α -induction for bi-unitary connections

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Introduction
A fusion category [13] has recently emerged as a new type of symmetry in a wide range of topics in mathematics and physics.Theory of operator algebras gives a nice framework to study this type of new symmetries, as exemplified by discovery of the Jones polynomial for knots [24] from the Jones theory of subfactors [23].We have three operator algebraic realizations of a fusion category based on endomorphisms of a type III factor [37], [38], bimodules over type II 1 factors [15,Chapter 9], and bi-unitary connections [1, Section 3].The last approach based on bi-unitary connections recently has renewed interest because of its relations to 2-dimensional statistical physics [9], [29].
A certain 4-tensor, a (finite) family of complex numbers indexed by four indices, is studied in physics literature such as [9], [41].This has been identified with a biunitary connection in the subfactor sense in [29], and further studies [30], [31], [32] have followed in this direction.Also see [20] for a recent development.Bi-unitary connections and related topics are also recently studied in [10], [11].This approach to a fusion category based on bi-unitary connections has advantage that everything is described with finite dimensional matrices and thus in principle computable on computers, while endomorphisms of a type III factor and bimodules over type II 1 factors are infinite dimensional.This computability is one reason physicists are interested in this approach recently.Our aim in this paper to study α-induction in this framework of bi-unitary connections.
The tensor functor α-induction originates in the operator algebraic studies of chiral conformal field theory [40].In this context, our basic object is a conformal net, which is a family of von Neumann algebras parametrized by intervals contained in the circle S 1 .It has a representation theory of superselection sectors and this has a structure of braiding [16], [17].If a conformal net has a certain finiteness property called complete rationality, we obtain a modular tensor category of its representations [34].See a short review [28] and a longer review [27] for a general theory of this topic.
In a classical representation theory, if we have a representation of a subgroup H ⊂ G, we have a method of induction to obtain a representation of G.We have a similar, but more subtle, induction machinery for representation theory of conformal nets, depending on braiding structure.Suppose we have an inclusion A ⊂ B of completely rational conformal nets and λ is a representation of A given as a Doplicher-Haag-Roberts endomorphism of A(I) for some fixed interval I in S 1 .Then we can extend this endomorphism to B(I) using the braiding.This was first defined by Longo-Rehren [40] and further studied by Xu [51] and Böckenhauer-Evans [4], [5], [6], where this was named as α-induction.
In a completely different setting, Ocneanu had a machinery of flat connections to study subfactors [43], [44], which consists of entries of a large unitary matrix.Then using a braiding structure of flat connections on the Dynkin diagrams of type A, he introduced a graphical calculus to construct new fusion categories related to the Goodman-de la Harpe subfactors [18] in [45].We have unified the two theories of α-induction and Ocneanu's graphical calculus in a fully general setting of braided fusion categories and proved various basic properties such as appearance of modular invariants in [7], [8] in the framework of endomorpshisms of type III factors.In this setting, α-induction is understood as a method of extending an endomorphism of a factor N to another factor M ⊃ N using braiding, where we do not assume anything about conformal field theory.This α-induction has been also understood in the language of bimodules and more abstract braided fusion category.
Our motivation for this paper is as follows.First, it is nice to have a concrete realization of α-induction in the setting of bi-unitary connections due to its finite

Preliminaries on braided fusion categories, subfactors and α-induction
We give our setting for α-induction for a subfactor N ⊂ M and endomorphisms of N as in [7].Let N be a type III factor.Throughout this paper, we assume the following for ∆ which is a finite set of mutually inequivalent irreducible endomorphisms of N of finite dimension.(See [7, Definition 2.1].)Assumption 2.1 We have the following for ∆.
(3) For any λ, µ ∈ ∆, the composition λµ decomposes into a direct sum of irreducible endomorphisms each of which is equivalent to one in ∆.
