数理解析研究所講究録
1077 巻 1999 年 105-139

105

An introduction to Ocneanu’s theory of double triangle algebras
nections on
for subfactors and classification of
the Dynkin diagrams$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{C}\mathrm{i}‘ \mathrm{b}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{o},\mathrm{n}$SATOSHI GOTO (SOPHIA UNIVERSITY)
後藤聡史 (上智大学)
January 19, 1999

1

Introduction

factor is one of the most important
The classification of subfactors of the hyperfinite
and stimulating problems in the theory of operator algebras since V. F. R. Jones initiatedhis celebrated index theory for subfactors in [16]. The strongest form of the classification
has been obtained by S. Popa based on his notion of strong amena bility in [26]. In thesubfactors with index less than four has one
early stage it was known that hyperfinite
as their principal graphs. For the completeandof the Dynkin diagrams
factor A. Ocneanu introduced the notion
classification of subfactors of the hyperfiniteof paragro up in [20]. And by the paragroup theory the classification of subfactors of thefactor with finite index and finite depth is reduced to the classification of
hyperfinite
nections on the (dual) principal graphs.
flat -unitary
The paragroup theory as well as its importance are now widely spread and more
and more people have become to work on the theory. The importance of paragrouptheory is not only because it is complete invariant for subfactors of the hyperfinite
factor with finite index and finite depth but also because it has deep relations to manyother theories in mathematics and mathematical physics. Actually it has been revealed
that there are striking relations between the paragroup theory and other theories suchas exactly solvable integrable lattice nuodels, quantun] groups, topological quantum field
theories (TQFT) both in the sense of Turaev-Viro ([32]) based on triangulation and in therational conformal field theories
sense of Reshetikhin-Turaev ([27]) based on surgery,(RCFT) in the sense of Moore-Seiberg ([19]) and so on. (See for example [6], [11], [9],[10], [22], [23], [28], [33], [34]. All of these relations are explained in [13].)subfactor theory at The Fields Institute
In 1995 A. Ocneanu gave a series of lecturesfrom April 19 to 25. In his lectures ([24]) he introduced a new algebra called
triangle algebra by using the notion of essential paths and extension of Kauffman-Lins’
Temperley-Lieb recoupling theory. He also gave many applications of his result. AInong
their applications he raised particularly five problems in his talks at Aarhus in June 1995,
which consists of one problem concerned with TQFT, one concerned with RCFT andthree concerned with subfactor theory. There he showed that his method gives essentially
one solution of them.
Among the solutions of the five problems the most fundamental result is the complete
classification of irreducible -unitary connections on the Dynkin diagrams An’ $Dn’ E6,7,8$
and other solutions will follow from it. More precisely the irreducible connections on the
Dynkin diagrams here means the irreducible connections on the four graphs which have the$\mathrm{I}\mathrm{I}_{1}$$\mathrm{I}\mathrm{I}_{1}$$A_{n},$ $D_{n},$ $E_{6},$ $E_{7}$$E_{8}$$\mathrm{I}\mathrm{I}_{1}$$\mathrm{I}\mathrm{I}_{1}$$bi$$co\mathrm{n}$$\mathrm{I}\mathrm{I}_{1}$$\mathrm{a}\mathrm{n}_{1}\mathrm{d}$$011$$\mathrm{d}o\mathrm{u}ble$$\mathrm{b}\mathrm{i}$106

Dynkin diagram and as the two horizontal graphs. (We call such a connection a K-L-unit
conn ection.) And the classification
the classification of irreducible
choice,-unitary connections up to auge
which is finer than the classificatioll up toisomorphisms.
The main purpose of this paper is to give a detailed proof of the classification of
irreducible connections on the Dynkin diagranls. As we mentioned in the beginningthe classification in more restricted case when the four graphs are all the same Dynkin
diagrams has been done in order to classify subfactors with index less than four. So theclassification of connections itself is very important for this purpose.
Another example in which -unitary connections on the Dynkin diagrams naturally
appear is the construction of a series of subfactors given by Goodman-de la Harpe-Jones([14]). These subfactors are called Goodman-de la-Jones bfactors. (We call them
GHJ subfactors in short. A. Ocneanu calls the salne subfactors Jones-Okam $oto$ subfactors
because S. Okamoto computed their principal graphs [25].) They are constructed from
A-K -unitary connections, where represents the Dynkin diagrams
andis one of)
the A-D-E Dynkin diagrams. The principalof these subfactors are easily obtainedby a simple method but the dual principal graphs as well as their fusion rules are much
more difficult to compute. The most important example is the subfactor with index
which is constructed from the embedding of the string algebra ofto that of , i.e., it-unitary connection. In this particular case it happens that it
obtained from an
is not very difficult to compute the dual principal graph (see [18], [13, Section 11.6]). But
it is more difficult to determine its fusion rule. Actually D. Bisch has tried to compute thefusion rule just from the graph but there were five possibilities and it turned out that therule can not be determined
the graph only. Some more information is needed
and Y. Kawahigashi obtained the fusion rule as an application of paragroup actions in[18]. In his lectures at The Fields Institute A. Ocneanu gave a solution to this problem ofdetermining the dual principal graphs and their fusion rules as one of some applicationsof his theory of double triangle algebra ([24]). In particular, the fusion rule algebra of all
K-K -unitary connection is used to determine the fusion rule of GHJ subfactors whichcorrespond to the Dynkin diagram . Here we would like to mention that some of recent
works has revealed a surprising relation between GHJ subfactors and conformal inclusions([37], [2], [3], [4]). Furthermore some generalization of the construction of GHJ subfactors
has been obtained by F. Xu $([3\dot{5}], [36])$ and J. B\"ockenhauer-D. E. Evans ([2], [3], [4]).
Another striking and unexpected observation which A. Ocneanu has found ([24]) is
the relation between fusion rule algebras of all K-K connections for a Dynkin diagram$K$ he obtained and affine
modular invariants corresponding to the graph . The
A-D-E classification of affine
invariants has been obtained in [7], (see also[31] . A. Ocneanu showed some interpretations of off-diagonal terms of these modular
invariant matrices corresponding toandin his lectures [24] by using a notionof essential paths. In December 1997, he introduced the notion of quantum
invariants which is the quantum version of Kleinian invariant and he showed another new
explanation of the off-diagonal terms. After A. Ocneanu’s work on the fusion rule algebrasof K-K -unitary connections essentially the same fusion rule algebras are constructedfrom conformal inclusions of-Witten models by F. Xu ([37]) and J.
B\"ockenhauer-D. E. Evans ([3], [4]) in flat cases. Note that the non-flat case, i.e., thecase ofandcan not be obtained fron] their approach using conformal inclusions.
Moreover the theory of double triangle algebra is recently used to generalize the result tothe case of conformal inclusions of
WZW models by J. B\"oCkenhauer-..D. E. Evans-Y.
$I\mathrm{i}’$$L$$\mathrm{h}\mathrm{e}\mathrm{l}\cdot \mathrm{e}1\coprod \mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$$\mathrm{b}i$$a\mathrm{r}y$$\mathrm{b}\mathrm{i}$$\mathrm{g}e$$\mathrm{b}\mathrm{i}$$H_{\dot{\mathrm{c}}}t1pe$$\mathrm{s}u$$A$$\mathrm{b}\mathrm{i}$$A_{n}$$I\iota’$$\mathrm{h}\mathrm{S}$$\mathrm{g}\mathrm{r}\mathrm{a}_{1}$$3+\sqrt{3}$$A_{11}$$E_{6}$$A_{11}- E_{6}\mathrm{b}\mathrm{i}$$\mathrm{i}\mathrm{s}_{l}$$\mathrm{f}\mathrm{r}\mathrm{o}\ln$$\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$\mathrm{b}\mathrm{i}$$I\iota’$$\mathrm{S}\mathrm{U}(2)$$I\mathrm{c}^{\nearrow}$$\mathrm{S}\mathrm{U}(2)_{1}\mathrm{n}\mathrm{o}\mathrm{d}_{\mathrm{U}}1\mathrm{a}\mathrm{r}$$)$$D_{n}$$E_{6,7,8}$$I_{1^{r}}l6inia\mathrm{n}$$\mathrm{b}\mathrm{i}$$\mathrm{S}\mathrm{U}(2)$$D_{odd}$$\mathrm{W}\mathrm{e}\mathrm{s}\mathrm{S}^{-}\mathrm{z}\mathrm{u}\min_{0}$$E_{7}$$\mathrm{S}\mathrm{U}(7t)$107
Kawahigashi ([5]). They showed that A. Ocneanu’s observation of the relation between
fusion rule algebras and modular invariant matrices holds true for some more general
WZW models.
cases including the case of
Now we give a brief outline of the contents in this paper. In the next section we will
give some definitions and terminologies concerning A. Ocneanu’s double triangle algebras
and we also fix some notations. Though all of the definitions of important notions suchas essential paths, gaps of finite graphs and chiral projectors are given in [24], we did notomit them for reader’s convenience because they are indispensable for the classificationof connections. We refer readers to [24] for more details.
In section 3 we define some operations on the set of connections such as direct sum,
conjugation, irreducible decomposition and composition (product). These operations are
first defined by A. Ocneanu in [24] and later M. Asaeda-U. Haagerup clarified the correspondence between these operations on connections and those on bimodules ([1]). In
order to deal with the system of connections closed under these operations we define
a notion of horizontally conjugate pair of connections and give a natural identification
between connections. We also give sonle equivalence relations on connections. We will
choice and total gauge choice and will make itticalemphasize the difference oftical gauge choice is the right equivalence relation to deal with a system
clear that theof connections. This point is also clarified by M. Asaeda-U. Haagerup ([1]). In order tomake the most of M. Asaeda-U. Haagerup’s notion of generalized open string bimodof connections.
we will show that Frobenius reciprocity holds for theSection 4 is devoted to show the correspondence between irreducible K-L bi-unitary-representations of the double triangle algebras on the graphs
connections and$K$ and. This is the most important tool to classify the irreducible connections. Thougha detailed proof of the correspondence is given in his original paper [24], some more details
are necessary for our purpose. So we will supply it here and as a corollary we show an)
special lllinimal central projections of the double
important correspondence between
triangle algebra and some irreducible -unitary connections on the Dynkin diagralns. We
also show the relation between the fusion rule algebra of K-K -unitary connectioroe andthe center of the double triangle algebra. Actually it turns out that the fusion rule algebra
is isomorphic to the center of double triangle algebra with different product from originalone. One will notice that the notion of horizontally conjugate pair and the equivalence
choice are both natural to deal with this correspondence.
tical
relation
Finally in section 5, the classification result is explained in each case of the Dynkin. In the procedure to get this result we also obtain the new fusion
diagrams
rule algebras which consists of all K-K -unitary connections. It also provides a simple
connections. Hence we get another proof of theandproof of the flatness offactor with index less than 4 bycomplete classification of subfactors of the hyperfinite
are also obtained easily.
this method. The flat part of non-fiat connections $D_{2n+1}$ andBy putting together all the cases of $A- D_{- E}$ we will obtain some important structuralof the fusion rule
result on the fusion rule algebras including a partial
algebra.
$\mathrm{S}\mathrm{U}(n)$$ve\mathrm{r}$$g\mathrm{a}u\mathrm{g}e$$v\mathrm{e}\mathrm{r}$$\mathrm{u}le$$\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\ln$$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{C}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}*$$L$$\mathrm{s}\mathrm{o}\mathrm{n}$$\mathrm{e}$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$ve\mathrm{r}$$ga\mathrm{u}ge$$A_{n},$ $D_{n},$ $E_{6,7,8}$$\mathrm{b}\mathrm{i}$$D_{2n},$$E_{6}$$E_{8}$$\mathrm{I}\mathrm{I}_{1}$$E_{7}$$\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{V}}\mathrm{i}\mathrm{t}\mathrm{y}$2Preliminaries and Notations

