Constructing a non-degenerate commuting square from equivalent systems of bimodules

Nobuya Sato ^{∗} Graduate School of Mathematical Sciences

University of Tokyo, Komaba, Tokyo, 153, JAPAN

          (e-mail address: nobuya@ms.u-tokyo.ac.jp)

Abstract

When we have two inclusions of AFD II_{1} factors A ⊂ B and C ⊂ D with finite index and finite depth and the system of the B-B bimodules is equivalent to that of the C-C bimodules in the sense of A. Ocneanu, we prove that we can construct a finite dimensional non-degenerate commuting square which produces the inclusions A ⊂ B and C^{opp} ⊂ D^{opp} by iterated basic constructions.

1991 Mathematics Subject Classification. Primary 46L37.

∗This work is supported by the JSPS Research Fellowships for Young Scientists.

1 Introduction

Since V. F. R. Jones has established the relation between subfactors and low dimensional topology, there has been much progress for subfactor theory. The most striking theory in combinatorial aspects of subfactor theory is Ocneanu’s paragroup theory [O1], which describes the quantum symmetry [O1, E-K3] of inclusions of approximately finite dimensional (AFD) II_{1} factors.

A paragroup consists of some data with axioms and its formulation is quite similar to that of exactly solvable lattice models except for the flatness axiom in paragroup theory [O1]. Surprisingly, this combinatorial object determines the AFD II_{1} subfactor completely, by Popa’s classification theorem [P]. Not only the interest to operator algebras, a paragroup also has its significance in other fields of mathematics and physics. Namely, paragroup theory is much related to exactly solvable lattice models [R], rational conformal field theory (RCFT) [B-G] in the sense of Moore-Seiberg [M-S], 3-dimensional topological quantum field theory (TQFT) [O3, T-V, E-K1] and quantum groups [W2, W3]. The basic common structure behind these theories, which we call “quantum symmetry” is some kind of a tensor category. For instance, a paragroup has the structure quite similar to a Hopf algebra [S1] and has “quantum doubles”, which has been first suggested by Ocneanu [E-K2]. This suggests a relation to quantum groups. Also, we can construct a 3-dimensional TQFT based on triangulations [O3, T-V, E-K1] and RCFT from a paragroup [E-K3]. These are also suggested by Ocneanu. The purpose of this paper is to study a paragroup from the viewpoint of a tensor category.

To construct subfactors, the method of commuting squares has been studied since their initiation [W1, G-H-J]. From a finite dimensional nondegenerate commuting square, we can construct two subfactors by iterating the basic construction in the horizontal and the vertical directions. In a tensor categorical viewpoint, two paragroups arising from these two subfactors have the same size. More precisely, they have same global indices [S2, Corollary 2.5]. (See [S2, Definition 2.3] for the definition of the global index for a subfactor. Also see [O3]. ) Or we can say that one is of finite depth if and only if the other is so [S2, Corollary 2.2].

As stated above, if we have an inclusion of AFD II_{1} factors, we can construct a 3-dimensional TQFT. Consider its relation to a construction of subfactors from a finite dimensional non-degenerate commuting square. The result is that two TQFT’s arising from two subfactors are complex conjugate to each other [S3, Theorem 2.4]. This leads us to the notion of the opposite equivalence of subfactors. We explain this notion in short. Ocneanu has introduced the notion of the equivalence between two systems of bimodules [O3, O4, O5]. Assume that we have finite systems of the A-A bimodules and the B-B bimodules. If there are A-B bimodules and the four types of bimodules A-A, A-B, B-A, B-B make a finite system, we say those two systems are equivalent. Also see Definition 2.2. That is, we can realize a paragroup which has all of the above systems of bimodules [O3]. In relation to 3-dimensional TQFT, the system of the A-A bimodules and that of the B-B bimodules give the same 3-dimensional TQFT [O3, E-K1]. We say that a subfactor A ⊂ B is equivalent to a subfactor C ⊂ D if the finite system of the B-B bimodules arising from A ⊂ B is equivalent to that of the C-C bimodules arising from C ⊂ D. Also, we say a subfactor A ⊂ B is opposite equivalent to C ⊂ D if A ⊂ B is equivalent to C^{opp} ⊂ D^{opp}. Returning to the finite dimensional non-degenerate commuting square, we know that two subfactors which we consider are opposite equivalent. (See the arguments after Definition 2.4 for more details.) Hence the opposite equivalence of subfactors is quite naturally associated with a finite dimensional non-degenerate commuting square. It is natural to ask what we can say if we have two subfactors which have the 3-dimensional TQFT’s complex conjugate to each other. This is our motivation for the present paper. Our starting point is two equivalent AFD II_{1} subfactors A ⊂ B and C ⊂ D with finite Jones index and finite depth. Then our main theorem states that we can construct a finite dimensional non-degenerate commuting square which produces the subfactors isomorphic to A ⊂ B and C^{opp} ⊂ D^{opp}, respectively. This is what we have expected. The techniques used to prove the theorem are quite tensor categorical, such as flatness and global index.

