Two subfactors arising from a non-degenerate commuting square II — Tensor categories and TQFT’s —

Nobuya Sato ^{∗} Department of Mathematical Sciences

University of Tokyo, Komaba, Tokyo, 153, JAPAN

Abstract

We prove that two subfactors N ⊂ M and P ⊂ Q arising from a non-degenerate commuting square have 3-dimensional topological quantum field theories (based on triangulations) complex conjugate to each other. Applying the asymptotic inclusion construction to each subfactor, we have new subfactors M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞} and Q ∨ (Q′ ∩ Q_{∞}) ⊂ Q_{∞}, then we also prove that the tensor categories of the M_{∞}-M_{∞} bimodules and the Q_{∞}-Q_{∞} bimodules are isomorphic to each other, if the two fusion graphs are connected. These results are based on our previous work and give a finer answer to a question raised by V. F. R. Jones in July, 1995.

1 Introduction

The theory of operator algebras has been related to many fields of physics since its initiation. In particular, theory of von Neumann algebras has recently interacted other fields of mathematics and theoretical physics. It is subfactor theory originated by V. F. R. Jones [J] in 1983 that makes these deep relations. He has found a new topological invariant of knots using subfactor theory. We have many relations to other fields of mathematics and theoretical physics such as topological quantum field theory (TQFT), quantum groups, exactly solvable lattice models, rational conformal field theory (RCFT) (in the sense of Moore-Seiberg [M-S]), etc. The important common structure behind these theories is a tensor category. For subfactor theory, it is A. Ocneanu that has given one of the explicit forms of a tensor category called a paragroup [O1] in 1987. Paragroup theory is essentially the theory of bimodules and their relative tensor products, and has similar structures to a finite group and its representation theory. A paragroup is a complete invariant for an AFD II_{1} subfactor with finite index under some good conditions ([P1], [P2]), and describes the symmetry of the subfactor. For this reason, a paragroup is sometimes called a “quantized Galois group” for a subfactor. He also axiomatized a paragroup by combinatorial data with which we are able to compare paragroup theory with other theories as stated above. Many papers on details of the theory are now available by contributions of others. On one hand, axioms of a paragroup [O1] are similar to those of the exactly solvable lattice models and its main body is given by four graphs and the function called biunitary connection which satisfies similar relations to Boltzmann weight. On the other hand, a paragroup arising from a subfactor gives a 3-dimensional topological quantum field theory of Turaev-Viro type based on triangurations as shown by Ocneanu. (See [T-V] and [E-K2].) He has also studied the “quantum double for a subfactor”, known as an asymptotic inclusion [E-K3]. de Boer and Goeree proved that the data of an RCFT gives those of a paragroup [B-G]. In general, we cannot build an RCFT from the data of a paragroup directly, but if we pass from an original subfactor to the asymptotic inclusion, we can construct data of an RCFT. (See [E-K4] section 13.5 for more details.) Thus subfactor theory is related to a wide range of recent mathematics and theoretical physics through paragroups and attracts us.

1991 Mathematics Subject Classification. Primary 46L37. Secondary 81R50, 81T40.

∗E-mail address: nobuya@ms.u-tokyo.ac.jp

V. F. R. Jones has recently raised the following question. Suppose that we have a finite dimensional non-degenerate commuting square with irreducible inclusion matrices as follows.

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

Iterating the basic constructions for the above commuting square, we get the following series of commuting squares of period 2.

R_{00} ⊂ R_{01} ⊂ R_{02} ⊂ ⋯ ⊂ R_{0∞}
∩ ∩ ∩ ∩
R_{10} ⊂ R_{11} ⊂ R_{12} ⊂ ⋯ ⊂ R_{1∞}
∩ ∩ ∩ ∩
R_{20} ⊂ R_{21} ⊂ R_{22} ⊂ ⋯ ⊂ R_{2∞}
⋮  ⋮  ⋮  ⋮
R_{∞0} ⊂ R_{∞1} ⊂ R_{∞2} ⊂ ⋯

Is there any relation between two subfactors R_{0∞} ⊂ R_{1∞} and R_{∞0} ⊂ R_{∞1}? In particular, is the finite depth condition for R_{0∞} ⊂ R_{1∞} equivalent to that for R_{∞0} ⊂ R_{∞1}? An answer to this question has been presented in our previous paper [S2] and an answer to the first question is that the above two subfactors have the same global indices, where a global index is the Jones index of the asymptotic inclusion subfactor of the original subfactor [O1]. The second answer is affirmative. These answers have been proved by using techniques of paragroup theory. Precisely, for proofs, we have used the notion of flatness, which is one of axioms of a paragroup. Flatness is a generalization of (co)associativity in a (finite dimensional) Hopf algebra. In fact, if the graphs appeared in the biunitary connection are of depth two, the biunitary connection satisfies flatness if and only if there exists a (non-trivial) unitary which satisfies the pentagonal relation. See [S1] for more details. Our goal in this paper is to investigate the relation in a more definite form.

