the L± and T± operators, i.e., Ba are Hermitian since they are tensor products of the L± and T± operators, i.e., A†h (v, p) = Ah? (v, p), Ba† (v, p) = Ba? (v, p). (4.5) † Ah (v, p) = Ah? (v, p), Ba† (v, p) = Ba? (v, p). (4.4) Theorem 4.3. Let H be a finite-dimensional semisimple Hopf algebra and H ⇤ its dual. Theorem 4.3. site Let (v, H be finite-dimensional semisimple Hopfviaalgebra and H ∗ its dual. Then each p) aadmits an M (H)-module structure the operators Ah (v, p) and ThenBeach site (v, p) admits an M (H)-module structure via the operators A (v, p) and h (v, p) given in Definition 4.2. a Ba (v, p) given in Definition 4.2. Proof. We use Figure 2 to proof Theorem 4.3. It is sufficient to show that (3.9) holds i Proof. Figure to proofby Theorem It is sufficient to H, show on We thisuse graph. We2 proceed a direct4.3. calculation, let h 2 b 2that H cop(3.9) andholds 2 H ⇤, cop i on this graph. We proceed by a direct calculation, let h ∈ H, b ∈ H and φ ∈ H ∗ ,

– 12 – – 12 –

2 1

p

7 6

5

v •

3 4

Figure 2. Graph representing seven copies of H ⇤ used in the proof of Theorem 4.3. Figure 2. Graph representing seven copies of H ∗ used in the proof of Theorem 4.3.

where i 2 {1, 2, .., 7}, then we have the LHS evaluated as where i ∈ {1, 2, .., 7}, then we have the LHS evaluated as 1 2 3 Ah Ba ( 1 ⌦ 1... ⌦ 7 ) = 7 hSa(3) , (1) ihSa(2) , (1) ihSa(1) , (1) i Ah Ba (φ ⊗ ... ⊗ φ ) ⇣ ⌘ 3  1 2 4 5 6 7 ⌦3 (2) 3 h (2) , φ = hSa , φ1 ihSa , φ2AihSa iA⌦ φ(2)1 ⌦⊗ φ2⌦⊗ φ⌦ ⊗⌦ φ4 ⊗ φ5 ⊗ φ6 ⊗ φ7 (3)

(2)

(1)

(1)

(1)

(1)

(2)

(2)

6

S

(5)

(3)

3

(1)

(6)

(1)

(3)

3

(2)

4

2

2 1

h

1 3 ihh(1) , 1(2)(3) S ihh(2) , S4 3(2)(1) ihh5 Sφ , S5 4(1) 3 3 4 (2)(1) (2)(3) , φ (3) (1) = hSa, φ(1) φ(1) φ(1) ihh=(1)hSa, , φ1(2)(3)(1) Sφ(1) ihh , Sφ φ ihh , Sφ φ ihh i (2) (3) (4) (2)(1) (2)(1) (2)(3) (1) (3) (3) (1) 5 5 6 6 7 7 hh(4) S3 ihh(6)4 , S⊗ φ(1)5 (3) 7 , 7 S 1 ihh(5)2, (3) φ (1) ⊗ φ hh , φ6 Sφ6 ihh , Sφ φ (3)iφ (1) ⊗ φ ⊗ ⊗iφ6 ⊗ φ7 1

1

(2)(2)

(2)

(2)(2)

(2)

(2)

(2)

(2)(2)

2

(2)

i

(2)

1 6 7 ⌦1 2(2) Sφ ⌦ 3 3(2)(2)φ3⌦ 4(2) ⌦4 φ5(2)4 ⌦ ⌦ 5 5 6 6 (2)(2) (2) (2) Sφ = hSa, φ(1) φ(1) φ(1) ihh, φ1(2)(3) Sφ Sφ φ Sφ φ Sφ7(1) φ7(3) i (2)(1) (2)(1) (2)(3) (1) (3) (3) (1) (3) (1) 1 2 3 1 1 hSa, S 3(2)(1) 3(2)(3) S 4(1) 4(3) 5(3) S 5(1) 5 6 7 S (1) φ (1) ⊗ (1) (2)(1) φ1 ⊗ φ2 ⊗ φ3 = ⊗ φ4 ⊗ φihh, ⊗ φ(2)(3) 1

(3)

3

(2)(2)

(2)

(2)

(2)

(2)

1 3 4 ⌦1 2(2) ⌦ ⌦ ⌦ 54 ⌦ 65 ⌦ 76 2 3 7 (2)(2) (2)(2) (2) ⊗ φ(2) ⊗ φ(2) ⊗ φ(2) ⊗ φ = ha ⊗ h, S(φ1(1) φ2(1) φ3(1) ) ⊗ Diφ ⊗ φ ⊗ φ , (2)(2) (2) (2)(2) (2) (2) (2) (2) = ha ⌦ h, S( 1(1) 2(1) 3(1) ) ⌦ Di 1(2)(2) ⌦ 2(2) ⌦ 3(2)(2) ⌦ 4(2) ⌦ where D = φ1(2)(3) Sφ1(2)(1) Sφ3(2)(1) φ3(2)(3) Sφ4(1) φ4(3) φ5(3) Sφ5(1) φ6(3) Sφ6(1) Sφ7(1) φ7(3) . where = 1(2)(3) S 1(2)(1) S 3(2)(1) 3(2)(3) S 4(1) 4(3) 5(3) S 5(1) 6(3) S 6(1) S 7(1) 7(3) . TheDRHS gives

5 (2)

⌦

(3)

6 (2)

⌦

6 (1)

S

7 (2)

,

7

7

(1)

(3)

(4.6)