The author is much indebted to Prof. Kawahigashi for stimulating discussions and comments on his work, and expresses his gratitude.

2 Equivalence of subfactors

Let us start with the definition of the finite system of the bimodules.

Definition 2.1 Let A be an AFD II_{1} factor. A finite system of A-A bimodules _{A}M_{A} is defined to be a set of irreducible A-A bimodules which is closed under the relative tensor product, irreducible decompositions and contragradients.

Recall that a finite system of bimodules gives a finite set of quantum 6jsymbols. We note that the system of A-A bimodules _{A}M_{A} and that of A^{opp}-A^{opp} bimodules _{A^{opp}}M_{A^{opp}} have the fusion rule algebras isomorphic to each other and finite sets of quantum 6j-symbols arising from these systems are complex conjugate to each other (up to gauge choices). In this case, we say that _{A^{opp}}M_{A^{opp}} is complex conjugate to _{A}M_{A}. (See [O3, E-K1] for the definition of the quantum 6j-symbols and their properties.)

We introduce the notion of the equivalence of finite systems of bimodules, which plays a crucial role in the present paper.

Definition 2.2 (A. Ocneanu, [O3]) Let A and B be AFD II_{1} factors. We say that a finite system of A-A bimodules _{A}M_{A} is equivalent to that of B-B bimodules _{B}M_{B}, if there exist A-B bimodules and the four types of bimodules A-A, B-B, A-B, B-A make a finite system. We denote this relation by AM_{A} ∼ _{B}M_{B}.

The finite system of four types of the bimodules in Definition 2.2 gives a data of a paragroup [O1, O3, E-K1]. We call the paragroup constructed in this way an “A-B paragroup”. (Or we may think this paragroup as that arising from the inclusion A ⊂ B.)

Note that the equivalence of two finite systems of bimodules indeed defines an equivalence relation. In fact, the reflexive law is trivial and the symmetric law follows that the conjugation operation gives the finitely many mutually inequivalent irreducible B-A bimodules. For the transitive law, assume that _{A}M_{A} ∼ _{B}M_{B} and _{B}M_{B} ∼ _{C}M_{C}. Hence we have the A-B and B-C paragroups with generators _{A}g_{B} and _{B}h_{C}, respectively. Then it is easy to see that _{A}g ⊗_{B} h_{C} is the generator of the A-C paragroup, which gives the finitely many mutually inequivalent irreducible A-C bimodules.

Let A ⊂ B be an inclusion of AFD II_{1} factors with finite Jones index and finite depth. Then we know that the system of the A-A bimodules _{A}M_{A} naturally arising from A ⊂ B is equivalent to that of the B-B bimodules BM_{B}. Using the above notation, we introduce the notion of the equivalence of two subfactors.

Definition 2.3 Let A ⊂ B and C ⊂ D be inclusions of AFD II_{1} factors with finite Jones index and finite depth. We say that these two subfactors are equivalent if _{B}M_{B} ∼ _{C}M_{C}.

Hence _{A}M_{A} ∼ _{B}M_{B} ∼ _{C}M_{C} ∼ _{D}M_{D} and we have the equalities of global indices of the systems of bimodules [[_{A}M_{A}]] = [[_{A}M_{B}]] = [[_{B}M_{A}]] = [[_{B}M_{B}]] = [[_{B}M_{C}]] = [[_{C}M_{B}]] = [[_{C}M_{C}]] = [[_{C}M_{D}]] = [[_{D}M_{C}]] = [[_{D}M_{D}]]. (The global index [[_{A}M_{A}]] of the finite system of the A-A bimodules _{A}M_{A} is defined by ∑_{X∈_{A}M_{A}}(dim_{A}X_{A})^{2}, where the summation runs through all the mutually inequivalent irreducible A-A bimodules in_{A}M_{A}. See [O3, E-K3] for more details.) Here we have used the notaion _{A}M_{B} for the A-B bimodules which makes _{A}M_{A} be equivalent to _{B}M_{B} by abuse of notation.

We list some instructive examples of equivalence of subfactors. Example 1 Let G be a finite group and H be its subgroup which is relatively simple. Two subfactors R ⊂ R ⋊ G and R ⋊ H ⊂ R ⋊ G obtained by the crossed product of an outer action of G on the AFD II_{1} factor R are equivalent because the systems of the R⋊G-R⋊G bimodules arising from both subfactors are equivalent by the Mackey machine of Ocneanu [O2, K-Y].