on finite graphs and the double triangle
In this section we give definitions of essential
algebras for the sake of completeness. We also fix some notations. We refer readers to[24] for the details.
$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{S}$108$\mathrm{r}n$

$=$

$c$

A

$B^{\Pi}A$$\mathit{1}\mathit{1}B$

Figure 1: Creation operator$c^{*}$$=$$v$

Figure 2: Annihilation operator
2.1.
projectors and Essential path .
a finite oriented bipartite graph. We denote even vertices of the graphbyand its odd vertices by.
We denote an abelian-algebra with basis[resp.
] by[resp.
)$B]$ . We use a notation $H$
for a Hilbert space with basis. Here Edge
represents the set of oriented edges of with orientation fromto
. The
Hilbert space $H$ defined as above bec.olnes an A-B binlodule with the action defined asfollows,
for
)
and
Edge
. Here[resp.
] means the source (starting point) [resp. range (end point)] of the edge
. The Hilbert space $H$ has the conjugate
spacewith the basis consists of the)orientation reversed edges. Then the conjugate Hilbert spacebecomesa B-A bimodule. We denote the adjacency matrix of by
and the Perron-Frobenius
eigenvector for
by which satisfy the condition. Here is the PerronFrobenius eigenvalue of.
Now we define the annihilation operator
by the following.
$We\mathrm{n}zl$$\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r}$$\mathrm{s}$$\mathcal{G}$$\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\mathcal{G}0$$\mathcal{G}$$\mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$$C^{*}$$u\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{0}\mathcal{G}$$e\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}^{(0,1}$$\mathcal{G}$$x\cdot\xi\cdot y=\delta_{x,S(}\delta\xi$$r(\xi),y\epsilon$$s(\xi)$$(0,1)\mathcal{G}$$\mathcal{G}$$\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{0}\mathcal{G}$$x\in \mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{0}}\mathcal{G},$$A$$v\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}^{1}\mathcal{G}$$y\in \mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$$\mathrm{v}_{\mathrm{e}\mathrm{r}\mathrm{t}^{1}}\mathcal{G}$$(0,1)\mathcal{G}$$\xi\in$$r(\xi)$$\xi$$\overline{H}$$\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{l}\cdot \mathrm{t}$$\overline{\xi}\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}^{(1,0}$$\mathcal{G}$$\overline{H}$$\mathcal{G}$$\triangle_{\mathcal{G}}$$\triangle_{\mathcal{G}}$$\triangle_{Q\mu}=\beta\mu$$\mu$$\beta$$\triangle_{\mathcal{G}}$$c\in \mathrm{H}_{0111}(AHB\otimes{}_{B}\overline{H}_{A}, A)$$c(\xi\otimes\overline{\eta})=\delta_{\xi,\eta}\mu(\gamma(\xi))^{1/}2\mu(S(\xi))-1/2$

for.

$s(\xi)$

Edge .
Its adjoint operator is called the creation operator which is given by the following.
$\xi,$$\eta\in$$\mathcal{G}$$c^{*}(_{X)}= \sum_{=\xi\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}Q,s(\xi)x}\xi\otimes\overline{\xi}\mu(r(\xi))^{1/}2(\mu S(\xi))^{-1/2}$.

We draw diagrams for the operators and as in Figures 1 and 2.
We remark that the composition of these operators
becomes scalar multiplication
operator . We define the Jones projection on the path spaces on the graph by $e=$
. For these operators we draw pictures as in Figures 3 and 4.
$c^{*}$$c$$c\cdot c^{*}$$\beta$$\mathcal{G}$$\beta^{-1}c^{*}\cdot c$$\Pi|$$c\cdot c^{*}=$$=\beta$$w$

Figure 3:

109

$H$

$\overline{LT}$$e=\beta^{-1}c^{*}\cdot c=\beta^{-1}$$Il$$t\mathrm{f}$Figure 4: The Jones projection$p_{n}=1-e_{1}\vee e_{2}$ $\mathrm{v}\cdots\vee en-1$

$=$

$=$

$|n$

.

Figure 5: Wenzl projector$p_{n}$, i.e. Path
We denote the set of paths of a graph with length by Path$s(\xik+1)=r(\xi_{\iota)}.$$\xi_{n})|\xi_{k}\in$$\xi=(\xi_{1},$Edge}, and denote the Hilbert space{)by. Note that times (relative) tensor
with orthonormal basis)
produce the path Hilbert space.
products
operators$c_{2,.n-1},.C$ alld that of
We can define the sequence of)on the Hilbert spacedepending on thethe Jones projections
position where they act.
$\xi_{2},$$()n\mathcal{G}=$$(n)\mathcal{G}$$n$$\mathcal{G}$$\mathcal{G},$$\cdots,$$\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$$\xi\in \mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{()}n\mathcal{G}$$n$$\mathcal{G}$$\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$${}_{A}H_{B}\otimes{}_{B}\overline{H}_{A}\otimes\cdots\otimes {}_{A}H_{B}(\mathrm{o}\mathrm{r}_{B}\overline{H}_{A})$$\mathcal{G}$$\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}/\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$c_{1},$$\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$$e_{1},$$e_{2},$$\mathcal{G}$$\ldots e_{n-1}$)
onis defined by $p_{n}=1-e_{1}\vee e_{2}\vee$Definition 2.1 The Wenzl projectors
draw the picture in Figure 5 for the enzl projector . The space ofV.
with length on a graph is defined byessential
point
lengthwithpaths
startingof a graphWe denote the space of essential.
and end point by...

$\mathrm{H}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(}n$$\mathcal{G}$$p_{n}$$\dot{\mathrm{W}}$$e_{n-1}.\dot{\mathrm{W}}\mathrm{e}$$p_{n}$$p\mathrm{a}ths$$n$$\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}(n)\mathcal{G}=p_{n}\cdot \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}$$\mathcal{G}$$\mathcal{G}$$y$$n,\acute{\mathrm{w}}\mathrm{i}\mathrm{t}\mathrm{h}$$x$$\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{x}(n,)\mathcal{G}y$

We remark that the space of essential paths can be defined as follows.
$\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}$–$\{\xi\in \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{(n})\mathcal{G}|e_{k}\xi=0\mathrm{f}\mathrm{o}1k=1,2, , \cdots, n-1\}$$=${$\xi\in \mathrm{H}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)\mathcal{G}|c_{k}\xi=0$

for $k=1,2,$$\cdots$$,$, $n-1$ }.