We further make an assumption on connectedness of certain bipartite graphs as follows, which will be necessary in the following Section.Note that this assumption depends on a choice of µ, an irreducible endomorphism of N in ∆.This is assumed throughout the paper, except for Section 6. Assumption 2.2 Consider a bipartite graph defined as follows.Let both even and odd vertex sets be labeled with the elements in ∆.The number of edges between the even vertex ν 1 and the odd vertex ν 2 is given by dim Hom(ν 1 µ, ν 2 ).We assume that this graph is connected.
Endomorphisms of N with finite dimension whose irreducible decompositions give endomorpshisms equivalent to ones in ∆ produce a fusion category, where objects are such endomorphisms and morphisms are intertwiners between endomorphisms.(See [7, Section 2.1] for more details on intertwiners.Also see [3] for a recent treatment of fusion categories and subfactors.See [13] for a more abstract and algebraic treatment of fusion categories.)We consider a subfactor N ⊂ M with finite index whose canonical endomorphism θ decomposes into a sum of endomorphisms equivalent to ones in ∆.Such a subfactor automatically has a finite depth.For a fixed N and a fusion category of such endomorphisms, an extension M of N is in a bijective correspondence to a Q-system as in [39,Section 6].In algebraic literature, this is often called a Frobenius algebra.(In this paper, we only consider C * -tensor categories.Also see [42,Theorem 2.3] for a bimodule formulation of a Q-system.) Since we assume to have a braiding, our fusion category is a braided fusion category.If a braiding is nondegenerate in the sense of [7,Definition 2.3], we say that this braided fusion category is a modular tensor category, but we do not assume this nondegeneracy in this paper.We have a notion of locality, ε(θ, θ)γ(v) = γ(v), for a Q-system corresponding to a subfactor N ⊂ M .(Here γ is the canonical endomorphism of M and an isometry v ∈ M satisfies vx = γ(x)v for x ∈ M and M = N v.) The name locality comes from locality of an extension of a conformal net in the setting of [40,Theorem 4.9] and it was called chiral locality in [7].A local Q-system is also called a commutative Frobenius algebra in algebraic literature.We deal with both local and non-local Q-systems in this paper.
The procedure called α-induction was defined in [40, Proposition 3.9] as follows.
In a very different setting, Ocneanu used Fig. 1 to represent a chiral generator in the double triangle algebra.(Also see [7,Fig. 47].)It was identified with the α-induction in [7,Theorem 5.3].(See [7,Section 4] for the double triangle algebra and [7, Fig. 4.1] for a graphical convention involving small half circles.)For various diagrams, we use the convention in [7, Section 3].In particular, our convention is as in Fig. 2, where T is an isometry in Hom(λ, µν) as in [7,Fig. 21].Note that rotation invariance of this type of diagrams is due to the Frobenius reciprocity for endomorphisms due to Izumi [21], [22].Also as in [7, Section 4], we draw a thin wire for an N -N morphism, a thick wire for an N -M or M -N morphism, and a very thick wire for an M -M morphism.For a thick wire, we can tell whether it stands for an N -M or M -N morphism from a diagram.We now produce a family of new bi-unitary connections from the setting as in the previous Section.We simply say a connection for a bi-unitary connection in this paper.