Example 2 More generally, let P ⊂ Q ⊂ R be inclusions of AFD II_{1} factors. Assume that the inclusion P ⊂ R has finite Jones index and finite depth. Hence the intermediate subfactors of P ⊂ R are also of finite depth by a theorem of D. Bisch [B]. Then we know that the system of the R-R bimodules arising from Q ⊂ R is a subsystem of that arising from P ⊂ R. Moreover, assume that both of the system of the R -R bimodules have the same global indices. Then these two subfactors are equivalent since so are the systems of bimodules. Hence, P ⊂ R and Q ⊂ R are equivalent in this case. Example 3 Let N ⊂ M be an inclusion of AFD II_{1} factors with finite Jones index and finite depth. Consider the asymptotic inclusion M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞} and the inclusion N ⊗ N^{opp} ⊂ M ⊗ M^{opp}. Assume that the fusion graph arising from the original inclusion N ⊂ M is connected. Then these two subfactors are equivalent because the system of the M ∨ (M′ ∩ M_{∞})- M ∨ (M′ ∩ M_{∞}) bimodules arising from the asymptotic inclusion is identified with that of the M ⊗ M^{opp}-M ⊗ M^{opp} bimodules arising from N ⊗ N^{opp} ⊂ M ⊗ M^{opp} [E-K2].

Next, we define the opposite equivalence of subfactors, which is naturally associated with a finite dimensional non-degenerate commuting square.

Definition 2.4 Let A ⊂ B and C ⊂ D be inclusions of AFD II_{1} factors with finite Jones index and finite depth. We say that these two subfactors are opposite equivalent if A ⊂ B is equivalent to C^{opp} ⊂ D^{opp}.

We will explain how the notion of opposite equivalence associates with a finite dimensional non-degenerate commuting square in more detail. Consider the following finite dimensional non-degenerate commuting square.

\begin{aligned}R_{00} & ⊂ & R_{01} \\ ∩ & & ∩ \\ R_{10} & ⊂ & R_{11} \end{aligned}

Then iterating the basic construction in the horizontal and vertical directions, we get two subfactors M_{0} ⊂ M_{1} and Q_{0} ⊂ Q_{1}, respectively. Passing to the paragroup theoretical viewpoint, we have the biunitary connection on the four finite bipartite connected graphs corresponding to the inclusion matrices in the initial commuting square. And by the string algebra construction, we have the following double sequence of finite dimensional C^{∗}-algebras.

A_{0,0}  ⊂  A_{0,1}  ⊂  A_{0,2}  ⊂ ⋯ ⊂ M_{0} = A_{0,∞}
∩  ∩  ∩  ∩
A_{1,0}  ⊂  A_{1,1}  ⊂  A_{1,2}  ⊂ ⋯ ⊂ M_{1} = A_{1,∞}
∩  ∩  ∩  ∩
A_{2,0}  ⊂  A_{2,1}  ⊂  A_{2,2}  ⊂ ⋯ ⊂ M_{2} = A_{2,∞}
∩  ∩  ∩  ∩
⋮  ⋮  ⋮  ⋮
∩  ∩  ∩
Q_{0} = A_{∞,0} ⊂ Q_{1} = A_{∞,1} ⊂ Q_{2} = A_{∞,2} ⊂ ⋯

Assume that either M_{0} ⊂ M_{1} or Q_{0} ⊂ Q_{1} is of finite depth. (Hence, by [S2, Corollary 2.2], the other is also of finite depth.) By Ocneanu’s compactness argument [O2, II.6], the flat part M_{0}′ ∩ M_{k} of the inclusion M_{0} ⊂ M_{1} is contained in A_{k,0} for each k. Denote ∨_{k=0}^{∞}M_{0}′ ∩ M_{k} by B_{k}. See the following diagram.

B_{0} ⊂ A_{0,0} ⊂ A_{0,1} ⊂ A_{0,2} ⊂ ⋯ ⊂ M_{0}
∩ ∩ ∩ ∩ ∩
B_{1} ⊂ A_{1,0} ⊂ A_{1,1} ⊂ A_{1,2} ⊂ ⋯ ⊂ M_{1}
∩ ∩ ∩ ∩ ∩
B_{3} ⊂ A_{2,0} ⊂ A_{2,1} ⊂ A_{2,2} ⊂ ⋯ ⊂ M_{2}
∩  ∩  ∩  ∩  ∩
⋮  ⋮  ⋮  ⋮  ⋮
∩  ∩  ∩  ∩
B_{∞} ⊂ Q_{0} ⊂ Q_{1} ⊂ Q_{2} ⊂ ⋯

In the proof of [S3, Theorem 2.4], we have shown that the system of the B_{∞}- B_{∞} bimodules arising from B_{∞} ⊂ Q_{1} is complex conjugate to that of the M_{0}-M_{0} bimodules arising from M_{0} ⊂ M_{1}. This implies that the subfactor B_{∞} ⊂ Q_{1} is opposite equivalent to M_{0} ⊂ M_{1}. Hence Q_{0} ⊂ Q_{1} is opposite equivalent to M_{0} ⊂ M_{1}.

