# The Quantum Symmetry in Nonbalanced Hopf Spin Models Determined by a Normal Coideal Subalgebra

Authors: Xin Qiaoling; Cao Tianqing
Journal: Journal of Mathematics
 (2021)

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## Abstract

For a finite-dimensional cocommutative semisimple HopfC∗-algebra H and a normal coideal ∗-subalgebra H1, we define the nonbalanced quantum double DH1;H as the crossed product of H with H1op^, with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed product AH1=⋯⋊H⋊H1^⋊H⋊H1^⋊H⋊⋯ as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of DH1;H on AH1, the field algebra can be obtained as the crossed product C∗-algebra. Moreover, we prove there exists a duality between the nonbalanced quantum double DH1;H and the observable algebra AH1.


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## Body

## 1. Introduction

A model consists of a mathematical description of the system and the energy function. Quantum chains as low-dimensional (1 + 1) models possess many interesting features, such as integrability and quantum symmetry. One of the simplest examples exhibiting quantum symmetry isG-spin models, introduced by K. Szlachanyi and P. Vecsernyes [1] in lattice field theories, where G is a finite group. G-spin models as a classical statistical systems have an order-disorder type of quantum symmetry, which is the quantum double DG. The quantum symmetry in G-spin models generalizes the ℤ2×ℤ2 symmetry which can sharply divide the ordered phase and the disordered phase in Ising models. Generally, if G is an Abelian group, G-spin models have a symmetry group G×G^, which is the direct product of the group G and the group of characters of G. Based on a field-theory analysis of G-spin models, Jiang and Guo [2] gave the concrete construction of a DG;N-invariant subspace in field algebra of G-spin models and proved the DG;N-invariant subspace is Galois closed if N is a normal subgroup of G. On the contrary, Xin and Jiang [3] generalized G-spin models to G-spin models determined by a normal subgroup N, in which the quantum double DN;G, and the field algebra determined by N are defined, and then, the observable algebra determined by N can be obtained as DN;G-invariant subalgebra. Based on these work, the quantum symmetry is given by the quantum double DN;G.Note thatG^, the disorder part of the quantum double DG, is Abelian, so G-spin models do not have Kramers–Wannier duality for non-Abelian group. For this reason, G-spin models can be extended to a larger class of models. In 1997, F. Nill and K. Szlachanyi [4] investigated a one-dimensional quantum chains of Hopf algebras, called Hopf spin models, i.e., for a finite dimensional Hopf C∗-algebra H, there is a copy of H on every lattice and a copy of H^ on every link, and they satisfy nontrivial commutation relations only if they are neighbour links and sites, where H^ is the dual of H. The two-side infinite crossed product is defined as the observable algebra of Hopf spin models, and the field algebra is the crossed product of the observable algebra by the comodule action. Subsequently, the authors [5] considered the Jones basic construction on Hopf spin models and constructed the crossed product of the field algebra by the dual of the quantum double for Hopf algebra, which is consistent with Jones basic construction for the field algebra and the observable algebra. In this paper, we consider the more general situation. For a finite-dimensional cocommutative semisimple Hopf C∗-algebra H and a normal coideal ∗-subalgebra H1, we define the quantum double DH1;H by means of the left H-module algebra structure on Hopf C∗-algebra H1op^, where H1op^ is the dual of the opposite Hopf C∗-algebra of H1. Since Hopf C∗-algebras on both sides of ⊗ in the quantum double DH1;H are different, we call DH1;H the nonbalanced quantum double. In particular, let H=ℂG be the group algebra of a finite group G and H1=ℂN be the group algebra of a normal subgroup N of G; then, H1 can be naturally regarded as the normal Hopf∗-subalgebra of H, and DH1;H reduces to DN;G [3]. Moreover, we will also describe the quantum symmetry in the corresponding quantum chains of Hopf C∗-algebras H and H1 called nonbalanced Hopf spin models: there is a copy of H^ on every lattice site and a copy of H1 on every link, and they satisfy some commutation relations, which generalizes the results established in [3, 4].The paper is organized as follows. In Section2, we introduce the nonbalanced quantum double and define a right comodule algebra action of nonbalanced quantum double DH1;H on the observable algebra AH1 determined by a semisimple normal Hopf ∗-subalgebra H1. It is known that there is one-to-one correspondence between right DH1;H-comodule algebra actions and left DH1;H^-module algebra actions, since DH1;H is finite dimensional, where DH1;H^ is the dual of DH1;H. Based on these, we can define the field algebra of nonbalanced Hopf spin models as the crossed product C∗-algebra of the observable algebra AH1 by DH1;H^ in Definition 5 and prove that the observable algebra is the -invariant subspace of the field algebra. In Section 3, we prove that is the commutants of AH,G and vice verse.In this paper, we work over a complex fieldℂ. All Hopf algebraic notations [6, 7] can be used in the following. For example, we denote by m,ι,S,Δ,ε,∗ the multiplication, the unit, the antipode, the comultiplication, the counit, and ∗-operation, respectively. And, we use the standard notation Δa=a1⊗a2 and △na=Δn−1∘id⊗Δa=a1⊗a2⊗⋯⊗an+1 in the Hopf algebra where there is an implicit summation on the right side and Δ∘id⊗Δ=Δ∘Δ⊗id.