We first recall the definition of a connection.(See [15,Section 11.3], [1, Section 3], [30, Section 2].There is an issue of connectedness of certain graphs and we follow conventions of [1, Section 3] on this matter.) We have four finite bipartite graphs G, G ′ , H, H ′ .The vertex set V 0 is common for even vertices of G and H. Similarly, the vertex sets V 1 , V 2 , V 3 are common for odd vertices of H and G ′ , even vertices of G ′ and H ′ , and odd vertices of G and H ′ , respectively.The four graphs satisfy some properties about the Perron-Frobenius eigenvalues and eigenvectors as in [30,Section 2].We choose edge ξ 0 , ξ 1 , ξ 2 , ξ 3 from the graphs H, G ′ , H ′ , G, respectively so that they make a closed square called a cell.A map W called a connection assigns a complex number to each of such cells.This complex value is represented with a diagram in Fig. 3.This map W satisfies axioms called bi-unitarity as in [30,Definition 2.2].(See also Fig. 8, 9, 10 below.)The name "bi-unitarirty" means that each number in Fig. 3 is an entry of a unitary matrix in two ways, one after normalization arising from the Perron-Frobenius eigenvector entries.Now we require that the graphs G and G ′ to be connected, but we do not require this for H and H ′ .(This convention is different from the one in [30], and the same as in [1].)We have an equivalence relation for connections on the same four graphs as in Remark after [1,Theorem 3].We call the graphs G, G ′ , H, H ′ the horizontal top graph, the horizontal bottom graph, the vertical left graph and the vertical right graph of W , respectively.We also call the vertices in V 0 , V 1 , V 2 , V 3 the upper left vertices, the lower left vertices, the lower right vertices and the upper right vertices, respectively.
The four graphs for a connection It is well-known that this bi-unitarity condition is characterized in terms of a commuting square as we now explain below.A commuting square of finite dimensional C * -algebras This notion was originally considered in [46].We say that this commuting square is nondegenerate if the span BC is equal to D. (Being nondegenerate is also sometimes said to be symmetric.)A nondegenerate square of finite dimensional C *algebras is described with a connection, for which we do not know yet whether it satisfies bi-unitarity, as in [15,Section 11.2].Then it has been proved in [49,Theorem 1.10] that the square gives a commuting square if and only the connection satisfies bi-unitarity.(Also see [15,Theorem 11.2].)In this sense, having a bi-unitary connection and having a nondegenerate commuting square of finite dimensional C * -algebras are the same thing.
A commuting square also naturally appears from a subfactor N ⊂ M with finite Jones index as Jones tower arising from the Jones basic construction [23].(Also see [15,Section 9.6] for such a commuting square.)If the original subfactor N ⊂ M is hyperfinite, of type II 1 and of finite depth, then the above commuting square recovers the original subfactor N ⊂ M by Popa's theorem [47].In this sense, a connection encodes complete information about such a subfactor.A connection corresponding to such a commuting square arising from a subfactor N ⊂ M satisfies a special extra property called flatness, which was introduced by Ocneanu [43], [44] and studied in [25].(See [15,Section 11.4] for more discussions on flatness.) We first define the connection W 1 (λ, µ), but we make a remark on one issue again.In this paper, we use operator algebraic realization of a braided fusion category and α-induction based on endomorpshisms of a type III factor as in [7] and [8], such as [7,Fig. 30], but this is simply because this was the first place where all necessary details were worked out, and we can equally use other formulations based on bimodules or abstract fusion categories.Such a choice of formulation does not cause any change in our results here.
We note that the left vertical bipartite graph in this Definition is defined as follows.One set of vertices is given by ∆ and the other is the same.The number of edges between the vertices ν 1 and ν 3 is given by dim Hom(λν 1 , ν 3 ).The other graphs are defined similarly, and this remark applies to all the Definitions of the connections in this Section.We often drop labels for the connection and/or intertwiners as long as no confusion arises, and simply draw a diagram in Fig. 6.
Figure 7: The diagram for the connection W 1 (λ, µ) in Fig. 5 We draw a diagram as in Fig. 7 which represents a complex number in the standard convention as in [7, Section 3].(Note that in Fig. 7, we drop orientations of wires which go from the top to the bottom.)This complex number is equal to the one represented by the composition of isometries as in Fig. 4 multiplied by d λ d µ d ν 1 d ν 4 by the standard convention in Fig. 2. Then unitarity of the connection W 1 (λ, µ) is represented as in Fig. 8, 9.Here the bars above the right squares denote the complex conjugates.Crossing symmetry for the connection W 1 (λ, µ) is given as in Fig. 10, where T1 and T4 stand for the Frobenius duals of T 1 and T 4 , respectively.This is a wellknown relation arising from the tetrahedral symmetry of the 6j-symbols as in [15,Definition 12.15], but we include a simple argument for this, because we need a similar argument later for W 3 (λ) as in Fig. 21.We first make a vertical reflection of the diagram in Fig. 7 to obtain Fig. 11.We then redraw Fig. 11 to obtain Fig. 12, which represents the complex number given by the diagram in Fig. 13 multiplied by d Figure 12: Redrawing of Fig.
