-Basic Construction from the Conditional Expectation on the Drinfeld Double

Abstract

Let be the Drinfeld double of a finite group and be the crossed product of and , where is a subgroup of . Then the sets and can be made -algebras naturally. Considering the -basic construction from the conditional expectation of onto , one can construct a crossed product -algebra , such that the -basic construction is -algebra isomorphic to .

1. Introduction

Index theory for subfactors was initiated by Jones ([1]) and has experienced rapid progress beyond the framework of operator algebras. For example, Jones’ index theory has found important applications in knots theory, conformal field theory, algebraic quantum field theory, and so on ([2–8]). For a nontechnical but broad overview of the subject including a lot of important connections with other areas, the readers can refer to [9].

Let be factors of type with finite Jones index and tr the faithful normal tracial state on . Denoted by the Hilbert space closure of is with respect to the norm . Then acts on by the left multiplication. The involution extends to an isometric conjugate linear operator on denoted by . The remarkable discovery of Jones is that the possible values for the index are , , or It is rather easy to construct a reducible inclusion of factors with any index value larger or equal to 4. All the values in the discrete series are realized by means of the basic construction. To be precise, let be a conditional expectation from onto associated with the trace, such that for and . The extension of to , denoted by , is the orthogonal projection of onto the closure of regarded as a subspace of . Then , the von Neumann algebra generated by and on , is called the basic construction, and , which is a perfect result. Subsequently Jones used the basic construction to obtain an increasing sequence of type factors, which is called the Jones tower, iteratively by adding the Jones projections which satisfy the Temperley-Lieb relations. Finally, Jones used this structure to construct an example for any possible index value below 4.

The Jones index theory for subfactors of type has been extended to unital -algebras by Watatani ([10]), and many interesting results of -index theory can be found in [11, 12]. Note that in [10] Watatani showed that if is an outer action of a finite group on a simple -algebra and is the condition expectation from onto the fixed point subalgebra given by the average over , then the basic construction is identified with the crossed product . It was also shown in [12] that if is a pure Hopf algebra acting outerly on a factor , then is a factor and therefore the Jones basic construction . However, different from the basic construction for type factors, the -basic construction does not have the concrete form in general, where is a conditional expectation of a -algebra onto a -subalgebra . The reason is that any factor of type possesses the faithful trace which is a state of this kind for which the Gelfand-Naimark-Segal construction may be performed, while for general -algebras, the existence of this functional is uncertain.

Letting be a finite group and its subgroup, denoted by , then is defined as the crossed product of , the algebra of complex functions on , and group algebra with respect to the adjoint action of the latter on the former. In particular, ; then is the Drinfeld double of . It was shown in [13] that there is the conditional expectation of index-finite type. In this paper, we prove that the -basic construction from the conditional expectation can be described as a crossed product .

The paper is organized as follows. In Section 2, we give a brief description of the -basic construction for -algebras, and we also collect the necessary definitions and facts about the set . In Section 3, we define an action of on , and obtain the resulting crossed product . Theorem 10 is our main result which means that the -basic construction is -algebra isomorphic to the -algebras .

2. The -Basic Construction of the Drinfeld Double

In this section, suppose that is a -algebra over and a -subalgebra with a common unit 1. Let us recall these definitions and facts which can be found in the works of Watatani ([10]) and Jiang ([13]).

Definition 1. A linear map is called a conditional expectation if it satisfies the following conditions:(1);(2)(bimodular property) for , , (3) is positive; i.e., is a positive element in for any .Actually, if is a conditional expectation from onto , then is a projection of norm one ([14]).

Definition 2. Let be a conditional expectation. A finite family is called a quasi-basis for if for all , Furthermore, if there exists a quasi-basis for , we call of index-finite type. In this case we define the index of by If is of index-finite type, then is in the center of and does not depend on the choice of quasi-basis ([10]).

In the following we recall the -basic construction from the conditional expectation .

Notice that is an -bimodule algebra; one can consider the module tensor product algebra , the product of which depends on as follows: for , Then turns out to be an algebra. And define an involution by for . The involution is well defined by considering conjugate operation in -algebras. Thus becomes an -algebra.

Recall that denotes the algebra of all bounded linear operators on a Hilbert space .