3 Constructing a commuting square from the

equivalent subfactors

We assume that the inclusions A ⊂ B and C ⊂ D satisfy the assumptions in Definition 2.3 in the sequel. Hence, we have the A-B paragroup naturally arising from the inclusion A ⊂ B and the A-D paragroup, which have the common system of the A-A bimodules. Then we can consider the following series of inclusions of finite dimensional von Neumann algebras.

End(g(ḡg)^{n−1} ⊗_{A} ∗_{A} ⊗ (hh̄)^{n−1}h) ⊂ End(g(ḡg)^{n−1} ⊗_{A} ∗_{A} ⊗ (hh̄)^{n})
∩  ∩
End((ḡg)^{n} ⊗_{A} ∗_{A} ⊗ (hh̄)^{n−1}h) ⊂ End((ḡg)^{n} ⊗_{A} ∗_{A} ⊗ (hh̄)^{n})

Here _{A}∗_{A} means that the A-A bimodule _{A}A_{A} and the D-A bimodule g and the A-B bimodule h stand for the generators of the A-D paragroup and of the A-B paragroup _{A}B_{B}, respectively.

Indeed, we have the quantum 6j-symbols from these inclusions. Hence they satisfy the tetrahedral symmetry. This implies the commuting square condition. (See [O3, E-K1] for quantum 6j-symbols and these arguments.)

Because the systems of bimodules which appear in the above commuting square are finite, we can easily know that it is a non-degenerate commuting square if n is large enough. From this commuting square, by iterating the basic construction, we can construct the Jones towers M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ ⋯ in the vertical direction and Q_{0} ⊂ Q_{1} ⊂ Q_{2} ⊂ ⋯ in the horizontal direction.

Lemma 3.1 The biunitary connection corresponding to the above non-degenerate commuting square is _{A}∗_{A}-flat and the higher relative commutant M′_{1} ∩ M_{k} of M_{1} ⊂ M_{2} coincide with End(\underbrace{⋯ ⊗ g ⊗ ḡ}_{k−folds} ⊗_{D}∗_{D}) for each k. In particular, the system of the M_{1}-M_{1} bimodules is naturally identified with that of the D-D bimodules.

Proof. Following a similar argument in [E-K1, page 556], it is easy to see that the biunitary connection we consider is _{A}∗_{A}-flat. We denote the string algebras constructed from this biunitary connection by A_{k,l} and denote End(\underbrace{⋯ ⊗ g ⊗ ḡ}_{k−folds} ⊗_{D}∗_{D}) by B_{k,−1}. We remark that A_{k,0} (resp. A_{k,−1}) in the following diagram gives the flat part M_{0}′ ∩ M_{k} of M_{0} ⊂ M_{1} (resp. the flat part M_{1}′ ∩ M_{k} of M_{1} ⊂ M_{2}) by Ocneanu’s compactness argument [O2, II.6].

A_{0,0}(= End(_{A}∗_{A})) ⊂ A_{0,1} ⊂ A_{0,2} ⋯ ⊂ M_{0}
∩  ∩  ∩  ∩
A_{1,−1}(= End(_{M_{1}}∗_{M_{1}})) ⊂  A_{1,0}  ⊂ A_{1,1} ⊂ A_{1,2} ⋯ ⊂ M_{1}
∩  ∩  ∩  ∩  ∩
A_{2,−1}  ⊂  A_{2,0}  ⊂ A_{2,1} ⊂ A_{2,2} ⋯ ⊂ M_{2}
⋮  ⋮  ⋮  ⋮  ⋮

Next, using the argument in [E-K1, page 556], it is easy to see that the partition function with ∗ =_{D} ∗_{D} in Fig. 3.1 has the value 1.

[figure: ∗; ⋮; ·; ∗; ·; ·; ⋯; ·; ∗; ·; ⋮; ∗; ·; ⋯; ·; ·; ·⊗g; ·⊗h; ·⊗h; ·⊗h; ·⊗ḡ; ḡ⊗·; g⊗·; Fig. 3.1]

This implies that the elements in B_{k,−1} commute with those in A_{1,l} for each k and l. Thus we know that B_{k,−1} is included in the flat part A_{k,−1} of M_{1} ⊂ M_{2} for each k. See the following diagram.