11 This shows we have crossing symmetry, Fig. 10.This, together with unitarity of the left hand side of Fig. 10, shows bi-unitarity of the connection W 1 (λ, µ).
We now prove that the fusion category of endomorphisms of N arising from ∆ and the one arising from the connections W 1 (λ, µ) for various λ and fixed µ are equivalent.This is where we need Assumption 2.2.Then we have the following theorem.
Theorem 3.2 Under Assumptions 2.1 and 2.2, the fusion category arising from the connections W 1 (λ, µ) for all λ ∈ ∆ is equivalent to the one arising from endomorphisms λ ∈ ∆ of N .
Proof.By the description of the intertwiners between open string bimodules in the proof of [1, Theorem 3], we have a natural injective linear map from Hom(λ ) is described with the higher relative commutants of the corresponding subfactor by the proof of [1, Theorem 3].These higher relative commutants are described with C k,−1 for the subfactor M 0 ⊂ M 1 in the proof of [48,Theorem 3.3].They are described with the intertwiner spaces of the N -N morphisms as in the proof of [48,Theorem 3.3].This means that the dimensions of the two intertwiner spaces Hom(λ 1 λ 2 , λ 3 ) and Hom(W 1 (λ 1 , µ))W 1 (λ 2 , µ)), W 1 (λ 3 , µ)) are the same, and thus the above natural linear map is surjective.The compositions of intertwiners in the two fusion categories are also compatible with this identification, so they are equivalent.□ Then we see that the structure of the braided fusion category of these endomorphisms of N passes to that of these connections.
We next introduce two types of connections.The first is the easier one W 2 (µ).
Definition 3.3 Let ν 1 , ν 2 ∈ ∆ and a 1 , a 2 be irreducible M -N morphisms arising from ∆ and the subfactor N ⊂ M .Fig. 14 represents the complex number , where a 1 and a 2 are M -N morphisms.We define the connection W 2 (µ) by this number, and use the diagram as in Fig. 15 to represent this connection.
Note that Remark after Definition 3.1 applies here again.For example, the left vertical bipartite graph in this Definition is defined as follows.One set of vertices is given by ∆ and the other is given by the representatives of the irreducible M -N morphisms.The number of edges between the vertices ν 1 ∈ ∆ and an M -N morphism a 1 is given by dim Hom(ιν 1 , a 1 ).We next introduce the other connection, W 3 (λ).
Definition 3.4 Let ν 1 , ν 2 ∈ ∆ and a 1 , a 2 be irreducible M -N morphisms arising from ∆ and the subfactor N ⊂ M .Fig. 16 represents the complex number , which is represented with the connection diagram in Fig. 17.We define the connection W 3 (λ) by this number.Here E ± is defined on page 455 below (14) in [7] and we recall Fig. 18, which is taken from [7, Fig. 30].The complex number represented by the diagram in Fig. 19 is equal to the one represented by the connection diagram in Fig. 17 multiplied by d ι d λ d ν 1 d a 2 by the standard convention in Fig. 2. Note that we drop orientations of wires which go from the top to the bottom in Fig. 19.
We make a vertical reflection of Fig. 19 to get Fig. 20.We again drop orientations of wires which go from the top to the bottom.Note that the complex number values given by these two diagrams are complex conjugate to each other.