The RHS gives B(h(1) aSh(2) ) Ah(3) (φ1 ⊗ ... ⊗ φ7 ) 1 1 3 3 B(h(1) aSh(2) ) Ah(3) ( 1 ⌦ ... ⌦ 7 ) = hh(3)(1) , S5 4(1) 4(3) ihh(3)(4) , 5(3) S 5(1) i 3 , 3 (3) S (1) ihh(3)(2) 4 ,S 4 (1) (3) ihh(3)(3) 5 = hh(3)(1) , φ1(3) Sφ1(1) ihh(3)(2) , Sφ φ ihh , Sφ φ ihh , φ Sφ i (3)(3) (3)(4) (1) (3) (1) (3) (3) (1) 6 6 7 7 hh(3)(5) 7 , 7 (3) S (1) ihh(3)(6) , S (1) (3) i hh(3)(5) , φ6(3) Sφ6(1) ihh(3)(6) , Sφ φ i⇣ ⌘ (1) (3)  4 1 2 3 5 6 7 B 3(1) aSh(2)4) (2)5 ⌦ ⌦ ⌦ ⌦ (2) ⌦ (2) ⌦ (2) B(h(1) aSh(2) ) φ1(2) ⊗ φ2 ⊗ (h φ(2) ⊗ φ(2) ⊗ φ(2) ⊗ φ6(2) ⊗(2)φ7(2) (2) = hh 4 , 41(3) S5 1(1) S5 3(1)6 3(3) S6 4(1) 74(3) 75(3) S 5(1) 6(3) S 6(1) S 7(1) 7(3) i φ φ Sφ(1) φ(3) Sφ(1) Sφ(1) φ(3) i = hh(3) , φ1(3) Sφ1(1) Sφ3(1) φ3(3) Sφ(3) (1) (3) (3) hS˜ a , 1 3 ihS˜ a , 2 ihS˜ a(1) , 3(2)(1) i (3) hS˜ a(3) , φ1(2)(1) ihS˜ a(2) , φ2(1) ihS˜ a(1) , (2)(1) φ(2)(1) i (2) (1) 1 4 ⌦ 5 2(2) ⌦ 6 3(2)(2) ⌦ ⌦ 5(2) ⌦ 6(2) ⌦ 7(2) 7 4 (2) ⊗ φ ⊗ φ ⊗ φ φ1(2)(2) ⊗ φ2(2) ⊗ φ3(2)(2) ⊗ φ(2)(2) (2) (2) (2) (2) 1 1 3 3 4 4 5 5 6 6 7 7 5 (1) 6 (3) S 7 7(3) S (1) (3) S (1) S (1) (3) i 4 , 4(3) S 5 (1) S 1 1 3 = 3 hh(3) (1) (3) φ i = hh(3) , φ(3) Sφ(1) Sφ(1) φ(3) Sφ(1) φ(3) φ(3) Sφ(1) φ(3) Sφ6(1) Sφ (1) (3) 2 3 1 2 3 4 a, 1(2)(1) i ⌦ ⌦ 5(2) ⌦ 6(2) ⌦ 7(2) 1 2 3 1 hS˜ 2 3 4 5 (2) ⌦ 6 (2)(2) ⌦ 7 (2)(1) (2) hS˜ a, φ(2)(1) φ(1) φ(2)(1) iφ(2)(2) ⊗ φ(2) ⊗ (1) φ(2)(2) ⊗ φ(2)(2) ⊗ φ ⊗ φ ⊗ φ (2) (2) (2) (2) 0 a ⌦ h(3) ,0S( 1(2)(1) 2(1) 3(2)(1) 1 2 = 3h˜ 1 )⌦D 2 i3 = h˜ a ⊗ h(3) , S(φ(2)(1) φ(1) φ(2)(1) ) ⊗ D i = ha ⊗ h, S(φ(2)(1) φ(1) φ(2)(1) ) ⊗ D0 i, = ha ⌦ h, S( 1(2)(1) 2(1) 3(2)(1) ) ⌦ D0 i, (4.7) – 13 – – 13 –

i

where a ˜ = h(1) aSh(2) and D0 = φ1(3) Sφ1(1) Sφ3(1) φ3(3) Sφ4(1) φ4(3) φ5(3) Sφ5(1) φ6(3) Sφ6(1) Sφ7(1) φ7(3) . 0 1 LHS 3 5 This isDequal where = 1toSthe S 3 computed S 4 4 above. S 5 6 S 6 S 7 7 . (3)

(1)

(1)

(3)

(1)

(3)

(3)

(1)

(3)

(1)

(1)

(3)

This is equal some to thesimple LHS computed We consider examples above. of graphs illustrating again these calculations in Appendix B. When the geometric operators do illustrating not act on the site, theycalculations essentially We consider some simple examples of graphs again these commute as states the following lemma. in Appendix B. When the geometric operators do not act on the site, they essentially commute as states the following lemma. Lemma 4.4. Let Γ be an arbitrary graph with v, w ∈ V and p, q ∈ F , h, g ∈ H and φ, ψ ∈ H cop , then Lemma 4.4. Let be an arbitrary graph with v, w 2 V and p, q 2 F , h, g 2 H and cop , (i)2AH(v,,p)then ◦ A (w, p0 ) = A (w, p0 ) ◦ A (v, p) if the two vertices v and w do not h

g

g

h

coincide. (i) A Ag (w, p ) = Ag (w, p ) Ah (v, p) if the two vertices v and w do not h (v, p) coincide. (ii) Ba (v, p) ◦ Bb (w, q) = Bb (w, q) ◦ Ba (v, p) if the two faces p and q do not coincide. 0

0

(ii) B 0 q) 0 Ba (v, p) if the two faces p and q do not coincide. (iii) Aah(v, (v,p) p) ◦ B Bbb(w, (v 0 ,q) p0 )==BBb (w, b (v , p ) ◦ Ah (v, p) at different sites.

0 0 (iii) Ah(i). (v, Suppose p) Bb (v 0there , p0 ) =exist Bb (vat , pleast ) Aone p) at connecting di↵erent sites. h (v, edge Proof. the vertices v and w, and the orientation is from v to w, then we have Proof. (i). Suppose there exist at least one edge connecting the vertices v and w, and

the orientation is from v to w, then hwe have Ah (v, p)(φ) = L+ (φ), Ag (w, p0 )(φ) = Lg− (φ). A (v, p)( ) = This by definition of Lh ± gives

This implies and and

Lh+ (

0

),

g

Ag (w, p )( ) = L ( ).

(4.5) (4.8)

A (A (φ)) = hg, φ(3) (Sφ(1) )ihh, Sφ(2)(1) φ(2)(3) iφ(2)(2) Ahh(Agg( )) = hg, (3) (S (1) )ihh, S (2)(1) (2)(3) i (2)(2) Agg (Ahh(φ)) φ(3) ihg, φ(2)(3) Sφ iφ . ( )) = hh, Sφ S (1) S (2)(1) i (2)(2) (1) (3) (2)(3) (2)(1) (2)(2)

be .AAcareful carefullook lookatatthe thetwo twoexpressions expressions shows shows A Ahh(A (Agg(φ)) ( )) = = Agg(Ahh(φ)), ( )), and this can be generalized for for any any number number of of edges edges connecting connecting vv and and w. w. generalized Suppose also also the the incident incident edges edges to to vv and and and and to to w w are are disjoint, disjoint, then then we we have have for for Suppose one incident edge one incident edge Agg(w)( (w)(φ) =L Lg+g+((φ), A )= ),

Ahh(v)( (v)(φ) =L Lh+h+((φ), A )= ),

(4.6) (4.9)

and obviously obviously these these two two operators operators are are commuting. commuting. and (ii) Consider Consider the the diagram diagram below below with with faces faces pp and and qq sharing sharing aa common common edge. edge. Then Then (ii) we have have we v • p

q

– 14 – – 14 –

that the face operators of the faces p and q are respectively a

a

a

b

b

b

Ba (φ1 ⊗ φ2 ⊗ φ3 ) = T+(3) (φ1 ) ⊗ T+(2) (φ2 ) ⊗ T+(1) (φ3 )

Bb (φ1 ⊗ φ2 ⊗ φ3 ) = T+(3) (φ1 ) ⊗ T+(2) (φ2 ) ⊗ T−(1) (φ3 ).