A_{0,0} ⊂ A_{0,1} ⊂ A_{0,2} ⊂ ⋯ ⊂ M_{0}
∩ ∩ ∩ ∩
B_{1,−1}(= End(_{D}∗_{D})) ⊂ A_{1,−1} ⊂ A_{1,0} ⊂ A_{1,1} ⊂ A_{1,2} ⊂ ⋯ ⊂ M_{1}
∩ ∩ ∩ ∩ ∩ ∩
B_{2,−1}  ⊂ A_{2,−1} ⊂ A_{2,0} ⊂ A_{2,1} ⊂ A_{2,2} ⊂ ⋯ ⊂ M_{2}
⋮ ⋮ ⋮ ⋮ ⋮ ⋮

We claim that A_{k,−1} = B_{k,−1} for each k. Note that the following commuting square for n large enough is the canonical commuting square that arises from M_{1} ⊂ M_{2}.

A_{n,−1} ⊂ A_{n,0}
∩  ∩
A_{n+1,−1} ⊂ A_{n+1,0}

Consider the inclusion matrix of B_{2n,−1} ⊂ A_{2n,−1}. In this inclusion, the Perron-Frobenius eigenvector μ for the odd vertices is the multiple of that for the even vertices by the Perron-Frobenius eigenvalue β. Hence [[_{M_{1}}M_{M_{1}}]] = ∑_{X}(βμ(X))^{2} = β^{2} ∑_{X} μ(X)^{2} = β^{2}[[_{D}M_{A}]], where the summation runs through all of mutually inequivalent irreducible D-A bimodules X. Because we have the equality [[_{M_{1}}M_{M_{1}}]] = [[_{A}M_{A}]] = [[_{D}M_{A}]], we have proved β = 1. This implies A_{k,−1} = B_{k,−1} for each k. The last statement of Lemma follows from the Popa’s classification theorem for strongly amenable subfactors [P]. So we are done. □

Remark 3.2 The proof of Lemma 3.1 is based on [E-K1, Sec. 5]. Readers should be aware that the original argument in [E-K1, page 556] is incomplete and the above argument in the accordance of the two systems using the global index is needed.

To state our main theorem, we consider the following commuting square, where the D-C bimodule g stands for the generator _{D}D_{C} of the C-D paragroup which canonically arises from C ⊂ D and the A-B bimodule h stands for the generator _{A}B_{B} of the A-B paragroup which canonically arises from A ⊂ B and the C-A bimodule X stands for the direct sum of the mutually inequivalent irreducible C-A bimodules.

End(g(ḡg)^{n−1} ⊗_{C} X_{A} ⊗ (hh̄)^{n−1}h) ⊂ End(g(ḡg)^{n−1} ⊗_{C} X_{A} ⊗ (hh̄)^{n})
∩  ∩
End((ḡg)^{n} ⊗_{C} X_{A} ⊗ (hh̄)^{n−1}h) ⊂ End((ḡg)^{n} ⊗_{C} X_{A} ⊗ (hh̄)^{n})

It is easy to see that all of the inclusion matrices are irreducible because of finiteness of systems of bimodules appearing in the commuting square.

Thus this commuting square is non-degenerate. Thus we have the biunitary connection arising from the above commuting square and denote it by W. By the string algebra construction, we get the following double sequences of finite dimensional C^{∗}-algebras A_{k,l} = End(\underbrace{⋯ ⊗ ḡ ⊗ g}_{k−folds} ⊗ _{C} ∗_{A} ⊗ \underbrace{h ⊗ h̄ ⋯}_{l−folds}), where _{C}∗_{A} is an irreducible C-A bimodule _{C}X_{A}. See the following diagram.

A_{0,0}(= End(_{C}∗_{A})) ⊂  A_{0,1}  ⊂  A_{0,2}  ⊂ ⋯ ⊂ M_{0} = A_{0,∞}
∩  ∩  ∩  ∩
A_{1,0}  ⊂  A_{1,1}  ⊂  A_{1,2}  ⊂ ⋯ ⊂ M_{1} = A_{1,∞}
∩  ∩  ∩  ∩
A_{2,0}  ⊂  A_{2,1}  ⊂  A_{2,2}  ⊂ ⋯ ⊂ M_{2} = A_{2,∞}
⋮  ⋮  ⋮  ⋮
∩  ∩  ∩
Q_{0} = A_{∞,0}  ⊂ Q_{1} = A_{∞,1} ⊂ Q_{2} = A_{∞,2} ⊂ ⋯

Then we have our main theorem. Theorem 3.3 In the above notations, the subfactors M_{0} ⊂ M_{1} and Q_{0} ⊂ Q_{1} are isomorphic to C^{opp} ⊂ D^{opp} and A ⊂ B, respectively. Proof. Assume the subfactor A ⊂ B is equivalent to C ⊂ D. We denote End(\underbrace{⋯ ⊗ ḡ ⊗ g}_{n−folds} ⊗ _{C}∗_{C}) (resp. End(\underbrace{⋯ ⊗ g ⊗ ḡ}_{n−folds} ⊗ _{D}∗_{D})) by C_{n,−1} (resp.