The following Mod erated Pascal rule is quite useful to count a dimension of essential
paths.
$\dim \mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(n+)}a,x1\mathcal{G}=\xi\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}\mathcal{G}\sum_{\xi r()=x},$

dinl$\mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{a,S(\xi}(n)\mathcal{G}-\dim \mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}(n-1)\mathcal{G})a,x$

For the proof of this rule, see [24, Section 5].
2.2. Extension of recoupling model and the double triangle algebra. Next we define anextended model of Kauffman-Lins’ recoupling theory ([17]) from a viewpoint of subfactor
theory by using the notion of essential paths. First we remark that the recouplingby using the fusion rule algebra of sector
for $q=e^{i\pi/N}$ a root of unity can be
fionl the Jones’ subfactor with principal
arising-synlbols
quantum
bimodule)and(or$\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}$$1^{\cdot}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{i}_{\mathrm{Z}\mathrm{e}}\mathrm{d}$$6\mathrm{j}$110$\in \mathrm{H}_{0}\mathrm{m}(\sigma_{n},\otimes\sigma n’\sigma_{k})$.

Figure 6: An intertwiner
$b$

$d$

$\mathrm{F}\mathrm{i}\mathrm{g}\iota 11^{\backslash }\mathrm{e}7:\mathrm{R}^{1}‘ \mathrm{c}()\iota 11)1\mathrm{i}_{1}$represent an intertwiner
as ingraph $A_{N-1}$ . For example the trivalenttois an irreducible sector (or bimodule). Herein$A_{N-1}$
type
prillcipal
graph
the
of
Jones’
the distinguishedj-th vertex
subfactors. The other notions in the recoupling theory stlch as -evaluations, tetrahedralin ternls of sectors and intertwiners
nets and (quantum) -symbols will bearising from the Jones’ subfactor. Especially we llave the recoupling as in Figure 7. Herethe coefficient is given as follows.
$\mathrm{v}\mathrm{e}\mathrm{l}\cdot \mathrm{t}\mathrm{e}\mathrm{x}$$\mathrm{H}\mathrm{o}\mathrm{m}(\sigma_{m}\otimes\sigma_{n}, \sigma_{k})$$\mathrm{F}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{l}\cdot \mathrm{e}6$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}11\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$$\sigma_{j}$$\mathrm{v}\mathrm{e}\mathrm{l}\cdot \mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}$$\mathrm{f}\mathrm{r}\mathrm{o}\ln$$\theta$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{l}\backslash \mathrm{e}\mathrm{t}\mathrm{e}\mathrm{d}$$6\mathrm{j}$$( \mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f})_{n}==\frac{\mathrm{T}\mathrm{e}\mathrm{t}\{\begin{array}{lll}Cl b nc d m\end{array}\}\triangle_{n}}{\theta((l,b_{\mathrm{t}}/\mathrm{t})\theta(_{C,d,n)}}$Here$\theta(a, b, c)$means the -evaluation and$\theta$Tet
represents a value of the tetrahedral net. (See [17], [24, Section 12].) The special casewhen $m=0$ is given in Figure 8 and we use this to define the convolution product of the
double triangle algebras.
Now fix a recoupling model which corresponds to a Perron-Frobenius eigenvalue
with the same Perron-Frobenius
and let $K$ be one of the Dynkin diagrams
as in the left handeigenvalue. We draw a picture for an essential path$A$$\beta$$A_{n},$ $D_{n},$ $E_{6,7,8}$$\xi\in \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(n)Kx,y$

”.

$|$

$|$

$=$

a

$h$

$\sum_{n}\frac{\triangle_{n}}{\theta(a,b,n)}$

$b$

$o$

Figure 8:

111
$v$

Figure 9:

Figure 10:
side of Figure 9 or we simply draw the picture in the right hand side. (See [24, Section10].) Then a double triangle algebra $A$ is defined as an algebra which elements are linear
combinations of pairs of essential paths as in Figure 10. We call a double triangle algebra
defined on the graphs $K$ and which correspond to upper and lower horizontal graphs
respectively a K-L double triangle alge . Two products are defined on this algebra.
One is. product defined as in Figure 11. The other product called convolution prod $uct$
the element of the right hand
. We
is defined as in Figure 12 and denotedside in this figure as in Figure 13 by using recoupling and the equality in Figure 14. The-operation for the convolution product on the double triangle algebra is given by Figure$L$$\mathrm{b}ra$$\mathrm{b}\mathrm{y}*$$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{P}\mathrm{o}\mathrm{s}\mathrm{e}$

$*$

15.

2.3 Ocneanu’s chiral projectors. We recall that special elements in $(A, *)$ is defined by$\delta_{\xi_{-},\eta+}$Figure 11:. product on the double triangle algebra

112

Figure 12: The convolution product on the double triangle algebra

Figure 13:
Figure 16 and Figure 17 and give the definition of Ocneanu’s chiral projectors.

Definition 2.2 The chiral projectors
and -which are central projections in the
double triangle algebra $(A, *)$ are defined as in Figure 18 and 19. The product of the twochiral projectors
is called the am bichiral projector and is denoted by.
$\Psi_{+}$$\Psi$$\Psi_{+}*\Psi_{-}$$\Psi_{\pm}$

2.4 Gaps on the Dynkin diagrams and inim central projections. The gap and O-gap
of a finite graph are numbers (positive integer or ) defined by the following. (See [24,Section 17])$\mathrm{m}$$\mathrm{a}l$$G$

gap $(c)$$\mathrm{O}-\mathrm{g}\mathrm{a}\mathrm{p}(c)$$\infty${

for all$\equiv$$\min$$\equiv$$\min\{n>0|\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}\mathrm{h}^{(n)}0,0G\neq 0\}$$n>0|\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(}|l)Ga,a\neq 0$$a\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}G$},.

Here represents the distinguished vertex of thegraph , i.e., the vertex ofwhich has the smallest Perron-Frobenius weight. The gaps and the -gaps of the Dynkin
diagrams are given in the following table.
$0$$\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{x}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}$

$G$

$G$

$0$

Figure 14:

113$=( \frac{\mu(y)\mu(_{\sim}^{\sim})}{\mu(x)\mu(lv)})^{1/2}$

Figure 15: The$*-\mathrm{o}\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}\mathrm{f}\mathrm{o}\mathrm{l}\backslash \mathrm{t}\mathrm{h}\dot{\mathrm{e}}$convolution product$x$$\alpha$$y$$x$$\overline{\alpha}$$y$$\equiv\sum_{x,y,\alpha}\overline{\mathrm{o}}_{x}^{1/2}\overline{\mathrm{O}}_{y\uparrow x}^{1}/2\mathrm{O}$

Figure 16:

Let $K$ be a connected finite bipartite graph and $(A, *)$ a double triangle algebra on $K$
endowed with the convolution product. The following proposition shows that there areon the double triangle
and.
.
two finite family of minimal central projections
$(A,
*)$
.
algebra$\{p_{k}^{+}$ $\}_{k}$$\{p_{k}^{-}$ $\}_{k}$Proposition 2.3 ([24, Proposition 17.3, Corollary 17.4]) Two
by Figure 20 and 21 are minimal central projections if$element\mathit{8}p_{k}+$$k<\mathrm{g}\mathrm{a}\mathrm{p}(K)/2$

and.

$p_{k}^{-}$

definedFor orthogonality of these projections we have the following propositions.

Proposition 2.4 ([24, Proposition 18.1]) The.
are mutually orthogonal if$\gamma’$? inimal central projections$k<\mathrm{g}\mathrm{a}\mathrm{p}(I_{1}’)/2$$x$$\alpha$$y$$\equiv\sum_{x,y,n\alpha},\overline{\mathrm{O}}_{xJ}1/2\overline{\mathrm{o}}_{1}^{1}\mathrm{O}_{7}/2l$$x$$\mathrm{F}\mathrm{i}\mathrm{g}\iota\iota \mathrm{r}\mathrm{e}17$:$\overline{\alpha}$$y$$p_{k}^{+}$(resp.

$p_{k}^{-}$

)

114

Figure 18:

Figure 19:
Proposition 2.5 ([24, Proposition 18.2]) The minimal central projectionsin the previous section are mutually orthogonal if $k\neq l$ and$p_{k}^{+}$$k+l<0- \mathrm{g}\mathrm{a}\mathrm{p}(K)$3and$p_{l}^{-}$

as.