## 2. The Field Algebra of Nonbalanced Hopf Spin Models

### 2.1. The Nonbalanced Quantum Double

It is known that the quantum doubleDH, originally introduced by Drinfeld for a Hopf algebra H [8], plays an important role in the field of mathematical physics, and the quasi-triangular structure leads to a braiding in the category of representations and many ensuing applications. In particular, when H is the group algebra for a finite group, the quantum double reduces to an interesting crossed product algebra CG⋊G, where CG denotes the algebra of complex valued functions on G and the action is the conjugation [9]. Subsequently, several alternative descriptions of the quantum double have appeared in the literature. S. Majid [10] and F. Hausser and F. Nill [11, 12] independently introduced the quantum double of a finite dimensional quasi-Hopf algebra. In 2003, D. Bulacu and S. Caenepeel [13] constructed the quantum double for quasi-triangular quasi-Hopf algebras. In 2004, L. Delvaux and A. Van Daele [14] gave the Drinfeld double of multiplier Hopf algebras, which generalized the usual quantum double for Hopf algebras. In the usual quantum double DH=Hop^⊗H, two (generalized) Hopf C∗-algebra on both sides of ⊗ are the same. In this section, we hope to consider the similar theory when Hopf C∗-algebra are different. For this aim, we will generalize the quantum double to the nonbalanced quantum double by replacing DH by DH1;H=H1op^⊗H, where H1 is a normal Hopf ∗-subalgebra of H. In [15], Chen considered the nonbalanced quantum double DH1;H=H1op^⊗H, where H is a finite-dimensional Hopf algebra and H1 is a Hopf subalgebra of H, and showed DH1;H is a Hopf algebra. On the basis of the work, we will prove that DH1;H is a Hopf C∗-algebra by using the way different from [15]. Let us recall the following definition.Definition 1.
LetM be a ∗-algebra and A be a Hopf C∗-algebra. Then, M is a left A-module algebra if there is a left action ⊳:A⊗M⟶M, a⊗m↦a⊳m, and satisfies the following conditions, for any a,b∈A,m,n∈M:(1)ab⊳m=a⊳b⊳m,1A⊳m=m,a⊳mn=a1⊳ma2⊳n,a⊳m∗=Sa∗⊳m∗,a⊳1M=εa1M.
IfM is a C∗-algebra, the map a⊳ is assumed to be norm continuous for all a∈A.Theorem 1.
IfM is a left A-module algebra, the crossed product M⋊A is a ∗-algebra with the ∗-algebra structure given for m,n∈M, a,b∈A by(2)m⊗an⊗b=ma1n⊗a2b,m⊗a∗=1M⊗a∗m∗⊗1A.
Moreover, ifM is a C∗-algebra and A is of finite dimension, then the crossed product M⋊A becomes a C∗-algebra.Proof.
Clearly,M⋊A has the ∗-algebra structure. We will show that M⋊A is a C∗-algebra in the following.
It follows from [16] that M has a faithful positive linear functional φM and A has an invariant functional φA such that(3)φA=φA∘SandφA⊗idΔa=id⊗φAΔa=φA1A,for all a∈A. Define the map θ on M⋊A as follows:(4)θm⊗a=φMmφAa.
Then,θ is a faithful positive linear functional on M⋊A. In fact,(5)θm⊗am⊗a∗=θm⊗aa1∗m∗⊗a2∗=θma1a1∗m∗⊗a2a2∗=θma1a1∗m∗⊗a2a2∗=φMma1a1∗m∗φAa2a2∗=φMma1a1∗φAa2a2∗m∗=φMmφAaa∗1Am∗=φMmm∗φAaa∗≥0,and θm⊗am⊗a∗=0 if and only if m⊗a=0.
By [17], we can construct the associated GNS representation of M⋊A. Denote by K the completion of M⋊A with respect to the inner product m,x=θx∗m. Let τ:M⋊A⟶BK be the left multiplication; then, M⋊A can embed isometrically into BK, and thus, it is a C∗-algebra.