We will establish a finer relation between the two subfactors by proving that these two subfactors have some kind of duality in TQFT. That is, their TQFT’s are complex conjugate to each other. In [K2], Y. Kawahigashi has discussed the relation between two subfactors arising from a special type of commuting square and he has proved that they have the same tensor category. Our result gives a fully general theorem of this type. We further prove that the two subfactors give the same “quantum doubles” in a sense that their asymptotic inclusions give the same tensor categories, as long as the two fusion graphs are connected. (This connectedness assumption holds for most cases.) Acknowledgement The author would like to thank Prof. Izumi for his comment on our previous paper [S2] which has led to Corollary 2.5 in this paper and also thank Prof. Kawahigashi for valuable discussions and his constant encouragement.

2 Tensor categories and TQFT’s

Assume that we have a finite dimensional non-degenerate commuting square with irreducible inclusion matrices. That is, we have four finite, bipartite, connected graphs G_{i} (i = 0, 1, 2, 3) and a biunitary connection W on these four graphs as in the following figure. We denote the distinguished vertex in the set of even vertices V_{0} of the graph G_{0} by ∗.

G_{2} G_{0} G_{3}  G_{1} W ∗

We apply the string algebra construction to the above data and get the double increasing sequence of finite dimensional von Neumann algebras (string algebras) {A_{k,l}}_{k,l≥0}. Thus we get AFD II_{1} factors M = ∨_{k=0}^{∞}A_{1,k}, Q = ∨_{k=0}^{∞}A_{k,1} and their subfactors N = ∨_{k=0}^{∞}A_{0,k}, P = ∨_{k=0}^{∞}A_{k,0}, respectively. See [O1] and [O3] for details of these constructions. In the present paper, we use the characters N, M, P and Q for the above constructed AFD II_{1} factors. We assume that the above constructed subfactors are of finite depth in the rest of the paper.

2.1 ∗−flat case

Assume that the biunitary connection W is flat with respect to the distinguished vertex in ∗. See [O1], [O3] and [K1] for flatness of the biunitary connection and its necessary and sufficient conditions.

Recall that the AFD II_{1} factor P constructed above consists of strings with pairs of paths as in the following picture.

[figure: ∗; ·]

We slightly modify these strings into open strings and introduce the notion of open string bimodules. These notions were already described in [O1, page 133] to prove the “Range Theorem”.

An open string on the vertical graph G_{3} is a pair of paths of the same length with the same endpoint and such that one of the paths starts from ∗ and the other starts from some even vertex on G_{3}. The following figure gives an open string on the graph G_{3}, where x is a even vertex on the graph G_{3}.

[figure: ∗; x; ·; ·]

We denote the linear space of open strings of length n spanned by the elements in the above figure with starting vertices ∗ and x by OpenString_{∗,x}^{(n)}. We can embed OpenString_{∗,x}^{(n)} into OpenString_{∗,x}^{(n+1)} as in the manner of embedding of the usual string algebras and OpenString_{∗,x}^{(n)} has the normalized trace tr compatible with the above embedding. Thus we have an increasing sequence of finite dimensional Hilbert spaces. We note that OpenString_{∗,x}^{(n)} admit the left and right actions of A_{n,0}. The algebra A_{n,0} acts naturally from the left and acts from the right in the following way. Because we have assumed that the biunitary connection W is flat with respect to ∗, we have the parallel transport Transp_{∗,x} from ∗ to x, where x is in the even vertices of G_{3}. (See [O3, II.5] for the definition of the parallel transport). Because the AFD II_{1} factor P is given by P = ∨_{n=0}^{∞}A_{n,0}, taking the completion of the union of OpenString_{∗,x}^{(n)} with respect to the inner product arising from tr, we get a P -P bimodule A(x).