(4.7)

A straightforward computation shows that the above operators commute. (iii) Consider a loop with two different sites (v, p) and (v 0 , p0 ) then the vertex operator for (v, p) and face operator for (v 0 , p0 ) are respectively h

h

Ah (φ ⊗ φ0 ) = L+(1) (φ) ⊗ L−(2) (φ0 ),

b

b

Bb (φ ⊗ φ0 ) = T+(2) (φ) ⊗ T+(1) (φ0 ).

(4.8)

With these operators, one can easily show Ah Bb (φ ⊗ φ0 ) = Bb Ah (φ ⊗ φ0 ). This can be generalised for any graph and easily shown that the vertex and face operators at two different sites commute. Hamiltonian We are now ready to define the Hamiltonian of the mirror bicrossproduct Kitaev model, provided we can construct projectors out of the geometric operators. Lemma 4.5. Let l and k be the Haar integrals for H and H cop respectively, then A(v) := Al (v, p), B(p) := Bk (v, p).

(4.9)

define a set of projectors. Proof. Given that there exist l ∈ H and k ∈ H cop satisfying definition (A.4), then we have l2 = l,

k 2 = k.

(4.10)

This implies for the vertex operator Al (v, p), we have A2 (v) = Al2 (v, p) = Al (v, p) = A(v),

(4.11)

showing Al (v, p) is a projector. Similarly, this applies to the face operator, i.e, B 2 (p) = B(p). The vertex and face operators were earlier shown to be Hermitian and in Lemma 4.4 were shown to be commuting. Hence the expression (4.12) is a projector and defines the Hamiltonian of the mirror bicrossproduct model. The Hamiltonian defining the M (H)-model is then given by X X H= (id − A(v)) + (id − B(p)) v∈V

(4.12)

p∈F

As a sum of Hermitian projectors, the Hamiltonian is Hermitian and diagonalizable. We would like now to determine a representation of the ground state PΓ of H. For this we are going to use the tensor network representation.

– 15 –

5 Tensor network representations for Bicrossproduct models 5 Tensor network representations for Bicrossproduct models We build the tensor network representation for theBicrossproduct mirror bicrossproduct model of 5 now Tensor network representations for We now build the tensor network representation for the mirror bicrossproductmodels model of

Section 4. Our starting point is to provide the diagrammatic framework for the tensor Section Ourthe starting point is torepresentation provide themodel tensor network built on Γ, decorated by H ∗ .the Wediagrammatic recall weframework consider Σfor without any We now4.states build tensor network for thethat mirror bicrossproduct of ⇤ network states built on , decorated by H . We recall that we consider ⌃ without any boundary simplicity. First we going the to determine a diagrammatic calculus. We Section 4.for Our starting point is toare provide diagrammatic framework for the tensor boundary for simplicity. First we are going to determine a diagrammatic calculus. ⇤which will define then the notion of tensor trace allows to evaluate quantities. With network states built on , decorated by H . We recall that we consider ⌃ without We any will define then theable notion of tensor trace allows evaluate quantities. With all this, wefor willsimplicity. be toFirst define of the to Hamiltonian as the invariant boundary wethe areground goingwhich tostate determine a diagrammatic calculus. We all this, we will be able to define the ground state of the Hamiltonian as the invariant space of thethen operators A(v) and B(p). trace which allows to evaluate quantities. With will define the notion of tensor space of the operators A(v) and all this, we will be able to defineB(p). the ground state of the Hamiltonian as the invariant 5.1 Diagrammatic scheme for tensor network states space of the operators A(v) and B(p). 5.1 Diagrammatic scheme for tensor network states To each oriented edge of the graph Γ, we associate a tensor as indicated below. To oriented edge ofscheme the graph , we associate a tensor 5.1each Diagrammatic for tensor network states To each oriented edge of the graph , we associate a tensor •

(5.1) • (5.1) Here the black dot represents the orientation of the edge inherited from the underlying Here the black dot represents the orientation of the edge inherited from the underlying graph (physical edge). The black arrows (virtual edges) attached to tensor represents Here the black dot represents the arrows orientation of the edge attached inherited to from the underlying graph (physical edge). The black (virtual edges) tensor represents the indices of the tensor. We place a clockwise oriented virtual loop in each face of graph (physical TheWe black arrows (virtualoriented edges) attached to tensor represents the indices of theedge). tensor. place a clockwise virtual loop in each face of the graph .of A virtual loop determines a face p,oriented to which we associate an element the indices the tensor. We place a clockwise virtual loop in each face of the graph cop Γ. A virtual loop determines a face p, to which we associate ⇤an element athe H cop , whereas we associate to each physical edge the element 2 H . p 2 graph e ∗ . A virtual loop determines a face edge p, to the which we associate ap ∈ H , whereas we associate to⇤ each physical element φ ∈ H .an element cop For any ap 2 H cop and we define theiredge canonical pairing eby2the elementary e 2 Hto ∗ , each ap 2For H cop , whereas we associate physical the element H ⇤elementary . any ap ∈ H and φe ∈ H , we define their canonical pairing by e the diagram cop ⇤ For any ap 2 H3. and e 2 H , we define their canonical pairing by the elementary diagram in Figure ap diagram ap := h e , ap i. • e (5.2) := h e , ap i. • e Di↵erent orientations of graph edges and loops are related using the relevant antipode. (5.2) ap related using Di↵erent orientations of graph aedges and loops are the ∗relevant antipode. p cop Figure 3. The elementary pairing between H

•

ap

e

:=

ap • S( e )

and H .