Semisimple Hopf algebras are intensively studied since they have important applications in topological invariants of knots and manifolds, quantum field theory, and so on. In fact, a Hopf algebra can be recovered from a normal Hopf subalgebra and some additional cohomological data [18]. As a result, normal Hopf subalgebras are an important tool in the classification of semisimple Hopf algebras. In 2012, B. Sebastian [19] studied normal left coideal subalgebras of semisimple Hopf algebras. In 2020, the authors [20] proved that, for a semisimple Hopf algebra, there is a one-to-one correspondence between right group-like projections and left coideal subalgebras.Definition 2.
LetH be any finite-dimensional semisimple Hopf C∗-algebra:(1)
A coideal subalgebraK of H is a subalgebra K of H with ΔK⊆H⊗K+K⊗H.(2)
A Hopf subalgebraK of H is said to be normal if K is invariant with respect to the right and left adjoint action:(6)Sh1kh2∈K,h1kSh2∈K,for allh∈H,k∈K.(3)
A coideal subalgebraK which is also a normal ∗-subalgebra of H is said to be normal coideal ∗-subalgebra of H.Remark 1.
(1)
IfK is a coideal subalgebra, then K is also a semisimple Hopf subalgebra (see Lemma 4.0.2 in [21]).(2)
Suppose thatG is a finite group and N is a normal subgroup of G. In this case of H=ℂG, the group algebra of G, K=ℂN can be viewed as a normal coideal ∗-subalgebra of H.
Note that the quantum doubleDG of a finite group is the crossed product of CG by the group algebra ℂG in G-spin models, and ℂG is always cocommutative. In order to give the definition of the nonbalanced quantum double in nonbalanced Hopf spin models, we suppose that H is a cocommutative semisimple Hopf C∗-algebra of finite dimension and H1 is a normal coideal ∗-subalgebra of H in the following. The left coadjoint representation of H on the dual of opposite Hopf C∗-algebra H1op^ is given for a∈H1, x∈H, and f∈H1^ by(7)a,x⋅f=Sx2ax1,f,where ⋅,⋅ denotes the canonical pairing between H1 and H1^.Proposition 1.
The HopfC∗-algebra H1op^ is a left H-module algebra for the left coadjoint representation.Proof.
Suppose thatx,y∈H, f,g∈H1op^, and a∈H1; then, we have(8)a,xy⋅f=Sxy2axy1,f=Sy2Sx1ax1y1,f=Sx1ax1,y⋅f=a,x⋅y⋅f.
Hence,xy⋅f=x⋅y⋅f, which means that H1op^ is a left H-module. Subsequently, using the cocommutativity of H, we obtain(9)a,x⋅fg=Sx2ax1,fg=Sx2ax11,fSx2ax12,g=Sx4a1x1,fSx3a2x2,g=Sx2a1x2,fSx4a2x3,g=a1,x1⋅fa2,x2⋅g=a,x1⋅fx2⋅g.
In order to complete the proof, we have to checkx⋅f∗=Sx∗⋅f∗. Since ∗-structure of H^ is defined by a,f∗=Sa∗,f¯, then(10)a,x⋅f∗=Sx2ax1,f∗=SSx2ax1∗,f¯=Sx1Sax2∗,f¯=x2∗Sa∗Sx1∗,f¯=x2∗Sa∗Sx1∗,f¯=SSx∗2Sa∗Sx∗1,f¯=Sa∗,Sx∗⋅f¯=a,Sx∗⋅f∗,where we use the fact S2=id and S is an anti-coalgebramorphism [7].Definition 3.
The nonbalanced quantum doubleDH1;H of H and H1 is defined as the crossed product of H with H1op^, with respect to the left coadjoint representation of the first algebra acting on the second one:(11)DH1;H=H1op^⋊H.
The Hopf∗-structures are as follows, for f⊗x,g⊗y∈DDH1;H:(12)f⊗xg⊗y=fx2⇀g↼Sx1⊗x3y,multiplication,1DH1;H=εH1⊗1H,unit,Δf⊗x=f2⊗x1⊗f1⊗x2,comultiplication,εf⊗x=f1H1εx,counit,Sf⊗x=Sx2⇀Sf↼x1⊗Sx3,antipode,f⊗x∗=x2∗⇀f∗↼Sx1∗⊗x3∗,∗−operation,where Sweedler’s arrows ⇀ (↼) denote the transpose of right (left) multiplication.Remark 2.
DH1;H is semisimple. Indeed, since H1 is semisimple, so is H1op^ [22]. Let T and t be a unique integral in H1op^ and H such that εH1op^T=1 and εHt=1; then, εDH1;HT⊗t=εH1op^TεHt=1, and(13)T⊗tf⊗x=Tt2⇀f↼St1⊗t3x=Tf1,St1f3,t2εf2⊗t3x=Tf1,St1f2,t2⊗t3x=Tεt1f1H1⊗t2x=Tf1H1⊗tx=Tf1H1⊗tεx=εf⊗xT⊗t.
Maschke theorem tells that any finite dimensional Hopf algebraK is semisimple iff there is a nonzero integral λ∈K such that εKλ≠0. Hence, DH1;H is semisimple.