Definition 2.1 We call the above introduced P -P bimodules {A(x)}_{x∈V_{0}} open string bimodules.

Remark 2.2 We note that the above notion of open string bimodules is equivalent to Ocneanu’s surface bimodules in [E-K3, page 210] if biunitary connection is flat with respect to ∗ and in this case, we identify both notions.

Theorem 2.3 The tensor category arising from the subfactor N ⊂ M is complex conjugate to that of the subfactor P ⊂ Q if the biunitary connection W is flat with respect to ∗. In particular, they have the TQFT’s complex conjugate to each other in this case.

Proof.

We can construct the open string bimodules of the P -P bimodules as stated above. On the other hand, we have the surface bimodules of the N-N bimodules and we identify these surface bimodules with the open string bimodules {A_{c}(x)}_{x∈V_{0}} as noted in the above Remark 2.2. Note that the right action of N on A_{c}(x) is given by the parallel transport Transp_{∗,x}^{c} via the biunitary flat connection W_{c} arising from the canonical commuting square. We already know these N-N open string bimodules A_{c}(x) are irreducible [E-K3, Theorem 2.1].

To show the statement, we claim that they have the same parallel transports.

First we have the following inclusions by Ocneanu’s compactness argument [O3, II.6]. Note that M′_{j} ∩ M_{k}’s (j, k ≥ 0) make the canonical commuting square. So we have a biunitary connection which is flat with respect to two ∗’s, hence a paragroup.

A_{0,0} ⊂ A_{0,1} ⊂ A_{0,2} ⊂ ⋯ ⊂ N
∩ ∩ ∩ ∩
M′ ∩ M ⊂ A_{1,0} ⊂ A_{1,1} ⊂ A_{1,2} ⊂ ⋯ ⊂ M
∩ ∩ ∩ ∩ ∩
M′_{1} ∩ M_{1} ⊂ M′ ∩ M_{1} ⊂ A_{2,0} ⊂ A_{2,1} ⊂ A_{2,2} ⊂ ⋯ ⊂ M_{1}
∩ ∩ ∩ ∩ ∩ ∩
M′_{1} ∩ M_{2} ⊂ M′ ∩ M_{2} ⊂ A_{3,0} ⊂ A_{3,1} ⊂ A_{3,2} ⊂ ⋯ ⊂ M_{2}
∩  ∩  ∩  ∩  ∩  ∩
⋮  ⋮  ⋮  ⋮  ⋮  ⋮

Because we can identify the initial vertex ∗ in V_{0} with that in M′ ∩ M, we may think that ∗ in M′ ∩ M is the initial vertex of the string algebras {A_{k,l}} (k ≥ 1, l ≥ 0). Then any vertical string arising from M′ ∩ M_{n} (n ≥ 0) commutes with any horizontal string in A_{k,l} (k = 1, l ≥ 0) because A_{k,l} ⊂ M. Again, we can identify the initial vertex ∗ in M′ ∩ M with that in M′_{1} ∩M_{1}, so we may think that ∗ in M′_{1} ∩M_{1} is the initial vertex of the string algebras {A_{k,l}} (k ≥ 2, l ≥ 0). Then any vertical string arising from M′_{1}∩M_{n} (n ≥ 1) commutes with any horizontal string in A_{k,l} (k = 2, l ≥ 0) because A_{k,l} ⊂ M_{1}. By continuing this procedure, we can conclude that any string arising from M′_{j} ∩ M_{k} (k ≥ j) commute with any string in A_{j+1,l} (l ≥ 0). Thus we can deduce the following identity of the partition function by a similar argument to the implication from (1) to (2) in the proof of Theorem 2.1 in [K1].

[figure: ∗; ∗; ·; ·; ·; ·; ρ_{+}; ρ_{−}; σ_{+}; σ_{−}; ξ_{+} · η_{+}; ξ_{−} · η_{−}; W_{c}; W; = δ_{ξ_{+},ξ_{−}}δ_{η_{+},η_{−}}C_{ρ,σ}]

Here the length of paths are given by |ξ_{+}| = |ξ_{−}| = m, |η_{+}| = |η_{−}| = n (m, n ≥ 1) and · means the concatenation of paths and each constant C_{ρ,σ} does not depend on the paths ξ_{±}, η_{±}.

Again, a similar argument to the implication from (2) to (3) in the proof of Theorem 2.1 in [K1] works and we get the following identity.