(5.3) := • • S( ) e Different orientations of graph edges and loops are relatede using the relevant antipode (5.3) ap S(ap ) as shown in Figure 4. ap := S(a•p ) • e e (5.4) := • • e e where all these basic diagrams are assumed to be invariant under arbitrary rotations, (5.4) for instance where all these basic diagrams are assumed to be ainvariant under arbitrary rotations, p for instance ap • e = • e • e = ap • e (5.5) ap (5.5) – 16 – – 16 – – 16 –

graph (physical edge). The black arrows (virtual edges) attached to tensor represents diagram the indices of the tensor. We placeapa clockwise oriented virtual loop in each face of the graph . A virtual loop determines a face p, to which we associate an element ⇤ := hedge ap 2 H cop , whereas we associate to each e , athe p i. element e 2 H . • physical e (5.2) For any ap 2 H cop and e 2 H ⇤ , we define their canonical pairing by the elementary Di↵erent orientations of graph edges and loops are related using the relevant antipode. diagram aapp ap • •

e e

:= h e , ap i. := • S( e )

(5.2) (5.3) Di↵erent orientations of graph edges and loops are related using the relevant antipode. ap S(ap ) ap ap := • • e:= e (5.4) • • S( e ) e (5.3) where all these basic diagrams are assumed to be invariant under arbitrary rotations, Figure 4. The antipode is used to change the orientation. ap S(ap ) for instance ap := •• e • e e = (5.4) All these basic diagrams are assumed to be invariant • under e arbitrary rotations. For instance see Figure 5. where all these basic diagrams are aassumed to be invariant under arbitrary rotations, p (5.5) for instance ap •

e

ap

= – 16 –

•

e

(5.5)

Figure 5. Rotating the diagram does not change their value.

– 16 – Let us now discuss how we can extend these diagrams to higher numbers of edges Let us now discuss how we can extend these diagrams to higher numbers of edges or or faces. faces. • First if the face p has more edges than e in its boundary we extend the above • First if the face p has more edges than e in its boundary we extend the above diagrams as follows. Consider another edge e0 which shares a common vertex 0 diagrams as follows. Consider another edge e which shares a common vertex with e. We define a glueing operation as in Figure 6. with e. Then we define a glueing operation by (2)

ap ap •

•

e0

:=

•

X

(1)

ap

(ap )

e

e

•

•

e0

=h

e0 e , ap i

(5.6)

where the arrows indicate the order in which the coproduct of

p

is applied to

Figure 6. Glueing edges with the coproduct of H cop : The arrows indicate the order in which the basic diagrams. The red dot denotes the origin for this coproduct. the coproduct of ap is applied to the basic diagrams. The red dot denotes the origin for this Hence in particular, if the face p have many edges for boundary, we first perform a coproduct.

clockwise multiplication of all the elements associated with the edges surrounding the face p and then pair the result with the element ap ap •

e

:= h 1e · · · – 17 –

n e , ap i.

• Second, the edge e will be in general adjacent to two faces, since there are no boundaries. So for any edge e with adjacent faces p, q, we pick e 2 H ⇤ and ap , aq 2 H cop and define a glueing operation by

• First if the face p hasamore than •edges p X e ain(1) its boundary we extend the above • e0 p diagrams as follows. Consider another edge e0 which shares vertex := = h e0a ecommon , ap i • with e. Then we define a glueing operation by • (ap ) e e (5.6) (2) ap where the arrows indicate the order in which •the 0 coproduct of p is applied to e ap The (1) • red Xdenotes the basic diagrams. dot the origin for this coproduct. a • e0 p In particular, if the face:=p have many edges for= boundary, h e0 e , ap i we first perform a Hence in particular, if the face p have many edges for boundary, firstsurrounding perform a clockwise multiplication of all(athe elements edges • • associated with thewe p) clockwise of the all the elements associated e pair the face multiplication p and then result withe the elementwith ap . the edges surrounding (5.6) the face p and then pair the result with the element ap where the arrows indicate the order in which the coproduct of p is applied to ap denotes the origin for this coproduct. the basic diagrams. The red dot := edges h 1e · ·for · neboundary, , ap i. Hence in particular, if the face p have we first perform a • many e clockwise multiplication of all the elements associated with the edges surrounding the face p and thebe result with the elementtoaptwo faces, since there are no • Second, thethen edgepair e will in general adjacent boundaries. So for any aedge e with adjacent faces p, q, we pick e 2 H ⇤ and •apSecond, theand edgedefine e willap glueing be in general adjacent , aq 2 H cop operation by to two faces, since there are no 1 n boundaries. So for any •edgee e := withh adjacent p, q, we pick φe ∈ H ∗ and p i. e · · · e , afaces a p cop ap face glueing operation as in Figure 7. If the faces ap , aq ∈ H and define the p and havee many we have to puttotogether Figure and Figure (1) • Second, the qedge will beedges, in general adjacent faces, since6 there are no 7. If X two • e furthermore oneany loopedge is outgoing we have to alsop,consider the antipode ⇤ following •e with adjacent e := boundaries. So for faces q, we (2) pick e 2 H and • Figure 4. ( e) e ap , aq 2 H cop and define a glueing operation by aq ap aq ap (5.7) (1) where the order of coproduct is determined of the underlying X by• the orientation e := graph edge. Hence, in •particulare if the faces p and q have many edges, we have (2) • ( ) e e to put together (5.6) and (5.7). If furthermore one loop is outgoing we have to also consider the antipode aq following (5.3). aq (5.7) 3 An element e can be split into two parts according to where the order offaces coproduct determined bycopthe orientation of the underlying Figure 7. Glueing with theis coproduct of H : The order of coproduct is determined (1) (2) graph edge. Hence, in particular if the faces p and q have many edges, we have ((S ⌦ id) graph )( edge. (5.8) by the orientation of the underlying e) = S e ⌦ e . to put together (5.6) and (5.7). If furthermore one loop is outgoing we have to 3 This is similar to the ket-bra representation of a matrix. also consider the antipode following (5.3).

3 3 An element split into two according to to An element φebe can be split intoparts two parts according e can (1)

3

(2)

((S ⌦ id) )( –) 17 = S– e ⌦ (1)e . (2) ((S ⊗ id) ◦ e∆)(φe ) = Sφe ⊗ φe .

(5.8)

(5.1)

This is similar to the ket-bra representation of a matrix.