### 2.2. The Observable Algebra and the Field Algebra

Let us continue to assume thatH is a finite-dimensional cocommutative semisimple Hopf C∗-algebra and H1 is a normal coideal ∗-subalgebra of H. Consider 1-dimensional lattice, which is composed of the lattice sites and links. We use even (odd) integers to denote lattice sites (links). There is a copy A2i of H on each lattice site and a copy A2i+1 of H1^, the dual of H1 on each link.Definition 4.
The quasi-local observable algebraAH1loc of nonbalanced Hopf spin models determined by the normal Hopf ∗-subalgebra H1 is a unital algebra generated by A2ia,A2i+1φ:a∈H,φ∈H1^,i∈ℤ subject to(14)AB=BA,A∈Ai,B∈Aj,i−j≥2,A2i+1φA2ia=A2ia1a2,φ1A2i+1φ2,A2iaA2i−1φ=A2i−1φ1φ2,a1A2ia2,where ⋅,⋅ denotes the canonical pairing between H and H^.
LetAH1n,m be a unital ∗-subalgebra of AH1loc generated by Ai,n<i<m. Using the C∗-inductive limit [17], AH1loc can be extended to a C∗-algebra AH1 called the observable algebra of nonbalanced Hopf spin models determined by a normal coideal ∗-subalgebra H1.Proposition 2.
For an intervalΛ of length 2, the map ρΛ:AH1Λ⟶AH1Λ⊗DH1;H given by(15)ρ2i−1,2iA2i−1φA2ia=A2i−1φ1A2ia2⊗φ2⊗a1,ρ2i,2i+1A2iaA2i+1φ=A2ia1A2i+1φ4⊗φ2⊗a3a2,φ3Sa4,φ1,where a∈Handφ∈H1^ define a right comodule algebra action of DH1;H on AH12i−1,2i (AH12i,2i+1) with respect to Δop (Δ).Proof.
Let us show thatAH12i−1,2i is a right DH1;H-comodule algebra with respect to ρ2i−1,2i. It suffices to check that the map ρ2i−1,2i satisfies the following relations:(16)ρ2i−1,2i⊗id∘ρ2i−1,2i=id⊗Δop∘ρ2i−1,2i,id⊗ε∘ρ2i−1,2i=id,ρ2i−1,2i∘m=m∘ρ2i−1,2i⊗ρ2i−1,2i,ρ2i−1,2i1AH12i−1,2i=1AH12i−1,2i⊗1DH1;H.
Now, let us compute by evaluating both sides on an elementA2i−1φA2ia in AH12i−1,2i:(17)ρ2i−1,2i⊗id∘ρ2i−1,2iA2i−1φA2ia=ρ2i−1,2iA2i−1φ1A2ia2⊗φ2⊗a1=A2i−1φ1A2ia3⊗φ2⊗a2⊗φ3⊗a1=A2i−1φ1A2ia2⊗Δφ2⊗a1=id⊗Δop∘ρ2i−1,2iA2i−1φA2ia,id⊗ε∘ρ2i−1,2iA2i−1φA2ia=A2i−1φ1A2ia2⊗εφ2⊗a1=A2i−1φ1A2ia2⊗φ21Hεa1=A2i−1φA2ia.
Hence,AH12i−1,2i is a right DH1;H-comodule. It remains to prove that ρ2i−1,2i is a ∗-algebra homomorphism. Using the commutative relations in Definition 4, we obtain(18)ρ2i−1,2iA2i−1φA2iaA2i−1ψA2ib=ρ2i−1,2iA2i−1φA2i−1ψ1ψ2,a1A2ia2A2ib=ρ2i−1,2iA2i−1φψ1ψ2,a1A2ia2b=A2i−1φψ11A2ia2b2ψ2,a1⊗φψ12⊗a2b1=A2i−1φ1ψ1A2ia3b2ψ3,a1⊗φ2ψ2⊗a2b1.
Similarly, we have(19)ρ2i−1,2iA2i−1φA2iaρ2i−1,2iA2i−1ψA2ib=A2i−1φ1A2ia2A2i−1ψ1A2ib2⊗φ2⊗a1ψ2⊗b1=A2i−1φ1A2i−1ψ1ψ2,a4A2ia5A2ib2⊗φ2ψ4ψ5,a2ψ3,Sa1⊗a3b1=A2i−1φ1ψ1A2ia5b2⊗φ2ψ4⊗a4b1ψ5,a1ψ2,a2ψ3,Sa3=A2i−1φ1ψ1A2ia4b2⊗φ2ψ3⊗a3b1ψ4,a1ψ2,εa21H=A2i−1φ1ψ1A2ia3b2⊗φ2ψ2⊗a2b1ψ3,a1,where we use the cocommutativity of H and ∑aa1a2Sa3=∑aa1εa21H=a.
By induction, we can define a comodule algebra action ofDH1;H on AH1Λ for any finite interval Λ⊂ℤ. It is known that if AH1Λ is a right DH1;H-comodule algebra, then AH1Λ is a left DH1;H^-module algebra via(20)ρξ:AH1Λ⟶AH1Λ,A↦idA⊗ξρA,for any ξ∈DH1;H^, A∈AH1Λ, where DH1;H^ is the dual of DH1;H. By Theorem 1, we can obtain the crossed product C∗-algebra of finite dimension AH1Λ⋊DH1;H^, denoted by ℱH1Λ. Let Λn be an increasing sequence of intervals; then, the natural embeddings ιn:ℱH1Λn⟶ℱH1Λn+1 are norm preserving.Definition 5.
On the nonbalanced Hopf spin models, the field algebraℱH1 is defined as the C∗-inductive limit for finite dimensional C∗-algebras ℱH1Λn.
Using the uniqueness ofC∗-inductive limit, the field algebra ℱH1 is actually the crossed product C∗-algebra AH1⋊DH1;H^ with respect to the comodule algebra. For convenience, denote by A,ξ the generating element in ℱH1.
For anyX∈DH1;H and A,ξ∈ℱH1, we define α:DH1;H×ℱH1⟶ℱH1 given by(21)αX⊗A,ξ=A,ξ1X,ξ2.
From now on, we suppressα and write XA,ξ for αX⊗A,ξ.Proposition 3.
The field algebraℱH1 determined by a normal Hopf ∗-subalgebra H1 is a left DH1;H-module algebra.Proof.
This follows from straightforward computations.Remark 3.
(1)
In Remark2, we have shown that T⊗t is the unique integral in DH1;H. Moreover, we have the following fact:(22)AH1=F∈ℱH1:T⊗tF=F.(2)
LetH1 and H2 be normal cosemisimple Hopf ∗-subalgebras of H. If H1⊆H2, then AH1⊆AH2.