[figure: ∗; ∗; ·; ·; ∗; ∗; ·; ξ_{+}; ξ_{−}; W_{c}; W; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; W_{c}; W; = δ_{ξ_{+},ξ_{−}}]

Especially, we have the following identity.

[figure: ∗; ∗; ·; ·; ∗; ∗; ·; ∗; ∗; η; η; W_{c}; W; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; ξ; ξ; W_{c}; W; = 1]

The above partition function is represented as a summation for paths ζ_{+}, ζ_{−} on the graph G_{3}.

[figure: ∗; ∗; ·; ∗; ∗; ·; ·; ·; ∗; ∗; ∗; ∗; η; η; W_{c}; W; ζ_{+}; ζ_{−}; ζ_{+}; ζ_{−}; ∑ζ_{+},ζ_{−}; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; ξ; ξ; W_{c}; W; = 1]

The second term in the left hand side is equal to δ_{ζ_{+},ρ_{+}}, δ_{ζ_{−},ρ_{−}} because the biunitary connection W_{c} arises from the canonical commuting square and this is flat with respect to ∗ [K1, Theorem 2.1]. Thus we get the following identity.

[figure: ∗; ∗; ·; ·; ∗; ∗; W; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; ξ; ξ; W_{c}; W; = 1]

Adjusting the length of paths of ξ, we get the following identity, where |ξ| = |η| = 2n, n ∈ N and˜ means the reversed path and x ∈ V_{0}.

[figure: ∗; ∗; ·; ·; ∗; ∗; ρ_{+}; ρ_{−}; ·; ·; ρ_{+}; ρ_{−}; ξ; ξ; η̃; η̃; x; x; W_{c}; W; = 1]

Again, a similar argument to the implication from (3) to (2) in the proof of Theorem 2.1 in [K1] works and we get the desired equality.

[figure: ∗; ∗; ·; ·; x; x; ·; ·; ρ_{+}; ρ_{−}; σ_{+}; σ_{−}; ξ; ξ; W_{c}; =; ∗; ∗; ·; ·; x; x; ·; ·; ρ_{+}; ρ_{−}; σ_{+}; σ_{−}; η; η; W]

Finally, we can construct the subfactor from the paragroup with biunitary connection W_{c} and we know that this subfactor is anti-isomorphic to that of N ⊂ M by J ·^{∗} J. This induces that the biunitary connection W_{c} is complex conjugate to that arising from N ⊂ M. So we know that the quantum 6jsymbols arising from N ⊂ M are complex conjugate to those of P ⊂ Q [E-K2, Theorem 3.1]. This completes the proof. □

2.2 General case

Next we consider the case in which the biunitary connection W is not flat with respect to ∗.

Theorem 2.4 The TQFT arising from the subfactor N ⊂ M is complex conjugate to that of the subfactor P ⊂ Q.

Proof. Set B_{k} = N′ ∩ M_{k}. We note that by Ocneanu’s compactness argument [O3, II.6], the inclusions B_{k} ⊂ A_{k+1,0} hold. Because we have assumed that the subfactor N ⊂ M is of finite depth, we get inclusions of AFD II_{1} factors B_{∞} ⊂ P ⊂ Q, where B_{∞} = ∨_{k=−1}^{∞}B_{k}.

We know that the fusion rule algebra of the B_{∞}-B_{∞} bimodules arising from B_{∞} ⊂ Q naturally contains that arising from B_{∞} ⊂ P as a fusion rule subalgebra. By the coincidence of the global indices of B_{∞} ⊂ Q and B_{∞} ⊂ P shown in [S2, Corollary 2.6], we know that these two fusion rule algebras are the same.

We also claim that the tensor category of the B_{∞}-B_{∞} bimodules arising from B_{∞} ⊂ Q gives a complex conjugate of that arising from the N-N bimodules for N ⊂ M. To prove this, we consider the following commuting square of period 2 as in [S2].

N′ ∩ M_{k−1} ⊂ A_{k,1}
∩  ∩
N′ ∩ M_{k} ⊂ A_{k+1,1}

We have the biunitary connection for this commuting square and denote this by W_{f} · W, where W_{f} is the biunitary connection arising from the following commuting square of period 2 [E-K1, Proposition 3.1].

N′ ∩ M_{k−1} ⊂ A_{k,0}
∩  ∩
N′ ∩ M_{k} ⊂ A_{k+1,0}

It has been shown that W_{f} · W is flat with respect to the distinguished vertex ∗′ in the set of even vertices V′_{0} of the principal graph of N ⊂ M corresponding to the irreducible bimodule _{N}N_{N} [S2, Theorem 2.2].