Our Ourcontraction contractionrule ruletherefore thereforebecomes becomes ap •

– 17 – := hS

e

aq

(1)

e

(2)

· · · , ap ih

e

· · · , aq i. (5.9)

similar the ket-bra representation of acan matrix. To3 This the isleft and to right adjacent face of e, we assign

ap

e

and S

(1)

e

respectively.

ap • – 18e –⇠

aq

(2)

(1)

S e

•

(2) e

aq

(5.10)

If we need to change the orientation of the edge, we use the antipode of H ⇤ as a direct

Our contraction rule therefore becomes ap ap (1) (2) • e := hS e · · · , ap ih e · · · , aq i. (1) (2) • e := hS e · · · , ap ih e · · · , aq i. aq aq

(5.9) (5.9) (2) (1) (2) (1) leftright andadjacent right adjacent of can e, we can assign φe S and Sφe respectively. To theTo leftthe and face offace e, we assign and respectively. e e (2) (1) To the left and right adjacent face of e, we can assign e and S e respectively. ap ap ap (1) ap S e • e ⇠ (1) • (2) e S e ⇠ • • e (2) aq aq e (5.10) aq aq (5.10) ∗ a direct If If wewe need to to change thethe orientation of of thethe edge, wewe useuse thethe antipode of of H ⇤Has need change orientation edge, antipode as a direct of of (5.3). Ifconsequence we need to change the orientation of the edge, we use the antipode of H ⇤ as a direct consequence Figure 4. consequence of (5.3). ap ap ap ap • • S( e ) e := • • S( e ) e := aq aq (5.11) aq aq (5.11) These tensors are then evaluated using the tensor trace which is nothing but the graphical weare justthen set up. The fully contracted tensor network, is a but complex These rules tensors evaluated using the tensor trace which iswhich nothing the These tensors are then evaluated using the tensor trace which is nothing but the number, rules for a we certain ground state of contracted the bicrossproduct model on the isgraph can graphical just set up. The fully tensor network, which a complex graphical rules we just set up. The fully contracted tensor network, which is a complex be interpreted as a collection virtual loops in the faces of that been suitably number, for a certain ground of state of the bicrossproduct model onhave the graph can number, for a certain ground state of the bicrossproduct model on the graph Γ can glued togetherastoaform the physical degrees be interpreted collection of virtual loopsofinfreedom. the faces of that have been suitably be interpreted as a collection of virtual loops in the faces of Γ that have been suitably glued together to form the physical degrees 4of freedom. glued together form Hopf the physical freedom. Definition 5.1. to(Dual tensor degrees trace .) ofThe dual Hopf tensor trace associated ⇤⌦|E| cop⌦|F | 4 with the graph the function ttr : trace H .) ⌦ H ! C, tensor trace associated Definition 5.1. is(Dual Hopf tensor The dual Hopf Definition 5.1. (Dual Hopf tensor⇤⌦|E| trace4 .) cop⌦|F The |dual Hopf tensor trace associated with the graph is the function :H ⌦H ! C, O ttrO with the graph Γ is the function ttrΓ : H ∗⊗|E| ⊗ H cop⊗|F | → C, a 7 ! ttr ({ }; {a (5.12) e p }) OOeOOp e2E e p2Fap 7 ! ttr ({ e }; {ap }) (5.12) φe ap 7−→ ttrΓ ({φe }; {ap }) (5.2) e2E

p2F

p∈F which is defined via diagramse∈E and the evaluation rules (5.2), (5.6) and (5.7). which is defined viavia diagrams andand thethe evaluation rules (5.2), (5.6) and (5.7). which is defined diagrams evaluation rules given in Figures 3, 6 and 7. 4

We refer to this as dual Hopf tensor trace as it is dual to the one defined in [? ] in the sense that ⇤op ⇤ cop (H4 We ⌦ Hrefer ) to = this H⇤ ⌦ Hdual . Hopf tensor trace as it is dual to the one defined in [? ] in the sense that as 5.2 ⇤opTensor network representation for the bicrossproduct model ⇤ ⇤ cop (H ⌦ H ) = H ⌦ H .

We now use the tensor trace to construct the quantum states that will define the protected space. Definition 5.2. Let φe , ψe ∈ H ∗ and ap , bq ∈ H cop . Let Γ the graph embedded in the surface Σ. The Hopf tensor network state on the graph Γ is given by X O (1) (2) |ΨΓ ({φe }; {ap })i := – 18ttr ({φ }; {a }) |φe i. (5.3) p –Γ e e∈EΣI e) – –(φ18 e∈EΣI

4

We refer to this as dual Hopf tensor trace as it is dual to the one defined in [2] in the sense that (H ⊗ H ∗op )∗ = H ∗ ⊗ H cop .

– 19 –

e •

• •

•

•

•

•

ap •

•

•

• •

•

•

•

•

Figure Figure3.8. Example Exampleofofa atensor tensornetwork, network,forforananhexagonal hexagonalgraph graph .Γ.

5.2 Tensor network forbicrossproduct the bicrossproduct model Theorem 5.3. (Groundrepresentation state of the mirror model). Let η and k the Haar ∗ cop integrals of H H trace respectively. Thethe ground state states of the that mirror bicrossproduct We now use the and tensor to construct quantum will define the model is protected space. |ΨΓ i := |ΨΓ ({ηe }; {kp })i (5.4) Definition 5.2. Let ∗ e , e 2 H ⇤ andcop ap , bq 2 H cop . Let the graph embedded in the where ⌃. for The φe , ψHopf and anetwork e ∈ Htensor p , bq ∈ H surface statewe onhave the graph is given by O (1) XX O (2) (2) (1) (5.5) |φ }; {a }) }; {a })i := ttr ({φ p e p Γ e | |Ψ ({Γ ({φ }; {a })i := ttr ({ }; {a }) | ee i. i. (5.13) e p p e e∈Γ e2E ⌃I

( e(φ ) e) e2E⌃e∈Γ I

Proof. Let us recall the Hamiltonian of the mirror bicrossproduct model is a sum of Theorem 5.3. (Ground state of the mirror bicrossproduct model). Let ⌘ and k the Haar local commuting terms A(v) and B(p). Hence it is sufficient to show that the operators integrals of H ⇤ and H cop respectively. The ground state of the mirror bicrossproduct A(v) and B(p) leave the state |ΨΓ i invariant individually. model is Consider a face p with a boundary consisting of n edges. A face of Γ, decorated by i := | consisting ({⌘e }; {k Consider a face p with a |boundary ofpn})i edges. A face of , decorated (5.14) by the Haar integral k, leads to the contribution |ΨΓ i given by the Haar integral ⇤k, leads to the contribution | i given by where for e , e 2 H and ap , bq 2 H cop we have X··· O (1) (2) | ({ e }; {ap })i := ttr ({ e }; {ap }) | e i. (5.15) · (e2 · ·e )•