Operations and equivalence relations of connections, conjugate pairs andAsaeda-Haagerup’s generalized open string bimodules

In this section we define some operations on the set of connections such as direct sum,composition, irreducible decomposition and conjugation etc. (See [24, Section 20].) We
also define some equivalence rela.tions OI1 it. 1Ve sllall give a natural identification ofconnections and define the above operations on the set of equivalence classes of connections
with the identification. These operatiolls are originally defined by Ocneanu ([24, Section20]). Later Asaeda and Haagerup ([1])
the uotion of generalized open string
bimodules which is aof openbimodule of Ocneanu ([20]) alld Sato([30]) and they clarified the relation between connections and bimodules. We rernarkthat the identification of connections given in this section is different Asaeda-Haagerup’s
setting. Here we will also give their original definitions for reader’s convenience.
$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{l}\cdot \mathrm{o}\mathrm{d}\iota \mathrm{t}\mathrm{C}\mathrm{e}\mathrm{d}$$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}_{0}11$$\mathrm{s}\mathrm{t}\mathrm{l}\cdot \mathrm{i}\mathrm{l}$Remark 3.1 In this paper connections on four graphs
as in Figure 22 arealways assumed to have connected horizontal graphsand . We do notthat
the vertical graphs andare connected. The word connection always means bi-unitaryconnection in this paper. So we will often use
word connection instead of ‘ $bi_{-un}i$ faryconnection’ for simplicity.
$\mathcal{G}0,$$\mathcal{G}_{0}$$\mathcal{G}_{1}$$\mathcal{G}_{1},$$\mathcal{G}2,$$\mathcal{G}_{2}$$\mathcal{G}_{3}$$\mathrm{a}\mathrm{s}\mathrm{s}\iota \mathrm{l}\mathrm{m}\mathrm{e}$$\mathcal{G}_{3}$$\mathrm{t}1_{1}\mathrm{e}$)First we define the notion of direct sunl,, irreducibility and conjugation on
the set of -unitary connections. (See [24, Section 20], [1, Section 3].)$\mathrm{c}\mathrm{o}\mathrm{n}$$\mathrm{P}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{b}\mathrm{i}$$\leq\Psi_{+}$$p_{k}^{+}\equiv$

Figure 20:

115$\leq\Psi_{-}$$p_{k}^{-}\equiv$

Figure 21:$\mathcal{G}_{0}$$\mathcal{G}_{1}$$\mathcal{G}_{3}$$\mathcal{G}_{2}$Figure 22:
be two -unitary conllections on
andDefinition 3.2 ([24, Definition 20.2]) Let, then a direct sum of these biand
four graphs$W$
on four graphs
unitary connections is a -unitary connection
defined by the following. See Figure 23.
$W_{2}$$W_{1}$$\mathcal{G}0,$$\mathcal{G}_{1},$$\mathcal{G}2,$$\mathcal{G}_{0},$$\mathcal{G}_{3}$$\mathcal{G}_{1’ 2}’\mathcal{G},$$\mathrm{b}\mathrm{i}$$\mathcal{G}’3\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}}}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{y}$$\mathcal{G}_{0},$$\mathrm{b}\mathrm{i}$$W_{1}(\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi 0}}.\mathrm{i}.\xi 1)$$W$$\mathcal{G}2,$$\mathcal{G}3\mathrm{u}\mathcal{G}_{3}’$if$\xi_{0}\in \mathcal{G}_{0},$ $\xi_{1}\in \mathcal{G}_{1},$$\xi_{2}\in \mathcal{G}2,$$\xi 3\in \mathcal{G}_{3}$if$\xi_{0}\in \mathcal{G}_{0},$ $\xi 1\in \mathcal{G}_{1}’,$$\xi_{2}\in \mathcal{G}_{2},$$\xi_{3}\in \mathcal{G}_{3}/$

,

,

( )
$\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi}}.0$$\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’,$$–$$\downarrow..\xi_{1}$$\{$$W_{2}(\xi_{3}\mathrm{i}_{\overline{\xi_{2}}}^{\underline{\xi_{0}}}.\mathrm{i}.\xi 1)$

,

,

$.$

.
$0$

.

We denote a direct sum.
by$W_{2}$$\mathrm{b}\mathrm{i}$,otherwise.

-unitary connection $W$ of two$\mathrm{b}\mathrm{i}$-unitary connections$W_{1}$

and$W_{1}\oplus W_{2}$Definition 3.3 ([24, Section 20]) Let $W$ be a connection on four graphs
which has the conlmon
and $W’$ a connection on other four graphs$\mathcal{G}0,$$\mathcal{H}_{0}=\mathcal{G}_{2},$$W_{1}$$\mathcal{G}_{2}$$\mathcal{H}_{2},$$\mathcal{G}_{1}\oplus \mathcal{G}_{3}’$$W_{2}$$\mathcal{H}_{3}$$\mathcal{G}_{0}$$\cdot \mathcal{G}_{0}$$\mathcal{G}_{0}$$\mathcal{G}_{3}$$\mathcal{H}_{1},$$\mathcal{G}_{1}’$$=\mathcal{G}_{3}\mathrm{u}\mathcal{G}_{3}’$$\mathcal{G}_{2}$$\mathcal{G}_{2}$Figure 23: Direct sum of two$W_{1}\oplus W_{2}$$\mathrm{b}\mathrm{i}$-unitary connections$\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’$$\mathcal{G}_{1},$$\mathcal{G}2,$$\mathcal{G}_{3}$116$\mathcal{G}_{0}$$W^{\prime/}$$\mathcal{G}’$$\mathcal{G}^{\prime/}$$\mathcal{H}_{2}$Figure 24: Composition (or product) of two connections, then we define a composition of thelll by a -unitary connection
graphobtained by connecting these graphs, making products of both connections and summingthem over all the common horizontal edges as in Figure 24. Here the vertical graphs
andof the composed connection
will change by this construction. We denote thecomposite connection
by $W\cdot W’$ or simply $WW’$ .
$\mathcal{H}_{0}=\mathcal{G}_{2}$$W^{\prime/}$$\mathrm{b}\mathrm{i}$$\mathcal{G}’$$W^{\prime/}$$\mathcal{G}^{\prime/}$$W^{\prime/}$Definition 3.4 ([24, Definition 20.3]) A -unitaryon four graphs
are called reducible if there exist two -unitary connections
andon four graphs
andsuch that the direct sum of them produces $W$
as in Figure 25 up to vertical gauge choice. We call a bi-unita.ry connection $W$ irreducible
if it is not reducible.
$\mathrm{b}\mathrm{i}$$\mathrm{c}\mathrm{o}\mathrm{n}11\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}W$$\mathrm{b}\mathrm{i}$$\mathcal{G}_{0},$$\mathcal{G}_{1},$$\mathcal{G}2,$$\mathcal{G}_{3}$$\mathcal{G}0,$$\mathcal{G}’1’ \mathcal{G}2,$$\mathcal{G}0,$$W_{1}$$\mathcal{G}_{1},$$\mathcal{G}2,$$\mathcal{G}_{3}$$W_{2}$$\mathcal{G}’3\mathrm{P}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$Remark 3.5 We remark that this definition of reducibility is the same as Asaeda-Haagerup’s([1, Section 3]). In general the reducibility up to vertical gauge choices is different fromthat up to total gauge choices. But the next
shows that both definition coincides
if two horizontal graphs
andas in Figure 22 are trees, i.e., if both graphs only
have single edges and have no cycle. (cf. [1, Relnarl\’’ Section 3].) Especially in the casewhen the horizontal graphs are both alllong the Dynkin diagrams
the twoequivalence relations coincide.
$\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{a}$$\mathcal{G}_{0}$$\mathcal{G}_{2}$$A_{n},$ $D_{n},$ $E_{6,7,8}$

Lemma 3.6 If one of the two horizontal
a tree, then any gauge choices on
can be forced to put on vertical gauge choice. In particular if the two horizontal
andare both trees, then two equivalence relation ‘total gauge choice’ and ‘uerticalgauge choice’ on the set of connections on the four graphs coincide.
$graph_{6}\mathcal{G}_{0}w$$\mathcal{G}_{0}$$\mathcal{G}_{0}$$graph_{\mathit{8}}$$\mathcal{G}_{2}$Proof Take a gauge choiceon an edge of . We can easily see thatthe same gauge choice can be given by taking a gauge choice on vertical graphs
andwhich consists only gauge on some edges ofandon some edges on . (This canbe shown by taking a vertical gauge step by step.) The existence of such a gauge
is assured by the fact that there is no multiple edge and no cycle on the graph .
$\alpha\in \mathrm{C}(|\alpha|=1)$$\mathcal{G}_{0}$$\mathcal{G}_{1}$$\alpha$$\mathcal{G}_{1}$$\mathcal{G}_{3}$$\alpha^{-1}$$\mathcal{G}_{3}$$\mathrm{c}\mathrm{h}_{0}\mathrm{i}\mathrm{C}\mathrm{e}\coprod$$\mathcal{G}_{0}$Remark 3.7 The condition in the above lemma that the graphdoes not have any cycle)$(n\geq 1)$ . It is easy tois necessary. Consider thewhen all the four graphs are
see that you can not force to put horizontal gauge choice to verticalin these examples.
that even if the two homzontal graphs consist of single edges, it may happen
that the above two equivalence
do not coincide.
$\mathcal{G}_{0}$$ca\mathit{8}e$$A_{2n+1}^{(1}$o.nes$Thi\mathit{8}\mathit{8}hows$$relation\mathit{8}$117;$\mathcal{G}_{1}$$W_{1}$$\mathcal{G}_{2}$$\mathcal{G}_{2}$$\mathcal{G}_{2}$$\mathcal{G}_{3}\oplus \mathcal{G}_{1}$$W_{2}$$W$$=\mathcal{G}_{1}\mathrm{u}\mathcal{G}_{1}’$$\mathcal{G}_{3}’$$\mathcal{G}_{4}$$\mathcal{G}_{4}$$\mathcal{G}_{4}$$\mathcal{G}_{3}\mathrm{u}\mathcal{G}_{3}’$Figure 25: Reducibility of a$\mathcal{G}_{0}$$\mathrm{b}\mathrm{i}$-unitary connection $W$$\overline{\mathcal{G}}_{0}$$\mathcal{G}_{3}$$\mathcal{G}_{3}$$\overline{\mathcal{G}}_{3}$$\tilde{\mathcal{G}}_{3}$$\mathcal{G}_{0}$$\tilde{\mathcal{G}}_{0}$Figure 26: Four connections associated to a connection$W_{0}$In this paper an equivalence relation on connections will mean vertical gauge choice
unless otherwise stated. Before we define a natural identification of connections we remark
first that four different connections which are transferred each other by renorlnalization
as in Figure 26, where represents
rule are associated to one -unitary connection
the graph with reversed orientation.
$\tilde{\mathcal{G}}$$W_{0}$$\mathrm{b}\mathrm{i}$$\mathcal{G}$) as in Figure 26
and(resp.
andDefinition 3.8 We say two connections
respectively.
andby
are horizontally (resp. vertically) conjugate. We denote. We
simply a conjugate connection of
We will call vertically conjugate connectionin
this
conjugationaverticalbydefine a conjugation operation on a set of connections
sense. The conjugation operation is often denoted by . So we will also adopt the notation
for a (vertically) conjugate connection of $W$ .
$W_{3}$$W_{0}$$W_{1}$$W_{0}$$\mathrm{t}\mathrm{h}\mathrm{e}\ln$$W_{0}^{h}$$W_{0}^{v}$$W_{0}$$W_{0}^{v}$$-.$$\overline{W}$In this paper, we regard two horizontally conjugate connections as the saIlue one. In
as a pair of two connections $(W_{0}, W_{1})$other words, we always consider a connection
which are horizontally conjugate each other. We call such a pair horizontally conjugate
pair. By this identification the relation ‘ vertically conjugate’ still make sense. Moreover,
oice’ and ‘ tical gauge choice’
we consider the two equivalence relation ‘ tot
on this identified set of connections. Again they are still equivalence relation on it.
In the following we consider the set of connections which have fixed common horizontal
graphs. We will use the following terminology. (See [24, Section 20].)$W_{0}$$\mathrm{a}lgau\mathrm{g}eCl1$$ve\mathrm{r}$bipartite graphs. A bi-unitary
be two connected
Definition 3.9 Let $K$ andasconnection on four graphs is called a K-L -unitary connection if it has the graph
in
as
27.
Figuregraph
horizontal
a
lower
as
an upper horizontal graph and the graph$L$$\mathrm{f}\mathrm{i}_{11}\mathrm{i}\mathrm{t}\mathrm{e}$$bi$$I\mathrm{i}^{\Gamma}$$L$