Lemma 3. Let be a conditional expectation. Let be a -representation of on a Hilbert space . Consider a conditional expectation . Then there exists a -representation such that for
For , set then Thus is a -seminorm on ([10]).

Definition 4. The completion of by the norm (after taking quotient by the ideal if necessary) is called the -basic construction from the conditional expectation . We denote this -algebra by .

Now assume that is a finite group with a subgroup and is a left coset representation of in , namely, and induces that , where is the unit of . Let us begin with the following definitions.

Definition 5. is the crossed product of and group algebra , where denotes the set of complex functions on , with respect to the adjoint action of the latter on the former.

Using the basis elements of , the multiplication rule is given as follows: The unit of is . becomes a unital -algebra by defining the -operation on the basis elements and extending antilinearly to .

The coproduct , the counit , and the antipode are defined by where , and . One can prove becomes a Hopf -algebra ([15]). There exists a unique element , called a cointegral, satisfying , , and . As a result, is semisimple with a natural -structure. Consequently it can be a -algebra of finite dimension.

In particular, if , then is the Drinfeld double of . For more information about one can refer to [16–18]. The main difference between and is that the former is a quasitriangular Hopf algebra while the latter is not ([19]).

Considering a linear map where , one can show that it is the conditional expectation from onto . Moreover, setting , then and is a quasibasis for . Thus . In this case, is of index-finite type.

Definition 6. The completion of with respect to the norm (after taking quotient by the ideal if necessary) is called the -basic construction from the conditional expectation of onto . We denote this -algebra by .

Note that is the unit of .

3. The Construction of a Crossed Product -Algebra

Let us continue to suppose that is a finite group and . Denoted by the set of all left cosets of , i.e., . Let and be the algebra of complex functions on and the group algebra over , respectively.

The set is a linear basis of where while the set is a linear basis of where Since is a finite group and , then can be regarded as a linear basis of , where we write instead of for notational convenience.

For any , we can define the map given on the basis elements of as which can be linearly extended in . One can verify that is an automorphic map, and then defines an automorphic action of on .

Assume that acts on a Hilbert space , and an action of on is given as above. Let We view as the Hilbert space of all square summable -valued functions on , and define: for , , .

Lemma 7. is a faithful -representation of on , while is a unitary representation of on . And

Definition 8. The associative -algebra on generated by is called the crossed product of by with respect to , and we denote it by (or simply by ).
Here and from now on, by and we always denote and , respectively.

Lemma 9. (1)The element in is a projection on , i.e., (2)For , we have (3)Let , then if and only if

Proof. (1) For any and , observe that and From the above equalities, we conclude that . Andwhich shows that .
(2) Let , we can compute that and (3) If , then and Thus commutes with .
Conversely, if and , then which implies that . This shows that .

From the proof of Lemma 9, we can obtain for any , .

Now we give the main theorem of this paper.

Theorem 10. There exists an isometric -isomorphism of onto

Proof. By the definition of , it suffices to show that there exists an -isomorphism of on .
The map is defined on the generators of by for , in . Since and commute, is a well defined map. Then it can be linearly extended in . Again since and Then Hence, combining with Lemma 9, we have that the map is a -homomorphism.
For , choose such that . This shows that is surjective. Finally, we will verify that is isometric.
Let be a map given as . It is easy to see that is a injective -homomorphism. Define by . Then is also injective. Again, by Lemma 3, there exists such that Since , for , where is a conditional expectation and , Thus, we have that . Now we have shown is an isometric -isomorphism.

Example 11. (1) Let (or ) be the group -algebra of (or ), namely, the -subalgebra of generated by the left regular representation of (or ). Consider the basic construction from the conditional expectation defined by where . Let be the action induced by translation from left, where stands for the group of all automorphism of . In [10], Watatani showed that the -basic construction is -algebra isomorphic to . This result is a special case of Theorem 10. In fact, there is an inclusion given by .
(2) If is a finite abelian group, then reduces to a symmetry group , where denotes the Pontryagin dual of . Let be a conditional expectation defined by We can define the map by for all . Then is an automorphic action of on . Then the -basic construction is -algebra isomorphic to . Furthermore, the -basic construction can be described as , where is the action induced by translation from left.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants Nos. 11701423, 61701343, and 61771294).