4 ⌘(2)

e2

···

n 4 ⌘(1) • ⌘(1) a4 an X Proof.| Let us recall the Hamiltonian of the mirror bicrossproduct model is a sum of 1 n n • p (a , ....., a )i = k it is sufficient to show that the operators (2) local commuting terms⌘A(v) and⌘B(p). Hence i ⌘3 • (2) A(v) and B(p) leave the state | i invariant individually. 1 3 ⌘(1) ⌘(1) • 1 3 a a 1 • 2 ⌘(2) 2 ⌘(2) ⌘(1)

a2

(5.16)

where we left the other faces specified by ai . Note that the first diagram actually encodes a number since all the tensors have closed legs, whereas the second diagram is the state. This contribution can written a–more explicit form as – –in 1920 – |

1 n p (a , ....., a )i =

X ⌘i

X

1 n (⌘(2)(2) · · · ⌘(2)(2) )(k) n Y

n Y j

j 1 n S⌘(2)(1) (aj )|⌘(1) i ⌦ · · · ⌦ |⌘(1) i

where we left the other faces specified by ai . Note that the first diagram actually encodes a number since all the tensors have closed legs, whereas the second diagram is the state. This contribution can written in a more explicit form as 1

n

|Ψp (a , ....., a )i = =

X ηi

X ηi

1

n

(η(2)(2) · · · η(2)(2) )(k) 1 n (η(3) · · · η(3) )(k)

n Y j

n Y j

j 1 n Sη(2)(1) (aj )|η(1) i ⊗ · · · ⊗ |η(1) i

j 1 n Sη(2) (aj )|η(1) i ⊗ · · · ⊗ |η(1) i,

(5.6)

where the Haar integrals η i = η ∈ H ∗ . We used (5.5) to write down the contribution to the state |ΨΓ i. To each edge on the left hand side of the above diagram, we labelled it by η(2) . We split η(2) according to (5.1), then assign η(2)(2) and Sη(2)(1) , to the left and right adjacent faces of each edge respectively. To each of the outer nontrivial faces, we evaluate them according to Figure 3. The state (5.6) is invariant under the action of B(p): Bk (p)|Ψp (a1 , ....., an )i =

X ηi

=

X ηi

=

X ηi

=

1 n 1 n (η(4) · · · η(4) )(k)(η(1)(1) · · · η(1)(1) )(k) 1 n 1 n (η(4) · · · η(4) )(k)(η(1) · · · η(1) )(k) 1 n 1 n (η(3) · · · η(3) )(k)(η(4) · · · η(4) )(k)

j

n Y j

j

j 1 n Sη(3) (aj )|η(1)(2) i ⊗ · · · ⊗ |η(1)(2) i

j 1 n Sη(3) (aj )|η(2) i ⊗ · · · ⊗ |η(2) i j 1 n Sη(2) (aj )|η(1) i ⊗ · · · ⊗ |η(1) i

n X Y 1 n 2 j 1 n (η(3) · · · η(3) ) (k) Sη(2) (aj )|η(1) i ⊗ · · · ⊗ |η(1) i j

ηi

=

n Y

n Y

X ηi

1 n (η(3) · · · η(3) )(k) 1

n

= |Ψp (a , ....., a )i.

n Y j

j 1 n Sη(2) (aj )|η(1) i ⊗ · · · ⊗ |η(1) i

We used the definition of the face operator of 4.2 in the first line. In the third line we perform a renumbering on η. The fifth line uses the property of the Haar integral. Next we consider a vertex v with n ingoing edges. The vertex contribution to ΨΓ is

– 21 –

=|

p (a

, ....., a )i.

(5.18)

We used the definition of the face operator of 4.2 in the first line. In the third line we perform a renumbering on ⌘. The fifth line uses the property of the Haar integral. Next we consider a vertex v with n ingoing edges. The vertex contribution to is 3 ⌘(2)

a•3

4 ⌘(2)

|

v (a

1

, ....., an )i =

X

•

•

a

⌘i

4

3 ⌘(1)

a2 ⌘ 2 (2)

• •

•

• • •

···

4 ⌘(1)

2 ⌘(1)

a1

• • •

1 an ⌘(2)

n ⌘(2)

1 ⌘(1) n ⌘(1)

(5.19)

whereonce onceagain againthe thefirst firstdiagram diagramis isactually actuallya anumber numbersince sinceallallthe thetensor tensorlegs legsare are where contractedand andthe thesecond seconddiagram diagramis isthe thestate. state.Written Writteninina amore moreexplicit explicitform formthe the contracted statecontribution contributionis is state n n X Y X Y j 1 1 n n 1 1 n n j · η j+1 j+1)(aj )|η |Ψ|v (a , ....., a )i = (Sη i ⊗ · · · ⊗ |η(1) ii (S⌘ · ⌘(2)(2) )(aj )|⌘ v (a , ....., a )i = (2)(1) (1)(1) i ⌦ · · · ⌦ |⌘ (2)(1) (2)(2) (1) η i ⌘ ij=1 j=1 n n XY

==

XY j j j 1 1 n n j · η j+1 j+1 (Sη )(a i ⊗ · · · ⊗ |η(1) ii (S⌘ ⌘(3) )(a)|η )|⌘ (2)(2) ·(3) (1)(1) i ⌦ · · · ⌦ |⌘ (1)

ηi

(5.7) (5.20)

j=1 ⌘ ij=1

This is then invariant under the action of A(v):

Al (v)|Ψv (a1 , ....., an )i n XY – 21 – j j+1 1 1 n n 1 n = (Sη(2) · η(3) )(aj )h(Sη(1)(1) )η(1)(3) , l(1) i · · · h(Sη(1)(1) )η(1)(3) , l(n) i|η(1)(2) i ⊗ · · · ⊗ |η(1)(2) i = = = =

η i j=1 n XY

j j+1 1 1 n n 1 n (Sη(2) · η(3) )(aj )h(Sη(1)(3) )η(1)(2) , l(1) i · · · h(Sη(1)(3) )η(1)(2) , l(n) i|η(1)(1) i ⊗ · · · ⊗ |η(1)(1) i

η i j=1 n XY

j j+1 1 n 1 n (Sη(2) · η(3) )(aj )h(η(2) ), l(1) i · · · h(η(2) ), l(n) i|η(1) i ⊗ · · · ⊗ |η(1) i

η i j=1 n XY

j j+1 1 n )(aj )h1, l(1) i · · · h1, l(n) i|η(1) i ⊗ · · · ⊗ |η(1) i (Sη(2) · η(3)

η i j=1 n XY ηi

=

j=1

j

j+1

(Sη(2) · η(3)

n XY j+1 1 n j i · η(3) )(aj )(l)|η(1) i ⊗ · · · ⊗ |η(1) )(a )h1, li|η(1) i ⊗ · · · ⊗ |η(1) i = (Sη(2) j

1

n

η i j=1

n XY j j+1 1 n )(aj )|η(1) i ⊗ · · · ⊗ |η(1) i = |Ψv (a1 , ....., an )i. (Sη(2) · η(3) η i j=1

We used the definition of the vertex operator of 4.2 in the first equality. We permute cyclicly the different components of η in the definition of the vertex operator of the first

– 22 –

equality. The third and fourth equalities use the counit property of a Hopf algebra. While the fifth and sixth equalities use the dual pairing properties of (A.4).