Note that we can naturally define the operations such as direct sum, conjugation andirreducible decomposition on the set of equivalellce classes of connections with the above118$I_{1}’$$W$$L$

Figure 27: K-L$\mathrm{b}\mathrm{i}$-unitary connection$I_{1^{od}}’d$$\overline{I1\prime}I\searrow’*_{f_{\mathrm{t}}}eve,n$$I_{1,*_{h}}^{\nearrow,I\mathrm{i}}ev\underline{enI_{1^{od}}^{\vee}\prime}d$Figure 28: A graph$I\iota’$and$\overline{I\iota^{r}}$identification. Before we define the colluposition (or product) operation on it we needsome more notations and terminology.
We often denote a graph $K$ with a distinguished vertex
by a pair. Let
be a connected finite bipartite graph with a distinguished vertex. A vertex ofwith the same (resp. different) colour as
is called even (resp. odd) vertex. The setof even and odd vertices are denoted byandrespectively. In the following, the notation $K$ will represent the graphwhenever we consider a horizontal graph$K$ with their even vertices on the left hand and odd
vertices on the right hand. The
$K$
graph
with reversed orientation, that is, one with their odd vertices on the left andeven vertices on the right will be denoted by . (See Figure 28.)
Note that when two horizontal graphs
and are connected finite bipartite with
distinguished vertices, four kinds of K-L -unitary connections will be distinguished by
the above notation which respects the orientations of the graphs. That is, there are fourkinds of K-L -unitary connections depending on which graphor and or they
actually have as horizontal graphs.
For a given connection we can naturally associate the index of its generalized open
string bimodule [1]. We call it an index of a connection. This value is the same as thesquare root of the index of subfactor constructed
the connection.
Now we define a composition (or product) of two (pairs of) connections.
$(I_{\mathrm{t}}’, *_{K})$$*_{K}$$(I\mathrm{i}’, *_{K})$$I\mathrm{i}^{r}$$*_{I\iota’}$$I\iota\prime odd$

$I\mathrm{i}\prime even$$(K, *_{I\iota’})$$\overline{\mathrm{A}’}$$I\iota^{r}$$L$$\mathrm{b}\mathrm{i}$$\overline{K}$$\mathrm{b}\mathrm{i}$$I\mathrm{i}^{r}$$L$$\overline{L}$$\mathrm{f}\mathrm{r}\mathrm{o}\ln$Definition 3.10 Let $(K, *_{K}),$ $(L, *_{L})$ andbe three connected finite bipartite
graphs with distinguished vertices. Let be a K-L -unitary connection and a L-M-unitary connection. We regard these as two pairs of connections
andas in the above setting. Define a(or product) of two (pairs of) connections
and by a (pair of) connection obtained by the following procedure. When andhave the common graph as bottom and top, then we just make aproduct
and regard this as a )of connectiolls. When one of andtop
have the graph as aoralld tlle other have the graph , then we
make a productor a product. Again we regard it as a pair of connections. A product of two (pairs of) connections and in the above sense will
) when it does not cause any confusion.
(or
be simply denoted by$(\mathit{1}\mathcal{V}I, *_{M})$$\mathrm{b}\mathrm{i}$$\beta$$\alpha$$(\alpha, \alpha^{h})$$\mathrm{b}\mathrm{i}$$(\beta, \beta^{h})$$comp_{\mathit{0}}\mathrm{s}iti_{\mathit{0}}\mathit{1}1$$\alpha$$\beta$$\beta$$\alpha$$L$$\mathrm{g}1^{\backslash }\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{s}1^{\cdot}\mathrm{e}\mathrm{s}_{\mathrm{P}}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}\mathrm{l}\mathrm{y}$$\alpha\cdot\beta$$\beta$$\mathrm{a}\mathrm{i}\mathrm{r}$

$1$

$L$

$(\alpha\cdot\beta, \alpha^{h}\cdot\beta^{h})$$\alpha$$\tilde{L}$$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{l}\overline{\mathrm{l}}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}1_{1}$$\alpha\cdot\beta^{h}$$\alpha^{h}\cdot\beta$$(\alpha\cdot\beta^{h}, \alpha^{h}\cdot\beta)$$\alpha$$\alpha\cdot\beta$$\beta$$\alpha\beta$

the above definition we are now ready to deal with a system of K-K bi-unitaryconnections which is closed under direct sum, product, conjugation and irreducible decomposition. The precise definition is given ill the following.
$i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$119$\mathcal{G}$$\mathcal{H}$$\mathcal{G}$$\mathcal{H}$Figure 29:

be a set of equivDefinition 3.11 Let $K$ be a connected finite bipartite graph andalence classes of horizontally conjugate pairs of K-K -unitary connections with respect
is called a system ofto the equivalence relation ‘vertical gauge choice’. Then a set
if it is closed
system-unitary
connectiona
K-K -unitary connections orunder direct sum, product, conjugation and irreducible decomposition. Another word
fusion rule algebra of K-K -unitary connections’ will often be used in the same meaning. A K-K -unitary connection system is said to be Finite if it contains only finitely
many irreducible connections. The rule of irreducible decomposition of products of two
irreducible connections in a system is called the fusion rule of the system.
$K\dot{\mathcal{W}}_{K}$$\mathrm{b}\mathrm{i}$$K\mathcal{W}_{K}$$\mathrm{b}i$$\mathrm{s}\mathrm{i}_{1}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{y}$‘$I_{1}^{\nearrow}- K\mathrm{b}i$$bi$$\mathrm{b}\mathrm{i}$Remark 3.12 Note that all the above operations are ) $ell$ -defined on a set of equivalenceclasses of horizontally conjugate pairs of K-K -unita connections with respect to thetical gauge choice’. It should be remarked that if we adopt anequivalence relationequivalence relation ‘total gauge choice’ instead of ‘vertical gauge choice’, composition of
two connection will not necessarily be well-defined.
$u$$\cdot$$bi$$7^{\cdot}y$$\ell_{Ve\mathrm{r}}$Example 3.13 Obviously there are two trivial exalllples of K-K -unitary connection
systems. One is a system which consists of only trivial (identity) connection as its irreducible object. The other is a system consisting of all K-K -unitary connections. For
exists a system generated by them.
any given family of K-K -unitaryinthissetting.
sellseakes
ofa
So the word generator$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{0}11\mathrm{S}\mathrm{t}\mathrm{h}\mathrm{e}1^{\backslash }\mathrm{e}$$\mathrm{b}\mathrm{i}$$\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{n}\overline{1}$Example 3.14 Let $N\subset M$ be a subfactor with finite index and finite depth. Then we
obtain a (fiat) -unitary connection $W$ on the four graphs as in Figure 29 by Ocneanu’s
Galois functor. Where is the principal graph alld $H$ is the dual principal graph. As. The composition ofusual we denote the (vertical) conjugate connection of $W$ by
$W$
and a
-unitary connection
produce aandtwo connections-unitary connections generated by. A systel]] of-unitary connection
is finite by the finite depth assumption. This can be regarded the same system as N$N$ bimodules arising from the subfactor $N\subset M$ .
connections
Another system ofalso corresponds to a system of M-M bimodule arising
generated by a connection
from the subfactor. This shows that for every subfactor with finite index and finite depth
we can associate two systems of -unitary connections. We will obtain many non-trivial
finite systems of connections in this way.
$\mathrm{b}\mathrm{i}$$\mathcal{G}$$\overline{W}$$\overline{W}=W^{v}$$\mathrm{b}\mathrm{i}$$\mathcal{G}- \mathcal{G}\mathrm{b}\mathrm{i}$$\overline{W}W$$W\overline{W}$$\mathcal{H}- \mathcal{H}$$W\overline{W}$$\mathcal{G}- \mathcal{G}\mathrm{b}\mathrm{i}$$\mathcal{H}- \mathcal{H}$$\overline{W}W$$\mathrm{b}\mathrm{i}$Example 3.15 Another fundamental example is a system generated by one bi-unitary
and $(L, *_{L})$ be two connectedconnection, i.e., a singly generated system. Let
finite bipartite graphs with the same Perron-Frobenius eigenvalue. Any K-L bi-unitarysubfactor $N\subset M$ by a string algebra construction.
connection $W$ yields a hyperfinite
This connection $W$ generates a systenl of four kinds of generalized open string$(I\iota^{r}, *_{K})$$\mathrm{I}\mathrm{I}_{1}$$\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{d}\mathrm{U}\mathrm{l}\mathrm{e}\mathrm{s}$120([1]) which has the same fusion rule as theofarising from the subfactor$N\subset M$ . By looking at the corresponding-unitary connections, we get a system of fourkinds of connections, i.e., K-K, K-L, L-K and L-L connections. Especially we obtain
K-K and L-L connection system in this way. If the subfactor $N\subset M$ has finite depth,then both systems will be finite. And if it has infinite depth, they become infinite systems.
$\mathrm{b}\mathrm{i}\mathrm{n}1\mathrm{o}\mathrm{d}_{\mathrm{U}\mathrm{l}\mathrm{e}}$$\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}}\mathrm{e}\ln$$\mathrm{b}\mathrm{i}$Remark 3.16 The procedure in Example 3.15 looks similar to that of Example 3.14. Actually the former is the special case of the latter. Here we remark that the latter is much
more general because the connections appear in the former example is always flat andinfinite system can not be obtained by the former procedure. Moreover a fiat connection
obtained by the Galois functor is very special because they have (dual) principal graphs as
the four graphs. The following example
this speciality of fiat connections obtainedby the Galois functor. Flat
obtained by Sato’s procedure ([29, Theorem 2.1])
which is a generalization of the example of [8] show that any finite depth subfactor generated by a (not necessarily flat) -unitary connection $W$ can be reconstructed by a different
flat connection
which is a horizontally composed connection byflat part con(See [29], [30]). Hence there are many examples of finite depth
nectionwhich are constructed by a flat connection that does not come from the Galois functor.
$show\mathit{8}$$connecti_{on}\mathit{8}$$bi$$W_{f}\cdot W$$it\mathit{8}$$W_{f}$$\mathit{8}ubfact_{\mathit{0}}rs$

Example 3.17 Let $K$ be one of the Dynkin diagrams. It is known that
there are at most two non-equivalent -unitary connections on the four graphs which
are all the same graph $K$ and trivially connected. (There are only one non-equivalentconnections in the case ofand exactly two mutually colnplex conjugate non-equivalentconnections in the case of) We call
fundamental
nections of the A-D$E$ Dynkin
diagrams. We denote one of them by $W$ and the other by . Let $N\subset M$ bea subfactor constructed by $W$ in the horizontal direction.
First we consider the system generated by a single connection $W$ . In this case thesystem is finite and has the same fusion rule as that of the system of bimodules arisingfrom the subfactor $N\subset M$ .
Next consider the system generated by the two -unitary connections
and . In
this paper we will mainly deal with this systen). It turns out that this system is finite andall the irreducible K-K -unitary connections appear in this system. (See section 5.)$A_{n},$ $D_{n},$ $E_{6,7,8}$$\mathrm{b}\mathrm{i}$$\mathrm{A}_{n}$$D_{n},$ $E_{6,7,8}.$$\mathrm{t}\mathrm{h}\mathrm{e}\ln$$co\mathrm{n}$$\overline{W}$$\overline{W}$$\mathrm{b}\mathrm{i}$$W_{\Gamma}$$\mathrm{b}\mathrm{i}$Remark 3.18 It a remarkable fact that there are only finitely many K-K irreducible-unitary connections on the Dynkin diagrams . This is one of the very special properties of the Dynkin diagrams. We should compare it to the following example. Consider thecase when all the four graphs are the same graph the principal graph of Goodman-de laHarpe-Jones subfactor with index(see Figure 30) which arise from an embedding
thealgebratothat. In this case there exists one paramofofeter family of (hence uncountably many) non-equivalent -unitaryon the. These connections are automatically irreducible by the criterion of Asaedafour
Haagerup [1, Corollary 2, Section 3] This means that even if we fix not only the twohorizontal graphs but all the four graphs it can happen that uncountably many irreducible
non-equivalent connections exist on the. The author would like to
his thanks
to Y. Kawahigashi for pointing out this example.
$i_{\mathit{8}}$$bi$$I\mathrm{i}^{r}$$a\mathit{8}$$\mathit{3}+\sqrt{3}$$A_{11}\mathit{8}tring$$E_{6}([\mathit{1}\mathit{4}], [\mathit{2}\mathit{5}])$$bi$$connecti_{on}\mathit{8}$$graph_{\mathit{8}}$$graph_{\mathit{8}}$$expres\mathit{8}$Remark 3.19 In this paper we mainly deal with the case when the graphis one of
the A-D-E Dynkin diagrams. It is natural to take the $usua,ldi\mathit{8}tinguiShedverteX*_{K}(i.e.$ ,$I\backslash ^{r}$

121

Figure 30: The principal graph of the$\mathrm{G}_{\mathrm{o}\mathrm{O}}\mathrm{d}\mathrm{n}1\mathrm{a}\mathrm{n}$-de la Harpe-Jones subfactor

the vertex with smallest Perron-Frobenius eigenualue) in these cases and we do so in thefollowing. We denote a hyperfinite
factor generated by the string algebra on the graph$K$ with the starting point
we willby the same notation . For a reversed graphagain
wealgebraanddenotestringits
point
startingaastotake the vertex nextofgenerating factor by the same . Then by generalized open string algebra construction
we can associate two different bimodules for a given horizontally conjugate pairof K-K -unitary connections. Again we deal with the pair of generalized open string
bimodules as a corresponding bimodule to the onginal pair of connections. By working on
this correspondence between pairs of connections and pairs of bimodule, we can show that
Frobenius reciprocity holds for the system of connections as in the next proposition.
$II_{1}$$\overline{I\mathrm{c}’}$$I\iota’$$*_{K}$$*_{K}$$\overline{l\mathrm{i}^{\Gamma}}$$([\mathit{1}f)$$bi$be three connected finite bipartite graphs with theandProposition 3.20 Let $K,$andbe three (pairs of) irreducible
same Perron-Frobenius eigenvalue. Let
respectively.
K-L,
and
K-M
L-M
are
which
-unitary connections
If appears times
and appears times intimes in, thenin the composite connection.
$L$$\mathit{1}\mathcal{V}I$$K\alpha L,$$L\beta_{M}$$\kappa\gamma_{\Lambda^{\prime I}}$

$bi$

$n$

$\gamma$$\alpha\beta$$\gamma\overline{\beta}$$\alpha appear\mathit{8}n$$n$$\beta$$\overline{\alpha}\gamma$

Proof Choose and fix distinguished vertices from even and odd vertices of each graphs$K,$
and $M$ . Then apply the correspondence between (pairs of) -unitary connectionsand (pairs of) generalized open string bimodules. We get the result from thereciprocity for bimodules ([22], [13, Section 9.8]).
$L$$\mathrm{b}\mathrm{i}$$\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}\square$

Let be a K-L -unitary connection. Then from the rule of irreducible) we get a graph which(orof finite product connections
an identity K-K
denotesHereis similar to the principal graph of a subfactor.
connection. We call this graph the principal fusion $grapl_{l}$ of a connection . Then we
can show the following by Asaeda-Haagerup’s criterion of irreducible decomposition ofconnections [1, Claim 1, Section 3]. This is wluat we observed in Example 3.14 and 3.15.

$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\ln_{\mathrm{P}^{\mathrm{O}}}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{b}\mathrm{i}$$\alpha$$Kid_{K}\cdot K\alpha L^{\cdot}L\overline{\alpha}K\ldots\kappa\alpha_{L}$$L\alpha_{K}$$I\mathrm{c}^{\prime id}IC$$\alpha$graph
Proposition 3.21 Let be a K-L -unitary connection. Then the principal
of is the same as the principal graph of the subfactor constructed from the connection.
$\alpha$$bi$$fu\mathit{8}i_{on}$$\alpha$$\alpha$4

Correspondence between connections and -representations of double triangle algebras$*$

and $(A, *)$ be the double
Let a graph $K$ be one of the Dynkin diagralnswith the convolution product. This is atriangle algebra corresponding to the graphfinite dimensional -algebra and is isomorphic to a finite direct sum of matrix algebras
as follows.
$A_{n},$ $D_{n},$ $E_{6},\tau,8$$I\iota’$$C^{*}$$(A, *)\cong\oplus_{i\in I}H_{i}\otimes\overline{H_{i}}\cong\oplus_{i\in I}\mathrm{E}\mathrm{n}\mathrm{d}(H_{i})$

.