Figure 20: A vertical reflection of Fig. 19 By redrawing Fig. 20, we have Fig. 21.We next introduce the connection W 4 (α + λ , µ), which is the α + -induced connection.Definition 3.5 Let a 1 , a 2 , a 3 , a 4 be irreducible M -N morphisms arising from ∆ and the subfactor N ⊂ M .Consider the diagram in Fig. 23.By composing isometries T 1 ∈ Hom(a 1 µ, a 2 ), T 2 ∈ Hom(α + λ a 2 , a 4 ), T 3 ∈ Hom(α + λ a 1 , a 3 ), and T 4 ∈ Hom(a 3 µ, a 4 ), we obtain a complex number T 4 T 3 α + λ (T * 1 )T * 2 ∈ Hom(a 4 , a 4 ).We define the connection W 4 (α + λ , µ) by this number and represent this as in Fig. 24.By a very similar argument to the one for bi-unitarity of W 1 (λ, µ), we obtain bi-unitarity of W 4 (α + λ , µ).We then have the intertwining Yang-Baxter equation for W 1 (λ, µ), W 3 (λ), W 3 (µ), and W 4 (α + λ , µ) as in Fig. 25, which was given in [26, Axiom 7].The meaning of this diagram is as follows.On the both hand sides of the identity, the six isometries are fixed for the six boundary edges of the hexagons in the same way.The left hand side means the summation of the product of the three connection values W 1 (λ, µ), W 3 (λ), and W 2 (µ) over all possible choices of the three isometries corresponding to the three internal edges of the hexagon.The right hand side means the summation of the product of the three connection values W 3 (λ), W 2 (µ), and W 4 (α + λ , µ) over all possible choices of the three isometries corresponding to the three internal edges of the hexagon.The both hand sides are equal because they are both equal to the composition of the six (co-)isometries corresponding to the six boundary edges of the hexagons.
Figure 25: The intertwining Yang-Baxter equation for W 1 (λ, µ), W 3 (λ), W 3 (µ), and We have seen α-induction applied to N ⊂ M produces bi-unitary connections.From this construction and the description of intertwiner spaces for connections as in [1, Theorem 3] again, we have the following theorem under Assumption 2.2 with the same arguments as in the proof of Theorem 3.2.Theorem 3.6 Under Assumptions 2.1 and 2.2 for a fixed µ, the fusion category arising from the connections W 4 (α ± λ , µ) for all λ ∈ ∆ is equivalent to the one arising from endomorphisms α ± λ of M given by α-induction.
Note that it is nontrivial that we have only finite many irreducible endomorphisms of M up to equivalence when we consider those arising from α ± λ .This has been proved in [7,Theorem 5.10].4 Rewriting W 4 (α + λ , µ) and switching of positive and negative braiding Though we have defined the α-induced connection W 4 (α + λ , µ) as in Fig. 23, we need to know full information about α + λ to compute this value.Since we are now in the process of defining the new connection W 4 (α + λ , µ) before knowing α + λ , we certainly hope to compute this number without using the information about α + λ .We show in this Section that this is possible.Fig. 26 represents the complex number for isometries T 1 ∈ Hom(a 1 µ, a 2 ), T ′ 2 ∈ Hom(a 2 λ, a 4 ), T ′ 3 ∈ Hom(a 1 λ, a 3 ), T 4 ∈ Hom(a 3 µ, a 4 ), which is represented with the connection diagram in Fig. 24.This composition corresponds to the diagram in Fig. 27 up to normalization constant, and we can redraw this as in Fig. 28.Note that we drop orientations of wires which go from the top to the bottom in Fig. 27 diagram gives W 4 (α + λ , µ), up to normalization constant, in terms of N -N and M -N morphisms and intertwiners including the braiding operators.This is what we asked for at the beginning of this Section.
