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

For a finite-dimensional cocommutative semisimple Hopf C∗-algebra H and a normal coideal ∗-subalgebra H1, we define the nonbalanced quantum double D(H1; H) as the crossed product of H with 􏽤 H op 1 , with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed productAH1 � · · ·⋊H⋊􏽣 H1⋊H⋊ H1⋊H⋊ · · · as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of D(H1; 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 D(H1; H) and the observable algebra AH1.


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 G-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 D(G). e quantum symmetry in G-spin models generalizes the Z 2 × Z 2 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 D(G; N)-invariant subspace in field algebra of G-spin models and proved the D(G; 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 D(N; G), and the field algebra determined by N are defined, and then, the observable algebra determined by N can be obtained as D(N; G)-invariant subalgebra. Based on these work, the quantum symmetry is given by the quantum double D(N; G).
Note that G, the disorder part of the quantum double D(G), 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. e 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 finitedimensional cocommutative semisimple Hopf C * -algebra H and a normal coideal * -subalgebra H 1 , we define the quantum double D(H 1 ; H) by means of the left H-module algebra structure on Hopf C * -algebra H op 1 , where H op 1 is the dual of the opposite Hopf C * -algebra of H 1 . Since Hopf C * -algebras on both sides of ⊗ in the quantum double D(H 1 ; H) are different, we call D(H 1 ; H) the nonbalanced quantum double. In particular, let H � C(G) be the group algebra of a finite group G and H 1 � CN be the group algebra of a normal subgroup N of G; then, H 1 can be naturally regarded as the normal Hopf *subalgebra of H, and D(H 1 ; H) reduces to D(N; G) [3]. Moreover, we will also describe the quantum symmetry in the corresponding quantum chains of Hopf C * -algebras H and H 1 called nonbalanced Hopf spin models: there is a copy of H on every lattice site and a copy of H 1 on every link, and they satisfy some commutation relations, which generalizes the results established in [3,4]. e 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 D(H 1 ; H) on the observable algebra A H 1 determined by a semisimple normal Hopf * -subalgebra H 1 . It is known that there is one-to-one correspondence between right Based on these, we can define the field algebra of nonbalanced Hopf spin models as the crossed product C * -algebra of the observable algebra A H 1 by D(H 1 ; 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 A (H,G) and vice verse.

e Nonbalanced Quantum Double.
It is known that the quantum double D(H), 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 C(G)⋊G, where C(G) 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] H) is a Hopf algebra. On the basis of the work, we will prove that D(H 1 ; H) is a Hopf C * -algebra by using the way different from [15]. Let us recall the following definition. (ab)⊳m � a⊳(b⊳m), If M is a C * -algebra, the map a ⊳ is assumed to be norm continuous for all a ∈ A.
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 φ M and A has an invariant functional φ A such that for all a ∈ A. Define the map θ on M⋊A as follows: en, θ is a faithful positive linear functional on M⋊A. In fact, 2 Journal of Mathematics 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〉 � θ(x * m). Let τ: M⋊A ⟶ B(K) be the left multiplication; then, M⋊A can embed isometrically into B(K), 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. Let H be any finite-dimensional semisimple Hopf C * -algebra: invariant with respect to the right and left adjoint action: for all h ∈ H, k ∈ K.
(3) A coideal subalgebra K which is also a normal * -subalgebra of H is said to be normal coideal * -subalgebra of H.

Remark 1
(1) If K is a coideal subalgebra, then K is also a semisimple Hopf subalgebra (see Lemma 4.0.2 in [21]). (2) Suppose that G 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.

Journal of Mathematics
In order to complete the proof, we have to check x · f * � (Sx * · f) * . Since * -structure of H is defined by 〈a, f * 〉 � 〈(Sa) * , f〉, then where we use the fact S 2 � id and S is an anti-coalgebramorphism [7].

Remark 2. D(H
Maschke theorem tells that any finite dimensional Hopf algebra K is semisimple iff there is a nonzero integral λ ∈ K such that ε K (λ) ≠ 0. Hence, D (H 1 ; H) is semisimple.

e Observable Algebra and the Field Algebra.
Let us continue to assume that H is a finite-dimensional cocommutative semisimple Hopf C * -algebra and H 1 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). ere is a copy A 2i of H on each lattice site and a copy A 2i+1 of H 1 , the dual of H 1 on each link.

Definition 4.
e quasi-local observable algebra A loc H 1 of nonbalanced Hopf spin models determined by the normal Hopf * -subalgebra H 1 is a unital algebra generated by where 〈·, ·〉 denotes the canonical pairing between H and H.
Let A n,m H 1 be a unital * -subalgebra of A loc H 1 generated by A i , n < i < m. Using the C * -inductive limit [17], A loc H 1 can be 4 Journal of Mathematics extended to a C * -algebra A H 1 called the observable algebra of nonbalanced Hopf spin models determined by a normal coideal * -subalgebra H 1 .
By induction, we can define a comodule algebra action of D (H 1 ; H) for any ξ ∈ D (H 1 ; H) (H 1 ; H). By eorem 1, we can obtain the crossed product C * -algebra of finite dimension A Λ H 1 ⋊D (H 1 ; H), denoted by F Λ H 1 . Let Λ n be an increasing sequence of intervals; then, the natural embeddings ι n : F On the nonbalanced Hopf spin models, the field algebra F H 1 is defined as the C * -inductive limit for finite dimensional C * -algebras F Λ n H 1 . Using the uniqueness of C * -inductive limit, the field algebra F H 1 is actually the crossed product C * -algebra A H 1 ⋊D(H 1 ; H) with respect to the comodule algebra. For convenience, denote by (A, ξ) the generating element in F H 1 .

Proposition 3.
e field algebra F H 1 determined by a normal Hopf * -subalgebra H 1 is a left D(H 1 ; H)-module algebra. Proof.
is follows from straightforward computations. □

Remark 3
(1) In Remark 2, we have shown that T ⊗ t is the unique integral in D (H 1 ; H). Moreover, we have the following fact:

Quantum Double Symmetry
e main objective of this section is to build a duality between the quantum double D (H 1 ; H) and the observable algebra defined in Section 2.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.