6

Outlook

In this article we proposed for the first time a Kitaev model built not on the Drinfeld double, but instead on the (mirror) bicrossproduct quantum group. Our construction is based on the use of a covariant system. As such it is therefore definitely different than the standard Drinfeld double construction which cannot be expressed in this formalism, except for some trivial graph. In a way the use of a covariant system makes the construction simpler since we always use a covariant action. In the Drinfeld double case, the action (for the L± ) is only covariant for the simplest graph, and it has to be modified for more general graphs, making the construction less straightforward. We have identified the vertex and face operators and showed that they are projectors as in the standard Kitaev model. We have also identified the ground state of the Hamiltonian, by introducing a tensor network representation. This new model opens up new directions to explore. • We have used a specific bicrossproduct model, namely the mirror one. It would be interesting to develop the construction for a more general bicrossproduct model H1 I/H2 , where Hi are some Hopf algebras associated to some finite groups for example. A case by case study could be useful to identify how a general bicrossproduct model can be used to define a general bicrossproduct Kitaev model. • The Kitaev model defined in terms of the Drinfeld double was shown to be related to the combinatorial quantization of Chern-Simons theory based on the Drinfeld double [17]. It would be interesting to see whether this result extends to the bicrossproduct case, namely that our model can be related to the combinatorial quantization of Chern-Simons theory based on the bicrossproduct quantum group. An important step to achieve this was the definition of a Hopf gauge theory for the Drinfeld double [30]. This provides another interesting question to address in the context of the bicrossproduct quantum group. Note that for this construction, we need to have a R-matrix. Thanks to the recent work [21], we do have some examples of bicrossproduct quantum group with an explicit expression for the R-matrix. • From the gravity perspective, the vertex and face operators are related to the Gauss constraint and the Flatness constraint, which are usually characterized in terms of symmetries by the Drinfeld double. It would be interesting to determine whether the bicrossproduct case has also some geometrical meaning. The semidualization we use in the present work seems to indicate naively that we dualize

– 23 –

somehow for example the Flatness constraint into another Gauss constraint, or vice versa. Investigations are currently underway to see if this argument can be made more rigorous.

Acknowledgments This research was partly supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

A A.1

Some relevant features of Hopf Algebras Star structure and dual pairing

Definition A.1. If k = C. Given an antilinear map ? : H → H satisfying the condition ?2 = id, (hg)? = g ? h? , ∀h, g ∈ H, (A.1) then it turns H into a ?-algebra. Hence H is a Hopf ?-algebra if condition (A.1) and the following are satisfied ∆h? = (∆h)?⊗? ,

(h? ) = (h),

(S ◦ ?)2 = id.

(A.2)

Definition A.2. Let A and H be Hopf-? algebras. A non-degenerate bilinear map h·, ·i : A × H −→ C, (a, b) 7−→ ha, hi

(A.3)

is called a dual pairing of A and H if it satisfies (i) : h∆(a), g ⊗ hi = ha, ghi, ha ⊗ b, ∆(h)i = hab, hi (ii) : ha, 1i = ε(a), h1, hi = ε(h)

(A.4)

(iii) : ha? , hi = ha, (Sh)? i.

Note that for the property A.4 we have extend the dual pairing on the tensor products by ha ⊗ b, g ⊗ hi = ha, gihb, hi, (A.5) and from the properties of the dual pairing it follows that hS(a), hi = ha, Shi.

– 24 –

(A.6)

A.2

Semisimplicity and Haar integrals

A Hopf algebra H is (semi)simple if it is semisimple as an algebra and it is co(semi)simple if H ∗ is (semi)simple. Theorem A.3. [30] Let H be a finite dimensional Hopf algebra over a field k of characterisic zero. Then H is semisimple if and only if S 2 = idH . Definition A.4. Let H be a Hopf algebra H. A (normalised) Haar integral in H is an element l ∈ H with h · l = l · h = (h)l for all h ∈ H and (l) = 1. Lemma A.5. [31] If H is finite-dimensional and semisimple, then H has a Haar integral. Lemma A.6. Let H be a Hopf algbera. 1. If l, l0 ∈ H are Haar integrals, then l = l0 2. If l ∈ H is a Haar integral, then ∆(n) (l) is invariant under cyclic permutations and S(l) = l. 3. If l ∈ H is a Haar integral, then the element e = (id ⊗ S)(∆(l)) is a separability idempotent in H, i.e. one has µ(e) = l(1) S(l(2) ) = 1, e.e = e and for all h ∈ H (h ⊗ 1) · ∆(l) = (1 ⊗ Sh) · ∆(l) ∆(l)(h ⊗ 1) = ∆(l)(1 ⊗ Sh). 4. If l ∈ H is a Haar integral, then κ : H ∗ ⊗ H ∗ → k, κ(α ⊗ β) = hα · β, li is a Frobenius form. 5. If l ∈ H is a Haar integral, then hα(1) , liα(2) = hα(2) , liα(1) = hl, αi1 for all α ∈ H ∗ .

B

Face and vertex operators for some simple examples of graph

Let us consider first the simplest graph Γe , with one vertex and one edge, as in Figure 9. The covariant system (H cop I/H, H ∗ ) provides a representation of H cop I/H for Γe . The covariant left action of H cop I/H on H ∗ given in (3.13) which makes H ∗ into a left H cop I/H-module is exactly the co-Schr¨odinger representation of H. Lemma B.1. The operators Lh+ (φ) = Ah (φ) := hh, (Sφ(1) )φ(3) iφ(2) ,

T+a (φ) = Ba (φ) := hSa, φ(1) iφ(2)

form a representation of H cop I/H on the minimal graph shown in Figure 9

– 25 –

(B.1)

Lemma B.1. The operators Lh+ ( ) = Ah ( ) := hh, (S

(1)

)

(3)

i

(2)

,

T+a ( ) = Ba ( ) := hSa,

(1)

i

(2)

(B.1)

form a representation of H cop I/H on the minimal graph shown in Figure 4

p

•

v

Figure 4. A minimal graph e as an H cop I/H-module on H. Figure 9. A minimal graph Γe as an H cop I/H-module on H.