## 2.1. The Nonbalanced Quantum Double

It is known that the quantum doubleDH, originally introduced by Drinfeld for a Hopf algebra H [8], plays an important role in the field of mathematical physics, and the quasi-triangular structure leads to a braiding in the category of representations and many ensuing applications. In particular, when H is the group algebra for a finite group, the quantum double reduces to an interesting crossed product algebra CG⋊G, where CG denotes the algebra of complex valued functions on G and the action is the conjugation [9]. Subsequently, several alternative descriptions of the quantum double have appeared in the literature. S. Majid [10] and F. Hausser and F. Nill [11, 12] independently introduced the quantum double of a finite dimensional quasi-Hopf algebra. In 2003, D. Bulacu and S. Caenepeel [13] constructed the quantum double for quasi-triangular quasi-Hopf algebras. In 2004, L. Delvaux and A. Van Daele [14] gave the Drinfeld double of multiplier Hopf algebras, which generalized the usual quantum double for Hopf algebras. In the usual quantum double DH=Hop^⊗H, two (generalized) Hopf C∗-algebra on both sides of ⊗ are the same. In this section, we hope to consider the similar theory when Hopf C∗-algebra are different. For this aim, we will generalize the quantum double to the nonbalanced quantum double by replacing DH by DH1;H=H1op^⊗H, where H1 is a normal Hopf ∗-subalgebra of H. In [15], Chen considered the nonbalanced quantum double DH1;H=H1op^⊗H, where H is a finite-dimensional Hopf algebra and H1 is a Hopf subalgebra of H, and showed DH1;H is a Hopf algebra. On the basis of the work, we will prove that DH1;H is a Hopf C∗-algebra by using the way different from [15]. Let us recall the following definition.Definition 1.
LetM be a ∗-algebra and A be a Hopf C∗-algebra. Then, M is a left A-module algebra if there is a left action ⊳:A⊗M⟶M, a⊗m↦a⊳m, and satisfies the following conditions, for any a,b∈A,m,n∈M:(1)ab⊳m=a⊳b⊳m,1A⊳m=m,a⊳mn=a1⊳ma2⊳n,a⊳m∗=Sa∗⊳m∗,a⊳1M=εa1M.
IfM is a C∗-algebra, the map a⊳ is assumed to be norm continuous for all a∈A.Theorem 1.
IfM is a left A-module algebra, the crossed product M⋊A is a ∗-algebra with the ∗-algebra structure given for m,n∈M, a,b∈A by(2)m⊗an⊗b=ma1n⊗a2b,m⊗a∗=1M⊗a∗m∗⊗1A.
Moreover, ifM is a C∗-algebra and A is of finite dimension, then the crossed product M⋊A becomes a C∗-algebra.Proof.
Clearly,M⋊A has the ∗-algebra structure. We will show that M⋊A is a C∗-algebra in the following.
It follows from [16] that M has a faithful positive linear functional φM and A has an invariant functional φA such that(3)φA=φA∘SandφA⊗idΔa=id⊗φAΔa=φA1A,for all a∈A. Define the map θ on M⋊A as follows:(4)θm⊗a=φMmφAa.
Then,θ is a faithful positive linear functional on M⋊A. In fact,(5)θm⊗am⊗a∗=θm⊗aa1∗m∗⊗a2∗=θma1a1∗m∗⊗a2a2∗=θma1a1∗m∗⊗a2a2∗=φMma1a1∗m∗φAa2a2∗=φMma1a1∗φAa2a2∗m∗=φMmφAaa∗1Am∗=φMmm∗φAaa∗≥0,and θm⊗am⊗a∗=0 if and only if m⊗a=0.
By [17], we can construct the associated GNS representation of M⋊A. Denote by K the completion of M⋊A with respect to the inner product m,x=θx∗m. Let τ:M⋊A⟶BK be the left multiplication; then, M⋊A can embed isometrically into BK, and thus, it is a C∗-algebra.