122

$=$
$\sum_{i}$(coef) (i)Figure 31:
Here the index
are labelled by nlinilnal central projections in $(A, *)$ andarecorresponding finite dimensional Hilbert spaces. This identification of elements in $(A, *)$is written as an extension of recoupling as in Figure 31.
Now we give the following theorem which lneans that we have only to find all theminimal central projections in $(A, *)$ in order to classify all irreducible K-K bi-unitary$i’ \mathrm{s}$$H_{i}’ \mathrm{s}$connections.
Theorem 4.1 There is-one correspondence between unitary equivalence classes$matri_{C}ial*$-representations of the K-K double triangle algebra $(A, *)$ andof irreducible
equivalence
K-K -unztary connections.
irreducibleof$one-t_{\mathit{0}}$$bi$$cla\mathit{8}seS$

Remark 4.2 Equivalence relation on -unitaryconsidered in this theorem$ith$)‘vertical gauge choice’. But it coincides
ttotal gauge choice’ in this case as westated in Remark 3.5. (See Lemma 3.6.)$bi$$connecti_{on}\mathit{8}$

$is$

$u$

Proof A proof for the construction of a -unitary connection for a-representationand vice versa is shown in [24, Section 15]. So we only have to show that two unitarilyequivalent -representation give two equivalent
up to vertical gauge choiceand vice versa. It is easy to see that two equivalent connections up to vertical gauge choicegive two unitarily equivalent -representation because the unitary matrix which transfer
one representation to the other is given by the unitary atrix of gauge choice. For theother direction, first we have to find the vertical edges connecting two horizontal graphs $K$ .
-representation of the K-K double triangle algebra $(A, *)$ . Because
Let be aas in Figure 32 are utually orthogonal projections, theelements
matricesare diagonalized with only and 1 in the diagonal entries by a certain
unitary. We draw edges connecting the vertices. and with the same nulnbers as that
. In this way we get the vertical edges conllectingof 1 in the diagonalized matrix
the two horizontal graphs. We label these vertical edges by sonle index set A. Then we
define a connection value of a rectangle as in Figure 33 by the number
for
A. Hereandis an
in $A$ as in Figure 34 andrepresents a-th entry of the. If we restrict the map$A$from
to the complex numbers
defined as above to),
we get the connection map. Now the -unitarity condition of this connection follows
easily from the fact thatis a -representation. (See [24, Section 15].) Note thatin this procedure to get a connection
fronu a given -representation , it is easy
reducible, then the connecting vertical edges as
to see that if thedefined on the fourwell as the connections
decompose into some irreducible
components which corresponds to the irreducible components of the -representation .
So the irreducibility is preserved by this correspondence. Now it is easy to see that the$\mathrm{b}\mathrm{i}$$\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}*$$*$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{S}$$*$$11\overline{1}$$\Phi$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}\mathrm{i}\mathrm{a}\mathrm{l}*$$a_{x,y}\in A(x, y\in \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}K)$$\mathrm{n}\overline{1}$$\Phi(a_{x,y})$

$0$

$r$

$y$

$\Phi(c\iota_{x,y})$$\Phi_{\lambda,\mu}(\xi\Theta^{\mathrm{x}}\eta)$$\xi,$$\eta\in \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}^{()}n(K)$$\Phi_{\lambda,\mu}(\xi\otimes\eta)$$\lambda,$$\xi\otimes\eta$$\mu\in$$(\lambda, \mu)$$\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{n}1\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}_{\mathrm{X}}\Phi(\xi\otimes\eta)$$\mathrm{C}$$\mathrm{E}_{\mathrm{S}}\mathrm{S}\mathrm{p}_{\mathrm{a}\mathrm{t}}1_{1(K}^{()}1$$\otimes \mathrm{E}\mathrm{s}\mathrm{s}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}^{(1)}(K)$$\mathrm{b}\mathrm{i}$$\Phi$$*$$W^{\Phi}$$\Phi$$*$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\dot{\mathrm{t}}\mathrm{i}_{\mathrm{o}\mathrm{n}}\Phi \mathrm{i}’ \mathrm{s}$$W^{\Phi}$$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\mathrm{s}\backslash$$*$$\Phi$

123$\mathrm{m}$$a_{x,y}$$\equiv$

Figure 32:$\xi$$u_{\lambda,\mu}(\xi, \eta)=$$db\downarrow\mu$$\lambda a\downarrow c$$|\overline{/}$Figure 33:-representation gives the two equivalent
two unitarilyaup to vertical gauge choice. Moreover one will notice
time.
salne
the
at
rise to two horizontally conjugate connections$\mathrm{t}\mathrm{l}\mathrm{T}\mathrm{a}\mathrm{t}$-unitary connections-representation$\mathrm{b}\mathrm{i}$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}*$$\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}*$$\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}_{\square }\mathrm{S}$Remark 4.3 The above correspondence in Theorem 4.1 hold true for the case of K-Ldouble triangle algebras and K-L -unitary connections. The proof is exactly the same as
the proof of Theorem 4.1. But we remark that we do not haue minimal central $pro_{J^{ect}}ions$as in Section 2 in the case of K-L double triangle algebras when $K\neq L$ .
$bi$$p_{k}^{\pm}$

Applying the above theorem to the concrete -representation corresponding to theas in Section 2, we get the following important result.
minimal central projection$*$$p_{1}^{\pm}$$E6,7,8$ . The minimal central
Corollary 4.4 Let $K$ be one of the Dynkin $diagram\mathit{8}A_{n},$tothe two mutually complex
correspondprojectionsof the K-K double triangle algebra
conjugate (flat) -unitary connections on the four graphs which are all the same Dynkin.
in the case of the Dynkin diagramin 35. In particular
diagram$D_{n},$$p_{1}^{\pm}$$bi$$Ka\mathit{8}$$p_{1}^{+}=p_{1}^{-}$$\mathrm{A}_{n}$Proof By looking at the shape of the nlinimal central projections , the vertical graphsof the corresponding irreducible K-K connections are the graph $K$ itself. Because we knowthat there are two mutually complex conjugate non-equivalent connections on the four
. And there is only one
is one ofgraphs as in Figure 35 when the graph. (See [13, Theorem 11.22]. We-unitary connection when the graph $K$ is one ofremark that Theorem 11.22 in [13] states the isomorphic classes of connections whereas it
is also true for the equivalence classes.) The difference of two connections corresponding$p_{1}^{\pm}$$I\iota’$$D_{n},$ $E_{6,7,8}$$A_{n}$$\mathrm{b}\mathrm{i}$$\xi\otimes\eta$$\equiv$

Figure 34:

124$I\iota’$

$K$

$K$

$I\iota’$Figure 35:

Figure 36:

toandis shown by the relation in Temperley-Lieb recoupling theory as in Figure
36 which represents the difference of positive and negative crossing. Where is a complex
number given bywith the Coxeter number . In particular it shows that theseconnections are mutually complex conjugate.
$p_{1}^{+}$$p_{1}^{-}$$\epsilon$$ie^{i\pi/}2h$$h$$\mathrm{t}\mathrm{w}\mathrm{o}\square$

Theorem 4.1 every
central projection in the K-K double triangle
algebra corresponds to an irreducible K-K connection. By the definition of productof two connections and the above correspondence, the. product of two minimal central
projections and corresponds to the product of two irreducible -unitary connections
and. So by decomposing the product connection
into irreducible ones andusing the correspondence between. irreducible connections and minimal central proj ections,
we get a linear combination of minimal central projections with positive integer coefficient.
This means that the center of the K-K double triangle algebra $Z=Z(A, *)$ is closed
under the . product operation. And this shows the fact that the fusion rule of K-K biunitary connections is given by the . product of corresponding minimal central projections.
So we get the following.
$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$$i^{\mathrm{F}\mathrm{r}}\mathrm{o}\mathrm{m}$$p$$W_{p}$$p$$W_{p}$$\mathrm{b}\mathrm{i}$$q$$W_{q}$$W_{p}\cdot W_{q}$Corollary 4.5 Let
be one of the Dynkin diagrams. Then the fusion rulealgebra of K-K -unitary connections is isomorphic to the center 2 of the K-K double
triangle algebra $(A, *)$ with . product, $i.e$ .
.
$I\mathrm{i}’$$A_{n},$ $D_{n},$ $E_{6,7,8}$$bi$$(Z, \cdot)$

5

Classification of irreducible$\mathrm{b}\mathrm{i}$5.1 Classification of irreducible A-A-unitary connections on the Dynkin diagrams$bi$-unitary connection .