Proof. We prove this lemma by showing that the operators Ah and Ba defined in (B.1) Proof. We prove this lemma by showing that the operators Ah and Ba defined in (B.1) satisfy the multiplication formula (3.9), that is, satisfy the multiplication formula (3.9), that is, X Ah Ba = X B(h(1) aSh(2) ) Ah(3) . Ah Ba = B(h(1) aSh(2) ) Ah(3) . (h) (h)

We proceed as follows; B(h(1) aSh(2) ) Ah(3) (φ) = B(h(1) aSh(2) ) hh(3) , (Sφ(1) )φ(3) iφ(2) – 25 – = hS(h(1) aSh(2) ), φ(2)(1) ihh(3) , (Sφ(1) )φ(3) iφ(2)(2) = hh(1) aSh(2) , Sφ(2)(1) ihh(3) , (Sφ(1) )φ(3) iφ(2)(2) = hh(1) aSh(2) ⊗ h(3) , Sφ(2)(1) ⊗(Sφ(1) )φ(3) iφ(2)(2) = Ah Ba (φ)

(B.2)

Notice that in the first and second equations we used the actions given in (B.1). In the fourth equation we used the pairing property (A.5). Consider now a graph Γ which consist of two edges connecting each other with associated Hilbert space HΓ = H ∗ ⊗ H ∗ as shown in Figure 10. Since the covariant action (3.13) defines representations of M (H) on H ∗ , we use it to define module structures on HΓ = H ∗ ⊗ H ∗ . It is important to note that the coproduct (3.10) of the bicrossproduct H cop I/H is not a tensor product one but rather complicated due to the presence of the coaction (3.7). The covariant actions on H ∗ which form the foundation for the construction of the vertex and face operators are then given by (3.12). Using the triangle operators of (4.2), we define respectively the vertex operator and the face operator, which can be seen as two different representations on HΓ at the site (v, p) of Figure 10. Ah (φ ⊗ φ0 ) := h(1) . (φ) ⊗ (φ0 ) / h(2) = Ba (φ ⊗ φ0 ) := a(2) . (φ) ⊗ a(1) . (φ0 ) =

– 26 –

X (h)

X (a)

h

h

a

a

L+(1) (φ) ⊗ L−(2) (φ0 ), T+(2) (φ) ⊗ T+(1) (φ0 ).

(B.3) (B.4)

Ba ( ⌦

0

) := a(2) . ( ) ⌦ a(1) . ( 0 ) =

X (a)

a

a

T+(2) ( ) ⌦ T+(1) ( 0 ).

(B.4)

Next we show that the operators Ah (v, p) and Ba (v, p) indeed define a representation of the algebra M (H) on H ⇤ ⌦ H ⇤ at the site (v, p). v • p

0

• v0 Figure 10. 5. A Figure A graph graph representing representing the the vector vector space space of of H H⇤∗⌦ ⊗H H⇤∗..

Next we show that the operators Ah (v, p) and Ba (v, p) indeed define a representation of the algebra M (H) on H ∗ ⊗ H ∗ at the site (v, p). – 26 in – (B.3) and (B.4) satisfy bicrossproduct Lemma B.2. The representations defined algebra relation (3.9) in the form X Ah Ba = B(h(1) aSh(2) ) Ah(3) . (B.5) (h)

Proof. We prove this lemma first for the loop given in Figure 10. This is done by a direct calculation, evaluating both side of the straightening formula on arbitrary elements φ, φ0 ∈ H ∗ . Evaluating the left hand side, we have   Ah Ba (φ ⊗ φ0 ) = Ah a(2) . φ ⊗ a(1) . φ0   0 0 = Ah hSa(2) , φ(1) iφ(2) ⊗ hSa(1) , φ(1) iφ(2)   = hSa, φ(1) φ0(1) i (h(1) . φ(2) ) ⊗ (φ0(2) / h(2) )   0 0 0 0 = hSa, φ(1) φ(1) i hh(1) , (Sφ(2)(1) )φ(2)(3) iφ(2)(2) ⊗ hh(2) , φ(2)(3) (Sφ(2)(1) )iφ(2)(2) = hSa, φ(1) φ0(1) ihh, (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) )iφ(2)(2) ⊗ φ0(2)(2)

= ha, S(φ(1) φ0(1) )ihh, (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) )iφ(2)(2) ⊗ φ0(2)(2)

= ha ⊗ h, S(φ(1) φ0(1) ) ⊗ (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) )iφ(2)(2) ⊗ φ0(2)(2) . (B.6)

We used the definitions of the actions given in (4.2) in the second and fourth equality. In the third and fifth equality we used the pairing property (A.5). The property (A.6) of the dual pairing was applied in the sixth equality with the property of dual pairing on tensor products (A.4) used in the last equality.

– 27 –

Computing the right hand side of the lemma, we have   B(h(1) aSh(2) ) Ah(3) (φ ⊗ φ0 ) = B(h(1) aSh(2) ) h(3)(1) . φ ⊗ φ0 / h(3)(2)   0 0 0 = B(h(1) aSh(2) ) hh(3)(1) , (Sφ(1) )φ(3) iφ(2) ⊗ hh(3)(2) , φ(3) (Sφ(1) )iφ(2)   = hh(3) , (Sφ(1) )φ(3) φ0(3) (Sφ0(1) )i a ˜(2) . φ(2) ⊗ a ˜(1) . φ0(2) = hh(3) , (Sφ(1) )φ(3) φ0(3) (Sφ0(1) )ihS˜ a, φ(2)(1) φ0(2)(1) iφ(2)(2) ⊗ φ0(2)(2)

= h˜ a, S(φ(1) φ0(1) )ihh(3) , (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) )iφ(2)(2) ⊗ φ0(2)(2)

= h˜ a ⊗ h(3) , S(φ(1) φ0(1) ) ⊗ (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) iφ(2)(2) ⊗ φ0(2)(2) = ha ⊗ h, S(φ(1) φ0(1) ) ⊗ (Sφ(2)(1) )φ(2)(3) φ0(2)(3) (Sφ0(2)(1) )i φ(2)(2) ⊗ φ0(2)(2) ,

(B.7)

where a ˜ = h(1) aSh(2) . The definitions of the actions (4.2) are used in the second and third equality. We used the dual pairing property (A.5) in the third and fourth equality. While in the sixth equality we did a renumbering using coassociativity of the coproduct of φ0 . In the last equality we used the relation ha = (h(1) aSh(2) )h(3) to establish the equivalence with the last equality of (B.6). The equivalence of (B.6) and (B.7) shows that the operators Ah and Ba define a representation of H cop I/H on the loop.

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