Semisimple Hopf algebras are intensively studied since they have important applications in topological invariants of knots and manifolds, quantum field theory, and so on. In fact, a Hopf algebra can be recovered from a normal Hopf subalgebra and some additional cohomological data [18]. As a result, normal Hopf subalgebras are an important tool in the classification of semisimple Hopf algebras. In 2012, B. Sebastian [19] studied normal left coideal subalgebras of semisimple Hopf algebras. In 2020, the authors [20] proved that, for a semisimple Hopf algebra, there is a one-to-one correspondence between right group-like projections and left coideal subalgebras.Definition 2.
LetH be any finite-dimensional semisimple Hopf C∗-algebra:(1)
A coideal subalgebraK of H is a subalgebra K of H with ΔK⊆H⊗K+K⊗H.(2)
A Hopf subalgebraK of H is said to be normal if K is invariant with respect to the right and left adjoint action:(6)Sh1kh2∈K,h1kSh2∈K,for allh∈H,k∈K.(3)
A coideal subalgebraK which is also a normal ∗-subalgebra of H is said to be normal coideal ∗-subalgebra of H.Remark 1.
(1)
IfK is a coideal subalgebra, then K is also a semisimple Hopf subalgebra (see Lemma 4.0.2 in [21]).(2)
Suppose thatG is a finite group and N is a normal subgroup of G. In this case of H=ℂG, the group algebra of G, K=ℂN can be viewed as a normal coideal ∗-subalgebra of H.
Note that the quantum doubleDG of a finite group is the crossed product of CG by the group algebra ℂG in G-spin models, and ℂG is always cocommutative. In order to give the definition of the nonbalanced quantum double in nonbalanced Hopf spin models, we suppose that H is a cocommutative semisimple Hopf C∗-algebra of finite dimension and H1 is a normal coideal ∗-subalgebra of H in the following. The left coadjoint representation of H on the dual of opposite Hopf C∗-algebra H1op^ is given for a∈H1, x∈H, and f∈H1^ by(7)a,x⋅f=Sx2ax1,f,where ⋅,⋅ denotes the canonical pairing between H1 and H1^.Proposition 1.
The HopfC∗-algebra H1op^ is a left H-module algebra for the left coadjoint representation.Proof.
Suppose thatx,y∈H, f,g∈H1op^, and a∈H1; then, we have(8)a,xy⋅f=Sxy2axy1,f=Sy2Sx1ax1y1,f=Sx1ax1,y⋅f=a,x⋅y⋅f.
Hence,xy⋅f=x⋅y⋅f, which means that H1op^ is a left H-module. Subsequently, using the cocommutativity of H, we obtain(9)a,x⋅fg=Sx2ax1,fg=Sx2ax11,fSx2ax12,g=Sx4a1x1,fSx3a2x2,g=Sx2a1x2,fSx4a2x3,g=a1,x1⋅fa2,x2⋅g=a,x1⋅fx2⋅g.
In order to complete the proof, we have to checkx⋅f∗=Sx∗⋅f∗. Since ∗-structure of H^ is defined by a,f∗=Sa∗,f¯, then(10)a,x⋅f∗=Sx2ax1,f∗=SSx2ax1∗,f¯=Sx1Sax2∗,f¯=x2∗Sa∗Sx1∗,f¯=x2∗Sa∗Sx1∗,f¯=SSx∗2Sa∗Sx∗1,f¯=Sa∗,Sx∗⋅f¯=a,Sx∗⋅f∗,where we use the fact S2=id and S is an anti-coalgebramorphism [7].Definition 3.
The nonbalanced quantum doubleDH1;H of H and H1 is defined as the crossed product of H with H1op^, with respect to the left coadjoint representation of the first algebra acting on the second one:(11)DH1;H=H1op^⋊H.
The Hopf∗-structures are as follows, for f⊗x,g⊗y∈DDH1;H:(12)f⊗xg⊗y=fx2⇀g↼Sx1⊗x3y,multiplication,1DH1;H=εH1⊗1H,unit,Δf⊗x=f2⊗x1⊗f1⊗x2,comultiplication,εf⊗x=f1H1εx,counit,Sf⊗x=Sx2⇀Sf↼x1⊗Sx3,antipode,f⊗x∗=x2∗⇀f∗↼Sx1∗⊗x3∗,∗−operation,where Sweedler’s arrows ⇀ (↼) denote the transpose of right (left) multiplication.Remark 2.
DH1;H is semisimple. Indeed, since H1 is semisimple, so is H1op^ [22]. Let T and t be a unique integral in H1op^ and H such that εH1op^T=1 and εHt=1; then, εDH1;HT⊗t=εH1op^TεHt=1, and(13)T⊗tf⊗x=Tt2⇀f↼St1⊗t3x=Tf1,St1f3,t2εf2⊗t3x=Tf1,St1f2,t2⊗t3x=Tεt1f1H1⊗t2x=Tf1H1⊗tx=Tf1H1⊗tεx=εf⊗xT⊗t.
Maschke theorem tells that any finite dimensional Hopf algebraK is semisimple iff there is a nonzero integral λ∈K such that εKλ≠0. Hence, DH1;H is semisimple.