We next show that the argument used in the proof of Theorem 2.3 works. That is, we shall prove the following identity, where |ξ| = |η| = 2n and x ∈ V′_{0}.

[figure: ∗′; ∗′; ·; ·; x; x; ·; ·; ρ_{+}; ρ_{−}; σ_{+}; σ_{−}; ξ; ξ; W_{c}; =; ∗′; ∗′; ·; ·; x; x; ·; ·; ρ_{+}; ρ_{−}; σ_{+}; σ_{−}; η; η; W_{f} · W]

We know that the parallel transport Transp_{∗′,x}^{c} via the biunitary connection W_{c} does not depend on the paths from ∗′ to x because W_{c} is flat with respect to ∗′. Also, we know that the parallel transport Transp_{x,y} via the biunitary connection W_{f} ·W does not depend on the paths from x to y because W_{f} · W is flat with respect to ∗′ [S2, Theorem 2.2]. By a similar argument to the implication from (2) to (3) in the proof of Theorem 2.1 in [K1], we have the following identity of the partition function.

[figure: ∗′; ∗′; ·; ·; ∗′; ∗′; ·; ξ; ξ; W_{c}; W_{f} · W; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; W_{c}; = 1]

Especially, we have the following identity.

[figure: ∗′; ∗′; ·; ·; ∗′; ∗′; ·; ·; ∗′; ∗′; ·; ξ′; ξ′; η; η; ζ; ζ; W_{c}; W_{f} · W; ρ_{+}; ρ_{−}; ρ_{+}; ρ_{−}; W_{c}; = 1]

Because the partition function of W_{c} surrounded by ∗′ can be dropped off in a similar way to that in the proof of Theorem 2.3, we get the desired equality. Thus we proved that the tensor category of the B_{∞}-B_{∞} bimodules arising from B_{∞} ⊂ Q gives the complex conjugate of that arising from the N-N bimodules for N ⊂ M. In particular, they have TQFT’s complex conjugate to each other.

By Ocneanu’s theorem [E-K2, Theorem 3.1], the former TQFT is the same as the TQFT arising from the Q-Q bimodules for B_{∞} ⊂ Q. The above coincidence of the global indices shows that this TQFT is also the same as the one arising from the Q-Q bimodules for P ⊂ Q. This completes the proof. □

Let N ⊂ M and P ⊂ Q be AFD II_{1} subfactors of both finite depth arising from a non-degenerate commuting square as above. Then we can deduce that the above Theorem 2.4 gives further information on the asymptotic inclusion subfactors M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞} and Q ∨ (Q′ ∩ Q_{∞}) ⊂ M_{∞} of N ⊂ M and P ⊂ Q, respectively.

Corollary 2.5 The tensor category of the M_{∞}-M_{∞} bimodules is isomorphic to that of the Q_{∞}-Q_{∞} bimodules, if the both fusion graphs are connected.

Proof. We know that the standard basis of the fusion rule algebra of the M_{∞}-M_{∞} bimodules arises from the irreducible decomposition of tensoring the M_{∞}-M ∨ (M′ ∩ M_{∞}) bimodules and the M ∨ (M′ ∩ M_{∞})-M_{∞} bimodules. We know that the center of Ocneanu’s tube algebra Tube M (resp. Tube N) arising from the M-M (resp. N-N) bimodules of the subfactor N ⊂ M is identified with the Hilbert space H_{S^{1}×S^{1}} associated to S^{1} × S^{1} in the TQFT [T-V] because we have assumed that the fusion graph is connected. (See [E-K3, section 3] for the definition of Ocneanu’s tube algebras and their properties.) Thus the center of the tube algebra Tube M and that of Tube N are isomorphic to each other. So we may think that the standard basis of the fusion rule algebra of the M_{∞}-M_{∞} bimodules arises from the irreducible decomposition of tensoring the M_{∞}-N ∨ (N′ ∩ M_{∞}) bimodules and the N ∨(N′∩M_{∞})-M_{∞} bimodules. A similar argument works for the Q_{∞}-Q_{∞} bimodules. As in [E-K3, Theorem 4.3], the irreducible decompositions of the M_{∞}-M_{∞} and Q_{∞}-Q_{∞} bimodules are given by the following figures, where π_{i}’s present the minimal central projections in tube algebras.

