#### Abstract

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

#### 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 is -spin models, introduced by K. Szlachanyi and P. Vecsernyes [1] in lattice field theories, where is a finite group. -spin models as a classical statistical systems have an order-disorder type of quantum symmetry, which is the quantum double . The quantum symmetry in -spin models generalizes the symmetry which can sharply divide the ordered phase and the disordered phase in Ising models. Generally, if is an Abelian group, -spin models have a symmetry group , which is the direct product of the group and the group of characters of . Based on a field-theory analysis of -spin models, Jiang and Guo [2] gave the concrete construction of a -invariant subspace in field algebra of -spin models and proved the -invariant subspace is Galois closed if is a normal subgroup of . On the contrary, Xin and Jiang [3] generalized -spin models to -spin models determined by a normal subgroup , in which the quantum double , and the field algebra determined by are defined, and then, the observable algebra determined by can be obtained as -invariant subalgebra. Based on these work, the quantum symmetry is given by the quantum double .

Note that , the disorder part of the quantum double , is Abelian, so -spin models do not have Kramers–Wannier duality for non-Abelian group. For this reason, -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 -algebra , there is a copy of on every lattice and a copy of on every link, and they satisfy nontrivial commutation relations only if they are neighbour links and sites, where is the dual of . 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 -algebra and a normal coideal -subalgebra , we define the quantum double by means of the left -module algebra structure on Hopf -algebra , where is the dual of the opposite Hopf -algebra of . Since Hopf -algebras on both sides of in the quantum double are different, we call the nonbalanced quantum double. In particular, let be the group algebra of a finite group and be the group algebra of a normal subgroup of ; then, can be naturally regarded as the normal Hopf-subalgebra of , and reduces to [3]. Moreover, we will also describe the quantum symmetry in the corresponding quantum chains of Hopf -algebras and called nonbalanced Hopf spin models: there is a copy of on every lattice site and a copy of on every link, and they satisfy some commutation relations, which generalizes the results established in [3, 4].

The paper is organized as follows. In Section 2, we introduce the nonbalanced quantum double and define a right comodule algebra action of nonbalanced quantum double on the observable algebra determined by a semisimple normal Hopf -subalgebra . It is known that there is one-to-one correspondence between right -comodule algebra actions and left -module algebra actions, since is finite dimensional, where is the dual of . Based on these, we can define the field algebra of nonbalanced Hopf spin models as the crossed product -algebra of the observable algebra by 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 and vice verse.

In this paper, we work over a complex field . All Hopf algebraic notations [6, 7] can be used in the following. For example, we denote by the multiplication, the unit, the antipode, the comultiplication, the counit, and -operation, respectively. And, we use the standard notation and in the Hopf algebra where there is an implicit summation on the right side and .

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

##### 2.1. The Nonbalanced Quantum Double

It is known that the quantum double , originally introduced by Drinfeld for a Hopf algebra [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 is the group algebra for a finite group, the quantum double reduces to an interesting crossed product algebra , where denotes the algebra of complex valued functions on 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 , two (generalized) Hopf -algebra on both sides of are the same. In this section, we hope to consider the similar theory when Hopf -algebra are different. For this aim, we will generalize the quantum double to the nonbalanced quantum double by replacing by , where is a normal Hopf -subalgebra of . In [15], Chen considered the nonbalanced quantum double , where is a finite-dimensional Hopf algebra and is a Hopf subalgebra of , and showed is a Hopf algebra. On the basis of the work, we will prove that is a Hopf -algebra by using the way different from [15]. Let us recall the following definition.

*Definition 1. *Let be a -algebra and be a Hopf -algebra. Then, is a left -module algebra if there is a left action , , and satisfies the following conditions, for any :If is a -algebra, the map is assumed to be norm continuous for all .

Theorem 1. *If is a left -module algebra, the crossed product is a -algebra with the -algebra structure given for , by**Moreover, if is a -algebra and is of finite dimension, then the crossed product becomes a -algebra.*

*Proof. *Clearly, has the -algebra structure. We will show that is a -algebra in the following.

It follows from [16] that has a faithful positive linear functional and has an invariant functional such thatfor all . Define the map on as follows:Then, is a faithful positive linear functional on . In fact,and if and only if .

By [17], we can construct the associated GNS representation of . Denote by the completion of with respect to the inner product . Let be the left multiplication; then, can embed isometrically into , and thus, it is a -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. *Let be any finite-dimensional semisimple Hopf -algebra:(1)A coideal subalgebra of is a subalgebra of with .(2)A Hopf subalgebra of is said to be normal if is invariant with respect to the right and left adjoint action: for all .(3)A coideal subalgebra which is also a normal -subalgebra of is said to be normal coideal -subalgebra of .