## 2.2. The Observable Algebra and the Field Algebra

Let us continue to assume thatH is a finite-dimensional cocommutative semisimple Hopf C∗-algebra and H1 is a normal coideal ∗-subalgebra of H. Consider 1-dimensional lattice, which is composed of the lattice sites and links. We use even (odd) integers to denote lattice sites (links). There is a copy A2i of H on each lattice site and a copy A2i+1 of H1^, the dual of H1 on each link.Definition 4.
The quasi-local observable algebraAH1loc of nonbalanced Hopf spin models determined by the normal Hopf ∗-subalgebra H1 is a unital algebra generated by A2ia,A2i+1φ:a∈H,φ∈H1^,i∈ℤ subject to(14)AB=BA,A∈Ai,B∈Aj,i−j≥2,A2i+1φA2ia=A2ia1a2,φ1A2i+1φ2,A2iaA2i−1φ=A2i−1φ1φ2,a1A2ia2,where ⋅,⋅ denotes the canonical pairing between H and H^.
LetAH1n,m be a unital ∗-subalgebra of AH1loc generated by Ai,n<i<m. Using the C∗-inductive limit [17], AH1loc can be extended to a C∗-algebra AH1 called the observable algebra of nonbalanced Hopf spin models determined by a normal coideal ∗-subalgebra H1.Proposition 2.
For an intervalΛ of length 2, the map ρΛ:AH1Λ⟶AH1Λ⊗DH1;H given by(15)ρ2i−1,2iA2i−1φA2ia=A2i−1φ1A2ia2⊗φ2⊗a1,ρ2i,2i+1A2iaA2i+1φ=A2ia1A2i+1φ4⊗φ2⊗a3a2,φ3Sa4,φ1,where a∈Handφ∈H1^ define a right comodule algebra action of DH1;H on AH12i−1,2i (AH12i,2i+1) with respect to Δop (Δ).Proof.
Let us show thatAH12i−1,2i is a right DH1;H-comodule algebra with respect to ρ2i−1,2i. It suffices to check that the map ρ2i−1,2i satisfies the following relations:(16)ρ2i−1,2i⊗id∘ρ2i−1,2i=id⊗Δop∘ρ2i−1,2i,id⊗ε∘ρ2i−1,2i=id,ρ2i−1,2i∘m=m∘ρ2i−1,2i⊗ρ2i−1,2i,ρ2i−1,2i1AH12i−1,2i=1AH12i−1,2i⊗1DH1;H.
Now, let us compute by evaluating both sides on an elementA2i−1φA2ia in AH12i−1,2i:(17)ρ2i−1,2i⊗id∘ρ2i−1,2iA2i−1φA2ia=ρ2i−1,2iA2i−1φ1A2ia2⊗φ2⊗a1=A2i−1φ1A2ia3⊗φ2⊗a2⊗φ3⊗a1=A2i−1φ1A2ia2⊗Δφ2⊗a1=id⊗Δop∘ρ2i−1,2iA2i−1φA2ia,id⊗ε∘ρ2i−1,2iA2i−1φA2ia=A2i−1φ1A2ia2⊗εφ2⊗a1=A2i−1φ1A2ia2⊗φ21Hεa1=A2i−1φA2ia.
Hence,AH12i−1,2i is a right DH1;H-comodule. It remains to prove that ρ2i−1,2i is a ∗-algebra homomorphism. Using the commutative relations in Definition 4, we obtain(18)ρ2i−1,2iA2i−1φA2iaA2i−1ψA2ib=ρ2i−1,2iA2i−1φA2i−1ψ1ψ2,a1A2ia2A2ib=ρ2i−1,2iA2i−1φψ1ψ2,a1A2ia2b=A2i−1φψ11A2ia2b2ψ2,a1⊗φψ12⊗a2b1=A2i−1φ1ψ1A2ia3b2ψ3,a1⊗φ2ψ2⊗a2b1.
Similarly, we have(19)ρ2i−1,2iA2i−1φA2iaρ2i−1,2iA2i−1ψA2ib=A2i−1φ1A2ia2A2i−1ψ1A2ib2⊗φ2⊗a1ψ2⊗b1=A2i−1φ1A2i−1ψ1ψ2,a4A2ia5A2ib2⊗φ2ψ4ψ5,a2ψ3,Sa1⊗a3b1=A2i−1φ1ψ1A2ia5b2⊗φ2ψ4⊗a4b1ψ5,a1ψ2,a2ψ3,Sa3=A2i−1φ1ψ1A2ia4b2⊗φ2ψ3⊗a3b1ψ4,a1ψ2,εa21H=A2i−1φ1ψ1A2ia3b2⊗φ2ψ2⊗a2b1ψ3,a1,where we use the cocommutativity of H and ∑aa1a2Sa3=∑aa1εa21H=a.
By induction, we can define a comodule algebra action ofDH1;H on AH1Λ for any finite interval Λ⊂ℤ. It is known that if AH1Λ is a right DH1;H-comodule algebra, then AH1Λ is a left DH1;H^-module algebra via(20)ρξ:AH1Λ⟶AH1Λ,A↦idA⊗ξρA,for any ξ∈DH1;H^, A∈AH1Λ, where DH1;H^ is the dual of DH1;H. By Theorem 1, we can obtain the crossed product C∗-algebra of finite dimension AH1Λ⋊DH1;H^, denoted by ℱH1Λ. Let Λn be an increasing sequence of intervals; then, the natural embeddings ιn:ℱH1Λn⟶ℱH1Λn+1 are norm preserving.Definition 5.
On the nonbalanced Hopf spin models, the field algebraℱH1 is defined as the C∗-inductive limit for finite dimensional C∗-algebras ℱH1Λn.
Using the uniqueness ofC∗-inductive limit, the field algebra ℱH1 is actually the crossed product C∗-algebra AH1⋊DH1;H^ with respect to the comodule algebra. For convenience, denote by A,ξ the generating element in ℱH1.
For anyX∈DH1;H and A,ξ∈ℱH1, we define α:DH1;H×ℱH1⟶ℱH1 given by(21)αX⊗A,ξ=A,ξ1X,ξ2.
From now on, we suppressα and write XA,ξ for αX⊗A,ξ.Proposition 3.
The field algebraℱH1 determined by a normal Hopf ∗-subalgebra H1 is a left DH1;H-module algebra.Proof.
This follows from straightforward computations.Remark 3.
(1)
In Remark2, we have shown that T⊗t is the unique integral in DH1;H. Moreover, we have the following fact:(22)AH1=F∈ℱH1:T⊗tF=F.(2)
LetH1 and H2 be normal cosemisimple Hopf ∗-subalgebras of H. If H1⊆H2, then AH1⊆AH2.