[figure: π_{k}; π_{i}; π_{j}; π_{k}; ∑_{k}; ⊗_{C}]

Because we have identified the TQFT of the N-N bimodules with that of the P -P bimodules, the coefficients and the 6j-symbols of these irreducible decompositions are the same. So we are done. □

3 Examples

Example 3.1 (A Goodman-de la Harpe-Jones subfactor) We consider a special type of the Goodman-de la Harpe-Jones subfactors based on Dynkin diagrams A_{11} and E_{6} [G-H-J, §4.5]. In the context of paragroup theory, this subfactor is constructed from the four graphs connected as in the following picture.

E_{6} A_{11} G_{1}  G_{2} W ∗

Here the graphs G_{1} and G_{2} are given by the following figures.

G_{1}  G_{2}

We have the biunitary connection W which is flat with respect to ∗ because of the flatness of Jones projections in string algebras arising from the graph A_{11}. By the string algebra construction, we get two subfactors N ⊂ M and P ⊂ Q as in the previous section. The subfactor N ⊂ M is called a Goodman-de la Harpe-Jones subfactor [G-H-J, §4.5]. Thus W satisfies the condition of Theorem 2.3 and when we apply the theorem, we know that the TQFT of the subfactor N ⊂ M is complex conjugate to that of P ⊂ Q. On the other hand, we already know that the subfactor P ⊂ Q is self antiisomorphic (i.e., (P ⊂ Q) ≅ (P^{op} ⊂ Q^{op})) because the subfactor P ⊂ Q with the principal graph A_{11} is uniquely determined [P1, Corollary 6.7]. So the above TQFT of N ⊂ M is the same as that of P ⊂ Q. Moreover, we know that the fusion rule algebra of the N-N bimodules arising from N ⊂ M is the same as that of the P -P bimodules arising from P ⊂ Q. This has determined the correct fusion rule for the subfactor among the tables in D. Bisch [B]. (See the fifth table in [B] for the this fusion rule.) These results were obtained by Y. Kawahigashi in his work on paragroup actions [K2, section 5] based on flatness. Our theorems generalize these results and it is an advantage of our method that it does not require any flatness assumption.

Example 3.2 (E_{7} commuting square) We consider the E_{7} commuting square. In the context of paragroup theory, we have four E_{7} graphs and the biunitary connection W. It is well known that there are two non-isomorphic biunitary connections on these four E_{7} graphs and also known that these are not flat with respect to ∗. We have two subfactors N ⊂ M and P ⊂ Q by the string algebra construction as above. Then we know that the TQFT of N ⊂ M is complex conjugate to that of P ⊂ Q. On the other hand, we already know that the subfactor N ⊂ M is self anti-isomorphic (i.e., (N ⊂ M) ≅ (N^{op} ⊂ M^{op})) because the subfactor N ⊂ M with the principal graph D_{10} is uniquely determined. (See [P1], [O1] and [K1].) So the above TQFT of N ⊂ M is the same as that of P ⊂ Q. The flat part {N′ ∩M_{k}}_{k≥−1} of N ⊂ M is given by Dynkin diagram D_{10} [E-K1, §3]. By an easy computation, we know that horizontal graphs which connect the original vertical graph E_{7} with the principal graph D_{10} of N ⊂ M are also D_{10} and E_{7}.

E^{7} E_{7} E_{7}  E_{7} W ∗ D_{10}  E^{7} D_{10} W_{f} ∗′

In the above four graphs, we have the biunitary connection W_{f} · W which is flat with respect to ∗′. The connecting relation between two horizontal graphs D_{10} and E_{7} are given by the following figure [S2, Figure 2].

∗′ ; D_{10} ; E_{7}

Thus we get the following principal graph of B_{∞} ⊂ Q [S2, Figure 3].

∗′

By Theorem 2.3, we also know that the tensor category of the B_{∞}-B_{∞} bimodules arising from B_{∞} ⊂ Q is the same as that of the N-N bimodules. Thus the TQFT of the subfactor B_{∞} ⊂ Q (which has the above graph as the principal graph) is the same as the D_{10} subfactor N ⊂ M. In particular, the fusion rule algebra of the B_{∞}-B_{∞} bimodules arising from B_{∞} ⊂ Q is the same as that of the N-N bimodules arising from the D_{10} subfactor.

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