C_{n,−2}) and End(_{A}∗_{A} ⊗\underbrace{h ⊗ h̄ ⋯}_{n−folds}) (resp. End(_{B}∗_{B} ⊗\underbrace{h̄ ⊗ h ⋯}_{n−folds})) by B_{−1,n} (resp. B_{−2,n}). Then we get the inclusions C_{n,−1} ⊂ A_{n,0} and C_{n,−2} ⊂ C_{n,−1} (resp. B_{−1,n} ⊂ A_{0,n} and B_{−2,n} ⊂ B_{−1,n}) for each n. (See the following diagram.)

B_{−2,1} ⊂ B_{−2,2} ⊂ ⋯ ⊂ N_{−2}
∩ ∩ ∩
B_{−1,0} ⊂ B_{−1,1} ⊂ B_{−1,2} ⊂ ⋯ ⊂ N_{−1}
∩ ∩ ∩ ∩
C_{0,−1} ⊂ A_{0,0} ⊂ A_{0,1} ⊂ A_{0,2} ⊂ ⋯ ⊂ M_{0}
∩ ∩ ∩ ∩ ∩
C_{1,−2} ⊂ C_{1,−1} ⊂ A_{1,0} ⊂ A_{1,1} ⊂ A_{1,2} ⊂ ⋯ ⊂ M_{1}
∩ ∩ ∩ ∩ ∩ ∩
C_{2,−2} ⊂ C_{2,−1} ⊂ A_{2,0} ⊂ A_{2,1} ⊂ A_{2,2} ⊂ ⋯ ⊂ M_{2}
⋮  ⋮  ⋮  ⋮  ⋮  ⋮
∩  ∩  ∩  ∩  ∩
P_{−2} ⊂ P_{−1} ⊂ Q_{0} ⊂ Q_{1} ⊂ Q_{2} ⊂ ⋯

Fig. 3.2

First, we show that the inclusion M_{0} ⊂ M_{1} is of finite depth. For this, we pay attention to the commuting squares (3.1) which produce the inclusion N_{−1} ⊂ M_{1}.

\begin{aligned}B_{−1,0}(= End(_{A}∗_{A})) & ⊂ & B_{−1,1} & ⊂ & B_{−1,2} & ⊂ & ⋯ & ⊂ & N_{−1} \\ ∩ & & ∩ & & ∩ & & & & ∩ \\ A_{1,0} & ⊂ & A_{1,1} & ⊂ & A_{1,2} & ⊂ & ⋯ & ⊂ & M_{1} \end{aligned}  (3.1)

Corresponding to this commuting square (3.2), we have the biunitary connection W_{1} · W which produces the double sequences of string algebras (3.1).

\begin{aligned}B_{−1,2n} & ⊂ & B_{−1,2n+1} \\ ∩ & & ∩ \\ A_{1,2n} & ⊂ & A_{1,2n+1} \end{aligned}  (3.2)

And it is easy to see that the biunitary connection W_{1} ·W is _{A}∗_{A}-flat. (Recall the definitions of B_{−1,n} and A_{1,n}.) Hence the inclusion N_{−1} ⊂ M_{1} has finite depth. Because the AFD II_{1} factor M_{0} is an intermediate subfactor of the inclusion N_{−1} ⊂ M_{1}, by a theorem of D. Bisch [B, Theorem 2.6], the inclusion M_{0} ⊂ M_{1} is of finite depth.

We prove that the algebras on the first vertical line in Fig 3.2 coincide with the higher relative commutants M′_{1} ∩ M_{k} of M_{1} ⊂ M_{2}. Using the argument in [E-K1, page 556], it is easy to see that the partition function in Fig. 3.1 with ∗ = _{D}D_{D} has the value 1. This implies that the elements in C_{n,−2} commute with those in A_{1,k} for each n and k [E-K1, page 556]. Thus we can get the inclusion C_{n,−2} ⊂ M′_{1} ∩ M_{n} for each n ≥ 1 with a help of Ocneanu’s compactness argument [O2, II.6].

Consider the inclusion C_{n,−2} ⊂ M′_{1} ∩ M_{n} for n large enough. Applying Lemma 3.1 to the biunitary connection W_{1} · W, we know that the system of the M_{1}-M_{1} bimodules arising from N_{−1} ⊂ M_{1} is identified with that of the D-D bimodules arising from C ⊂ D. Then we know that in the above inclusion the Perron-Frobenius eigenvector μ for the odd vertices is the multiple of that for the even vertices by the Perron-Frobenius eigenvalue β. By a similar argument to the one we have used to prove the accordance of A_{k,−1} = B_{k,−1} in the proof of Lemma 3.1, it is enough to prove that the global index [[M_{1} : N_{−1}]] of N_{−1} ⊂ M_{1} is equal to [[D : C]] of C ⊂ D for our purpose.