*Remark 1. *(1)If is a coideal subalgebra, then is also a semisimple Hopf subalgebra (see Lemma 4.0.2 in [21]).(2)Suppose that is a finite group and is a normal subgroup of . In this case of , the group algebra of , can be viewed as a normal coideal -subalgebra of .Note that the quantum double of a finite group is the crossed product of by the group algebra in -spin models, and is always cocommutative. In order to give the definition of the nonbalanced quantum double in nonbalanced Hopf spin models, we suppose that is a cocommutative semisimple Hopf -algebra of finite dimension and is a normal coideal -subalgebra of in the following. The left coadjoint representation of on the dual of opposite Hopf -algebra is given for , , and bywhere denotes the canonical pairing between and .

Proposition 1. *The Hopf -algebra is a left -module algebra for the left coadjoint representation.*

*Proof. *Suppose that , , and ; then, we haveHence, , which means that is a left -module. Subsequently, using the cocommutativity of , we obtainIn order to complete the proof, we have to check . Since -structure of is defined by , thenwhere we use the fact and is an anti-coalgebramorphism [7].

*Definition 3. *The nonbalanced quantum double of and is defined as the crossed product of with , with respect to the left coadjoint representation of the first algebra acting on the second one:The Hopf -structures are as follows, for :where Sweedler’s arrows () denote the transpose of right (left) multiplication.

*Remark 2. * is semisimple. Indeed, since is semisimple, so is [22]. Let and be a unique integral in and such that and ; then, , andMaschke theorem tells that any finite dimensional Hopf algebra is semisimple iff there is a nonzero integral such that . Hence, is semisimple.

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

Let us continue to assume that is a finite-dimensional cocommutative semisimple Hopf -algebra and is a normal coideal -subalgebra of . 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 of on each lattice site and a copy of , the dual of on each link.

*Definition 4. *The quasi-local observable algebra of nonbalanced Hopf spin models determined by the normal Hopf -subalgebra is a unital algebra generated by subject towhere denotes the canonical pairing between and .

Let be a unital -subalgebra of generated by . Using the -inductive limit [17], can be extended to a -algebra called the observable algebra of nonbalanced Hopf spin models determined by a normal coideal -subalgebra .

Proposition 2. *For an interval of length 2, the map given bywhere define a right comodule algebra action of on () with respect to ().*

*Proof. *Let us show that is a right -comodule algebra with respect to . It suffices to check that the map satisfies the following relations:Now, let us compute by evaluating both sides on an element in :Hence, is a right -comodule. It remains to prove that is a -algebra homomorphism. Using the commutative relations in Definition 4, we obtainSimilarly, we havewhere we use the cocommutativity of and .

By induction, we can define a comodule algebra action of on for any finite interval . It is known that if is a right -comodule algebra, then is a left -module algebra viafor any , , where is the dual of . By Theorem 1, we can obtain the crossed product -algebra of finite dimension , denoted by . Let be an increasing sequence of intervals; then, the natural embeddings are norm preserving.

*Definition 5. *On the nonbalanced Hopf spin models, the field algebra is defined as the -inductive limit for finite dimensional -algebras .

Using the uniqueness of -inductive limit, the field algebra is actually the crossed product -algebra with respect to the comodule algebra. For convenience, denote by the generating element in .

For any and , we define given byFrom now on, we suppress and write for .

Proposition 3. *The field algebra determined by a normal Hopf -subalgebra is a left -module algebra.*

*Proof. *This follows from straightforward computations.

*Remark 3. *(1)In Remark 2, we have shown that is the unique integral in . Moreover, we have the following fact:(2)Let and be normal cosemisimple Hopf -subalgebras of . If , then .

#### 3. Quantum Double Symmetry

The main objective of this section is to build a duality between the quantum double and the observable algebra defined in Section 2.

Theorem 2. *Suppose that is an irreducible representation of on the Hilbert space with a vacuum vector satisfying**Then, there is a unique -homomorphism from to such that*(1)*(2)**, , where the prime means commutant in and the bar denotes weak closure*

*Proof. *(1)For any , the map is defined by Firstly, it is well-defined. In fact, for any and , we have which shows that can imply for all . Moreover, is a -representation, i.e., and , which yields that from [23]. Hence, can be extended to by continuity. The uniqueness of is due to the fact . In order to check that , it is enough to verify it on the generators of the Hilbert space . For any ,(2)It is enough to check that , since any -algebra of finite dimension is always weakly closed. For any , , we obtainwhich shows for a dense subset in . It follows from the continuity of that in . Thus, .

It remains to show that . Set ; then,Using the result [17], one can concludes that is dense in with respect to the weak operator topology, which yields that there exists a net such that converges to , and then,Let be defined in Remark 2; then,which means , and thus, .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors are greatly indebted to Jiang Lining for his helpful discussions. This work was supported by the National Natural Science Foundation of China (Grants nos. 11701423, 61701343, and 61771294).