## 3. Quantum Double Symmetry

The main objective of this section is to build a duality between the quantum doubleDH1;H and the observable algebra defined in Section 2.Theorem 2.
Suppose thatΠ is an irreducible representation of ℱH1 on the Hilbert space ℋ=ΠℱH1Ω¯ with a vacuum vector Ω satisfying(23)Ω,ΠXFΩ=εXΩ,πFΩ,∀X∈DH1;H,F∈ℱH1.
Then, there is a uniqueC∗-homomorphism U from DH1;H to Bℋ such that(1)
UX1ΠFUSX2=ΠXF,∀X∈DH1;H,∀F∈ℱH1(2)
UDH1;H′=ΠAH1¯, UDH1;H=ΠAH1′, where the prime means commutant in Bℋ and the bar denotes weak closureProof.
(1)
For anyX∈DH1;H, the map UX:ΠℱH1Ω⟶ΠℱH1Ω is defined by(24)UXΠFΩ=ΠXFΩ,∀F∈ℱH1.Firstly, it is well-defined. In fact, for anyT,F∈ℱH1 and X∈DH1;H, we have(25)ΠTΩ,ΠXFΩ=Ω,ΠT∗XFΩ=Ω,ΠT∗X1FΩεSX2=Ω,ΠSX2T∗X1FΩ=Ω,ΠSX3T∗SX2X1FΩ=Ω,ΠSX2T∗εX1FΩ=ΠX∗TΩ,ΠFΩ.which shows thatΠFΩ=0 can imply ΠXFΩ=0 for all X∈DH1;H.Moreover,U is a ∗-representation, i.e., UX∗=UX∗ and UXY=UXUY, which yields that U≤1 from [23]. Hence, U can be extended to ℋ by continuity.The uniqueness ofU is due to the fact UXΩ=εXΩ.In order to check thatUX1ΠFUSX2=ΠXF, it is enough to verify it on the generators ΠℱH1Ω of the Hilbert space ℋ. For any T∈ℱH1,(26)UX1ΠFUSX2ΠTΩ=UX1ΠFSX2TΩ=ΠX1FSX2TΩ=ΠX1FX2SX3TΩ=ΠX1FΠεX2TΩ=ΠX1εX2FΠTΩ=ΠXFΠTΩ.(2)
It is enough to check thatUDH1;H′=ΠAH1¯, since any C∗-algebra of finite dimension is always weakly closed.For anyX∈DH1;H, T,F∈AH1, we obtain(27)UXΠFΠTΩ=UXΠFTΩ=ΠX1FX2TΩ=ΠεX1FX2TΩ=ΠFXTΩ=ΠFUXΠTΩ,which shows UXΠF=ΠFUX for a dense subset ΠℱH1Ω in ℋ. It follows from the continuity of U that UXΠF=ΠFUX in ℋ. Thus, ΠAH1¯⊆UDH1;H′.
It remains to show thatUDH1;H′⊆ΠAH1¯. Set Z∈UDH1;H′; then,(28)ZUX=UXZ,∀X∈DH1;H.
Using the result [17], one can concludes that ΠℱH1 is dense in Bℋ with respect to the weak operator topology, which yields that there exists a net Fα⊆ℱH1 such that ΠFα converges to Z, and then,(29)UX1ΠFα−ZUSX2⟶0,ΠXFα−εXZ⟶0.
LetX=T⊗t be defined in Remark 2; then,(30)ΠT⊗tFα⟶εT⊗tA=A,which means UDH1;H′⊆ΠAH,G¯, and thus, UDH1;H′=ΠAH1¯.



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