Because the biunitary connection W_{1} · W is _{A}∗_{A}-flat, we know that the equality [[M_{1} : N_{−1}]] = [[B : A]]. The global index [[B : A]] is equal to [[D : C]] because of the equivalence of the systems of the bimodules. This proves [[M_{1} : M_{−1}]] = [[D : C]]. Thus we proved C_{n,−2} = M′_{1} ∩ M_{n}, i.e., the dual principal case.

We prove that the algebras on the second vertical line in Fig. 3.2 are exactly the higher relative commutants M′_{0} ∩ M_{k} of M_{0} ⊂ M_{1}. For this, we note that the biunitary connection W_{1} which corresponds to the commuting squares (3.3) is _{A}∗_{A}-flat.

\begin{aligned}B_{−1,2n} & ⊂ & B_{−1,2n+1} \\ ∩ & & ∩ \\ A_{0,2n} & ⊂ & A_{0,2n+1} \end{aligned}  (3.3)

Applying Lemma 3.1 to this biunitary connection, we know that the system of the M_{0}-M_{0} bimodules arising from N_{−1} ⊂ M_{0} is identified with that of the C-C bimodules arising from C ⊂ D. By a similar argument in the dual principal case, it is enough to show the equality [[M_{0} : N_{−1}]] = [[D : C]]. Because the biunitary connection W_{1} is ∗-flat, we have the equality [[M_{0} : N_{−1}]] = [[B : A]]. This proves [[M_{0} : N_{−1}]] = [[D : C]]. Thus we proved C_{n,−1} = M′_{0} ∩ M_{n}, i.e., the principal case.

Similar arguments can be applied to the proofs of B_{−1,n} = Q′_{0} ∩ Q_{n} and B_{−2,n} = Q′_{1} ∩ Q_{n} for each n. This completes the proof of Theorem. □

Remark 3.4 By Theorem 3.3 with the argument after Definition 2.4, subfactors A ⊂ B and C ⊂ D are opposite equivalent if and only if they arise from a finite dimensional non-degenerate commuting square.

4 Example

We consider the most basic example. Let R be the AFD II_{1} factor and G = S_{3} be the symmetric group of order 3 which acts on R outerly. Consider the inclusion R ⊂ R ⋊ G = M which is obtained from the crossed product. Let H = S_{2} be the symmetric group of order 2. Then R⋊H = N is a subfactor of M. By example 1 in section 2, the system of the M-M bimodules arising from the inclusion R ⊂ M is identified with that of the M-M bimodules arising from N ⊂ M. We denote the mutually inequivalent irreducible bimodules in the system of the M-M bimodules by X_{1}, X_{2} and X_{3}, where X_{1}, X_{2}, X_{3} correspond to the trivial representation 1, the two dimensional representation σ and the sign representation ε of the irreducible unitary representations of S_{3}, respectively. Then looking at the inclusion R ⊂ M, we know that MX_{k} ⊗_{M} M_{R} decomposes into the irreducible M-R bimodule _{M}Z_{R} for each k = 1, 2, 3. Also, looking at the inclusion N ⊂ M, by the Mackey machine of Ocneanu [O1, K-Y], we know that _{N}M ⊗_{M} X_{k}_{M} decomposes into the irreducible N-M bimodules in the following way.

NM ⊗_{M} X_{1}_{M} = _{N}Y_{1}_{M},_{N}M ⊗_{M} X_{2}_{M} = _{N}Y_{1}_{M} ⊕_{N}Y_{2}_{M},_{N}M ⊗_{M} X_{3}_{M} = _{N}Y_{2}_{M},

where Y_{1} and Y_{2} correspond to the trivial representation 1 and the sign representation ε of the irreducible unitary representations of S_{2}. (Recall that this decomposition is indeed given by the induction-restriction graph between S_{3} and S_{2} with the Mackey machine in the representation theory of finite groups.) Hence, using the notations in the proof of Theorem 3.3, the inclusions A_{0,2n} ⊂ A_{0,2n+1} and A_{0,2n} ⊂ A_{1,2n} are described by the following Bratteli diagrams. See Fig. 4.1.