T 4 E + (λ, a 1 )  We next study what the effect of switching the positive and negative braiding is.Vertical reflection of Fig. 28 gives Fig. 29.The complex numbers given by these two Figures are mutually complex   We have the identity as in Fig. 30.We also show that we can rewrite W 3 (λ) into the form without using α + λ as follows.5 Locality of the Q-system and flatness of W 4 (α + λ , µ) In the case of Ocneanu's construction [45], he obtained flat connections only for A n , D 2n , E 6 and E 8 as we see in the next Section in detail.(Note that Assumption 2.2 does not hold for these examples.We will treat this issue in the following Section.Also see [15,Theorem 11.24].)It has been observed that these cases exactly correspond to the Q-systems with locality ε(θ, θ)γ(v) = γ(v) as in [8, Section 5] (where this property was called chiral locality).So we expect some relations between locality of the Q-system and flatness of the corresponding α ± -induced connection in general.We show that this is indeed the case in this Section.We now assume locality.Note that we then have irreducibility of ι : N → M by [4,Corollary 3.6].(Having a Q-system with locality in a general modular tensor category is enough in [4] rather than a conformal net.) We prove flatness of the connection W 4 (α + λ , µ) in the sense of Fig. 33, which is taken from [15,Fig. 11 Proof.Locality gives Fig. 35, where the triple vertices on the both hand sides represent γ(v).We then have the identity in Fig. 36, where small black circles represent co-isometries in Hom(θ, λ) and Hom(θ, µ).We next have the identity in Fig. 37 by a property of braiding and then the identity in Fig. 38 by rewriting θ with ῑι.Pulling the thick wires straight and rotating this diagram for 90 degrees Proof.We prove the identity in Fig. 33.Both horizontal and vertical sizes of the large diagram in Fig. 33, are supposed to be even, but we can take both of them to be 1 in our current setting, so we first give a proof for this case.Now our * vertex in Fig. 33 corresponds to ι.Because all the four corner vertices in Fig. 33 are now ι, we set a 1 = a 2 = a 3 = a 4 = ι in Fig. 28.We then have Fig. 39 and the complex value this diagram represents is equal to the one given by the partition function in Fig. 33 multiplied by d λ d µ d ι .
By Lemma 5.1, the value Fig. 39 represents is equal to the one Fig.40 represents.The latter is equal to d λ d µ d ι , so the complex value given by the partition function in Fig. 33 is 1.
When the horizontal and vertical sizes of the large diagram in Fig. 33

Examples
Ocneanu considered connections on A-D-E Dynkin diagrams in [45].We revisit this topic from our viewpoint now and would like to apply the results in Section 3.Here we have a nontrivial issue since Assumption 2.2 does not hold now for any choice of µ.
Let ∆ be the set of endomorphisms of a type III factor N corresponding to the Wess-Zumino-Witten model SU (2) k , where k is a positive integer called a level.(See [12, Subsection 16.2.3],for example.)We label the irreducible objects of the modular tensor category with 0, 1, 2, . . ., k, using the Dynkin diagram of type A k+1 as in Fig. 41, where the label 0 denotes the vacuum representation, that is, the identity automorphism of N .Such a system of endomorphisms with braiding has been constructed from a conformal net by Wassermann [50] and this braiding is nondegenerate by [34,Corollary 37]  Let G be one of the A-D-E Dynkin diagrams and choose the vertex with the smallest entry of the Perron-Frobenius eigenvector entry.We then have the Goodmande la Harpe-Jones subfactor as in [18,Section 4.5] or [15,Section 11.6], and the corresponding Q-system.If G is of type A, then this Q-system has index 1 and is trivial.If G is of type D, then the Q-system has index 2 and corresponds to a crossed product by Z/2Z.If G is E 6 or E 8 , then the Q-systems correspond to the subfactors arising from conformal embeddings SU (2) 10 ⊂ S(5) 1 and SU (2) 28 ⊂ (G 2 ) 1 by [8,Proposition A.3]. Also see [8,Appendix] for the case of E 7 .