X_{1}X_{2}X_{3} Y_{1} Y_{2}  Z Fig. 4.1

Note that the Perron-Frobenius eigenvalues of the vertical graphs in the non-degenetate commuting square which we have constructed in Theorem 3.3 coincide. Also note that the Perron-Frobenius eigenvalue of the left vertical graph of the Dynkin diagram A_{5} in Fig. 4.1 is equal to three. From these facts, we know that a finite connected bipartite graph with the Perron-Frobenius eigenvalue three and the unique odd vertex is the Dynkin diagram D_{4}. Hence we know that _{N}M ⊗_{M} Z_{R} decompose into the irreducible bimodules W_{1} ⊕ W_{2} ⊕ W_{3}. See Fig. 4.2.

X_{1}X_{2}X_{3} Y_{1} Y_{2}  Z W_{1}W_{2}W_{3} Fig. 4.2

From these data, we know that both _{N}Y_{1} ⊗_{M} M_{R} and _{N}Y_{2} ⊗_{M} M_{R} decompose into the irreducible N-R bimodules W_{1} ⊕ W_{2} ⊕ W_{3}. Hence, we get the following Bratteli diagrams corresponding to the non-degenerate commuting square in the proof of Theorem 3.3. See Fig 4.3.

X_{1}X_{2}X_{3} Y_{1} Y_{2}  Z W_{1}W_{2}W_{3} Fig. 4.3

Then by Theorem 3.3, the above graphs and the naturally associated quantum 6j-symbols give the biunitary connection W which produces the inclusions isomorphic to R ⊂ M and N^{opp} ⊂ M^{opp}. (In this case, we have the Dynkin diagram A_{5} in the above four graphs and this implies that W is flat with respect to X_{1}. Therefore, we know that the principal graphs of two subfactors arising from the non-degenerate commuting square by iterating the basic construction in the vertical and horizontal directions are given by the graphs in Fig. 4.1.)

References

[B] Bisch, D., A note on intermediate subfactors, Pac. J. Math. 163 (1994), 201–216.

[B-G] de Boer, J. and Goeree, J., Markov traces and II_{1} factors in conformal field theory, Comm. Math. Phys. 139 (1991), 267–304.

[E-K1] Evans, D. E. and Kawahigashi, Y., From subfactors to 3-dimensional topological quantum field theories and back —a detailed account of Ocneanu’s theory—, Internat. J. Math. 6 (1995), 537–558.

[E-K2] Evans, D. E. and Kawahigashi, Y., On Ocneanu’s theory of asymptotic inclusions for subfactors, topological quantum field theories and quantum doubles, Internat. J. Math. 6 (1995), 205–228.

[E-K3] Evans, D. E. and Kawahigashi, Y., Quantum symmetries on operator algebras, a book to appear.

[G-H-J] Goodman, F., de la Harpe, P., and Jones, V. F. R., “Coxeter graphs and towers of algebras”, MSRI publications 14, Springer, 1989.

[J] Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 1–25.

[K] Kawahigashi, Y., On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), 63–107.

[K-Y] Kosaki, H. and Yamagami, S., Irreducible bimodules associated with crossed product algebras. Internat. J. Math., 3 (1992), 661–676.

[M-S] Moore, G. and Seiberg, N., Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177–254.

[O1] Ocneanu, A., Quantized group, string algebras and Galois theory for algebras, in “Operator algebras and applications, Vol. 2 (Warwick, 1987),” London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, (1988), 119–172.

[O2] Ocneanu, A., “Quantum symmetry, differential geometry of finite graphs and classification of subfactors”, University of Tokyo Seminary Notes 45, (Notes recorded by Y. Kawahigashi), 1991.

[O3] Ocneanu, A., An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, preprint, (1991).

[O4] Ocneanu, A., Modular invariants, subfactors and quantum field theory, talk at Rome, July, 1996.

[O5] Ocneanu, A., The classification of connections and intermediate subfactors, talk at India, January, 1997.

[P] Popa, S., Classification of amenable subfactors of type II, Acta Math. 172 (1994), 352–445.

[R] Roche, Ph., Ocneanu cell calculus and integrable lattice models, Comm. Math. Phys. 127 (1990), 395–424.

[S1] Sato, N., Fourier transform for paragroups and its application to the depth two case, to appear in Publ. RIMS, Kyoto Univ.

[S2] Sato, N., Two subfactors arising from a non-degenerate commuting square —An answer to a question raised by V. F. R. Jones—, to appear in Pac. J. Math.

[S3] Sato, N., Two subfactors arising from a non-degenerate commuting square II —Tensor categories and TQFT’s—, to appear in Internat. J. Math.

[T-V] Turaev, V. G. and Viro, O. Y., State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992), 865–902.

[W1] Wenzl, H., Hecke algebras of type A_{n} and subfactors, Invent. Math. 92 (1988), 345–383.

[W2] Wenzl, H., Quantum groups and subfactors of type B, C, and D, Comm. Math. Phys. 133 (1990), 383–432.

[W3] Wenzl, H., C^{∗}-tensor categories from quantum groups, preprint, (1997).