We next choose λ = µ = 1 in the setting of Section 3 and consider the connection W 4 (α ± λ , µ).Since the irreducible N -M morphisms are labeled with the vertices of G by the arguments in [15,Section 11.6], all the four vertex sets for W 4 (α ± λ , µ) are also labeled with the vertices of G. Then the requirements for the Perron-Frobenius eigenvalues and the Perron-Frobenius eigenvector entries force all the four graphs of W 4 (α ± λ , µ) to be G, but Assumption 2.2 does not hold, since the horizontal top and bottom graphs are never connected, due to a Z/2Z-grading on the vertices of the Dynkin diagrams.This issue is resolved as follows.
The connection W 4 (α ± λ , µ) splits into two connections on mutually disjoint graphs, both of which are isomorphic to G. Then both of the two connections must be of the following form as given in [44].(Also see [15,Fig. 11 The horizontal top and bottom graphs for W 4 (α ± j , 1) always have exactly two connected components.This is still valid after we make irreducible decomposition of such connections.Let W be a pair of two such connections W a and W b arising from irreducible decomposition of W 4 (α ± j , 1) for some j and W ′ be a pair of two such connections W ′ a and W ′ b from W 4 (α ± k , 1) for some k.We can compose W a with exactly one of W ′ a and W ′ b , and compose W b with the other one, since they have the matching horizontal bottom and top graphs.For the fusion rules and intertwiner spaces of this composition product, we can use either of the two compositions and have the same results, by the same arguments to the ones in the proof of Theorem 3.6.In this way, we obtain a fusion category of connections which is equivalent to the one of M -M morphisms arising from α ± -induction.We now obtain the diagrams of decompositions of connections which are the same as in [6,Fig. 2,5,8,9] and [8,Fig. 40,42].These were originally found by Ocneanu for such connections.We also know that the results in the previous Section on flatness apply to these cases, though Assumption 2.2 does not hold now.For this type of computations, we do not need exact information of the connections, and simply having the graphs involved is often sufficient.See [19] for such computations.
We now discuss the issue of complex conjugate connections.For W 4 (α ± 1 , 1), the connection Fig. 42 is symmetric in j and m, so the effect of switching positive and negative braiding in Proposition 4.1 now amounts to taking simply complex conjugate connections.That is, W 4 (α + 1 , 1) and W 4 (α − 1 , 1) are mutually complex conjugate.Now consider the case of G = E 6 .The connections W 4 (α + 2 , 1) and W 4 (α − 2 , 1) are also mutually complex conjugate.Both of the connections W 4 (α + 3 , 1) and W 4 (α − 3 , 1) decomposes into two irreducible connections each.Since irreducible decomposition of a complex conjugate connection gives complex conjugate connections of those appearing in the irreducible decomposition in general, the complex conjugates of the two irreducible connections arising from W 4 (α + 3 , 1) appear in the irreducible decomposition of W 4 (α − 3 , 1).In this way, we see that switching the positive and negative braiding for the α-induced connections amounts to taking complex conjugate connections.The same argument also works for E 7 and E 8 .This was also observed by Ocneanu [45], but is special to the A-D-E Dynkin diagrams.The decomposition rules for E 7 as in [8,Fig. 42] follows from our computations in [14] showing that the principal graph of the subfactor arising from the E 7 connection is D 10 .
The modular tensor categories corresponding to the Wess-Zumino-Witten model SU (N ) k also have Z/N Z-grading, and a similar method to the above gives how to handle this issue.
We add a remark on the effect of Assumption 2.2.Even without this Assump-tion, our Definitions 3.1, 3.3, 3.4, 3.5 on our new connections make sense.The only problem is that if horizontal graphs are disconnected, we cannot compose two connections in a usual way in Theorem 3.6, and other results such as Theorem 5.2 are not affected.Another way of handling the issue in Assumption 2.2 is to make ∆ smaller.For example, if we choose λ = µ = 2 in the numbering of irreducible objects for the fusion category SU (2) k as in Fig. 41 and consider only irreducible objects numbered with even integers, then everything in the above Sections works fine.
7 Triple sequence of string algebras and another interpretation of the α-induction in terms of biunitary connections Let * be the vertex corresponding to the identity automorphism of N in ∆.We construct a triple sequence {A jkl } jkl as in [26,Section 2].This is a triple sequence version of the standard construction of the double sequence of string algebras in [15,Section 11.3].A new property we need for compatibility of identification is the Intertwining Yang-Baxter Equations as in [26,Axiom 7], and we now have this identity as in Fig. 25.
Note that for inclusions such as A 2j,2k,2l ⊂ A 2j,2k,2l+1 , we use unitary equivalences α + λ ι ∼ = ιλ, α + λ ι ∼ = ι λ, λῑ ∼ = ῑα + λ , and λῑ ∼ = ῑα + λ .These are compatible with ways of identification of strings with the connections.(The only nontrivial identification is done with W 3 (λ), where we use unitary equivalence of ιλ and α λ ι as in Fig. 16.) By taking unions over k and making the GNS-completions with respect to the compatible trace, we have a commuting square of hyperfinite type II 1 factors as in [26,Assumption 1.1].We see that our triple sequence of string algebras arise from this commuting square as in [26,Section 3].See [26,Section 5] for the case of the Dynkin diagrams of type A-D-E.
Our construction of α-induced connections from N ⊂ M uses information on all the N -M morphisms and their intertwiners.This is also true for Ocneanu's chiral generator picture in Fig. 1 in the double triangle algebra.This is theoretically fine, but we would like to have a method to obtain the α-induced connections purely in terms of connections.We discuss such a method at the end of this paper.
In the original setting of the set ∆ of irreducible endomorphisms of N , we fix µ ∈ ∆ and suppose we have a family of connections W 1 (λ, µ) for λ ∈ ∆ and the horizontal top and bottom graphs for all of them are the same finite bipartite graph G. Let * be the vertex of G corresponding to the identity automorphism in ∆.Our positive braiding gives equivalence of the two composite connections W 1 (λ 1 , µ) • W 1 (λ 2 , µ) and W 2 (λ 1 , µ) • W 1 (λ 1 , µ).This is given by Fig. 43, where S and T give unitary matrices giving vertical gauge choices arising from a positive braiding between λ 1 and λ 2 .Suppose we have a flat connection W 0 with respect to * (in the sense of [15,Definition 11.16]) with the horizontal top and bottom graphs being finite bipartite graphs G and H such that the composition W 0 • W0 decomposes into a direct sum of irreducible connections each of which is equivalent to one of W 1 (λ, µ).This is a connection version of our setting in Section 2 and our connection W 0 must be of the form W 2 (µ) for some ι.Construct a double sequence of string algebras {B kl } kl as in [15,Section 11.3] from the connection W 0 .That is, the commuting square is described with W 0 .We further construct a double sequence of string algebras {C kl } kl so that is described with W 1 (λ, µ) and is described with W 0 .Because of the braiding between W 1 (λ, µ) and W 0 • W0 , we have compatibility between

Figure 2 :
Figure 2: Graphical convention for normalization of an intertwiner

2 .
with a trace on D is characterized by the following mutually equivalent conditions.(See [15, Proposition 9.51], for example.) 1.The conditional expectation E B from D to B with respect to trace restricted to C coincides with the conditional expectation E A from C to A with respect to trace.The conditional expectation E C from D to C with respect to trace restricted to B coincides with the conditional expectation E A from B to A with respect to trace.

Figure 23 :a 3 a 4 a 1 a 2 Figure 24 :
Figure 23: The diagram for the connection W 4 (α + λ , µ) in Fig. 5 , 28.The complex number represented by the diagram in Fig. 28 is equal to the one represented by the connection diagram in Fig. 26 multiplied by d λ d µ d a 1 d a 4 .Since Fig. 28 does not involve α + λ , this

3 Figure 28 :
Figure 28: Redrawing of Fig. 27 conjugate.By comparing these two Figures, we have the following Proposition.