Let
be one of the Dynkin diagran]unitary connections.
$A$

$1S$

$A_{n}$. We first classify all (irreducible) A-A bi-

Proposition 5.1 Let
be the four graphs connected in Figure 22. Supposethat both the upper graphand the lower graphareand suppose there
a biunitary connection on the four graphs. Then the connecting vertical graphs
andare$\mathcal{G}0,$$\mathcal{G}_{1},$$\mathcal{G}_{0}$$\mathcal{G}2,$$\mathcal{G}_{3}$$a\mathit{8}$$\mathcal{G}_{2}$$A$$\iota s$$\mathcal{G}_{1}$$\mathcal{G}_{3}$125uniquely determined by the $ini$ tial condition, $i.e.$ , the condition of edges connected to the
distinguished $vertex*of$ the upper graph A. Moreover such a connection is unique up to
vertical gauge choice.
Proof Because the string algebra on the graph is generated only by Jones projections,
are uniquely determined by looking at the dimension ofandthe vertical graphs
to the initial condition, i.e., the vertices
essential paths with starting pointof the lower graph connected to the distinguished vertex of the upper graph A. (See[13, Section 11.6].) The connection on the four graphs can be decomposed into irreducible
ones. And an irreducible A-A connection have only one initial edge by the criterion in[1, Section 3, Claim 1] and the fact tluat it is autolnatically flat ([13, page 593]). The
choice of the initial edge is one-to-one correspondent to the vertex of the (lower) graph .
Hence the uniqueness (up to vertical gauge choic.e) of the connections on the four graphs
is proved by the uniqueness of irreducible connections corresponding to each vertex of. We know that there is at least one -unitary connection on the four graphs with an
initial edge corresponding to each vertex of the lower graph A. (Note that a corresponding
commuting square is given in [25].) So the proof will end to show that these are the only
irreducible -unitary connections up to vertical gauge choice. This is done by using thecorrespondence between -representations of the A-A double triangle algebras and A-A-unitary connections. More precisely we llave to sllow the following equality.
$A$$\mathcal{G}_{3}$$\mathcal{G}_{1}$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}\mathrm{n}\mathrm{d}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$

$A$

$A$

$\mathrm{b}\mathrm{i}$$A$$\mathrm{b}\mathrm{i}$$*$$\mathrm{b}\mathrm{i}$$a \in\sum_{\mathrm{e}\mathrm{r}\mathrm{t}A}(_{k0}^{n-1}\sum_{=}\dim \mathrm{E}_{\mathrm{S}}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}(oAk\backslash ))2=\sum_{=0}n-1k(_{a\in \mathrm{v}_{\mathrm{e}}\mathrm{I}\mathrm{t}A}\sum \mathrm{d}\mathrm{i}\ln \mathrm{E}\mathrm{S}\mathrm{S}\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}_{a}(k)A)2$

This equality can be easily obtained by a direct computation.

$\square$Remark 5.2 The above method also works for the classification of irreducible A-K biwith the same Coxeter number as
unitary connections for arbitrary Dynkin diagrams
A. The only different point is the last equaldy concerning the dimensions of the double
triangle algebras. In the case of general Dynkin diagram , we have to show the following
equality instead of the above one.
$I\iota’$$I\iota’$dinl$x \in \mathrm{v}\sum_{\mathrm{e}\mathrm{r}\mathrm{t}K}(_{k=}\sum^{m}\dim \mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}^{(}k0x)K\mathrm{I}^{2}=\sum_{k=0}^{n1}(_{a\in \mathrm{V}\mathrm{e}}\sum_{\mathrm{t}\mathrm{r}A}$$\mathrm{E}\mathrm{s}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}_{a)}(k.)A(_{x\in \mathrm{e}\mathrm{r}}\sum_{\mathrm{V}\mathrm{t}K}\dim \mathrm{E}\mathrm{S}\mathrm{S}\mathrm{p}_{\mathrm{a}}\mathrm{t}\mathrm{h}_{x}^{(k})K)$

which is the same as (theandHere $m$ is the maximal length of essential paths on
Coxeter $number$) $-2$ . This also can be shown by a direct computation in each case. But
here we give another proof based on estimates of the global index in the following.
$A$

$IC$

Remark 5.3 The uniqueness of irreducible A-K -unitary connection corresponding toeach vertex of $K$ does not seem to be obvious though we know the uniqueness of corresponding commuting square up to isomorphism. Here we remark that an isomorphic classan uomorphic class of connections (See [13, Definitionof commuting
10.11] for the definition of isomorphic connections) and it does not imply the uniquenessof equivalent connections up to (vertical) gauge choice.
$bi$$\mathit{8}quarecorre\mathit{8}ponds$

5.2 Classification of irreducible A-K -unitary $com$ lections.
Let be one of the Dynkin diagranls $An$ and one of the Dynkin diagramswith the same Coxeter number as . Before going into the details of the classification,$bi$$I\iota’$

$A$

$A$

$A_{n},$ $D_{n},$ $E_{6,7,8}$

126

we will show that a simple consideration on therule algebra leads anconsequence, that is, the system of -unitary connectiolls which consist of all irreducible
A-A, A-K, K-A and K-K connections are finite. It llleans that the numbers of all equivalence classes of irreducible A-K and K-K -unitary connections are finite. Moreover we
can measure its size explicitly. To sllow this first we need the next lemma.
$\mathrm{f}\mathrm{u}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{i}_{\ln_{\mathrm{P}^{\mathrm{o}\mathrm{r}}}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$be as above. Then all irreducible K-K (resp. A-A) bi-unitary
Lemma 5.4 Let and(resp.
) forconnections are obtained from the producttwo
irreducible A-K -unitary connections and .
$A$$I\mathrm{i}^{r}$$K\overline{\alpha}A$$bi$$\alpha$$A\alpha_{K}\cdot K\overline{\beta}_{A}$$A\beta\kappa$$\mathit{8}ome$$\beta$Proof We give a proof for the case of K-K -unitary connections because the same
proof also works for the case of A-A -unitary connections. This is an easy consequenceof Frobenius reciprocity. Let $K^{W}K$ be any irreducible K-K -unitary connection. Thenand make a product of them.
take (any) irreducible A-K -unitary connectionin the irreducible decomposition of the productTake an irreducible A-K connection. So we get, i.e., we haveby Frobenius.
$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$A\alpha_{K}$$A\beta_{K}$$A\alpha w_{K}\succ A\beta\kappa$$A\alpha_{KK}.w_{K}$$I_{1}^{\prime\overline{\alpha}}\beta_{K}\succ K^{W}K$$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{P}^{\Gamma \mathrm{O}\mathrm{C}}\mathrm{i}\mathrm{t}\mathrm{y}\square$

be one of the Dynkin diagrams. Then the numbers
Theorem 5.5 Let
A-K
equivalence
irreducible
andK-K-unitaryallclassesconnections are finite.
ofof
Moreover, they have the same global index
that of the system of all irreducible A-A-unitary connections.
$A_{n},$ $D_{n},$ $E_{6,7,8}$$l\mathrm{i}’$$bi$$a\mathit{8}$$bi$is shown in the previous section. So we consider
Proof The case when the graph isthe other cases. The system of all A-A -unitary connections are obtained from a finite
set of irreducible A-K connections by the previous lemnla. We choose and fix such a finite
set and consider the system generated by one A-K -unitary connection $A^{W}K$ which isthe (finite) direct sum of all the A-K connections we have chosen. It was shown thatthe set of all irreducible A-A connections are finite. So thiscontains only finitely
many different irreducible A-K connections because of the local finiteness of the principal
fusion graph of the generator $A^{W}K$ .
We claim that all irreducible A-K connections appear in this system. Otherwise wehave an .reducible A-K connection
which do not appear in this system. If we takeas an generator, we get a different system having the same set
an A-K connectionof irreducible A-A connections and strictly larger set of irreducible A-K connections than
before. Thus we get two systems of four kinds of connections consisting of irreducible A-A,
A-K, K-A and K-K connections which are both generated by one A-K connection. Now
it is easy to see that proof of the equality
for the estimates of global index for subfactor $N\subset M$ still works in the case of singly
generated connection system. Here the sulnnlations run over all irreducible. (See [13, Proposition 12.25] for the proof of
appear in the system generated by
subfactor case.) So we get a contradiction from the estilnates of the global index of thetwo systems. Hence the system must contain all the irreducible A-K connections.
Now the same argument shows that we have finitely many irreducible K-K connections and these are the all irreducible K-K connections by the previous lemma. So thesystem of four kinds of connections consisting of all irreducible A-A, A-K, K-A and K-K
connections is generated by one A-K connection . Applying the estimates of theindices of this system we get the result.
$I\mathrm{c}’$$A_{n}$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$$\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}}\mathrm{e}\ln$$\mathrm{i}\mathrm{r}$$A^{\tilde{L}}.K$$w\oplus z$$\sum_{N}\lambda_{N}[_{N}X_{N}]=\sum_{N}Y_{M}[_{N}Y_{M}]=\sum_{M}Z_{\Lambda i}[_{M}Z_{M}]$$\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{d}_{\mathrm{U}\mathrm{l}\mathrm{e}}$$N^{\mathrm{j}1I}\Lambda,I$$w$$\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}1\coprod$

127

By this estimates of global index we can easily classify all irreducible A-K bi-unitaryas the following proposition.
. $r:$ .
.
.
: :
...
be the four graphs connected.. as in Figure 22. Suppose
Proposition 5.6 Let
there is a biis $K$ andand the lower graphisthat the upper graphareandunitary connection on the four graphs. Then the connecting vertical graphs
uniquely determined by the initial condition, $i.e.$ , the condition of edges connected to theup $to.v;$ erticalerA.
distinguished vertex ofconnec..tions$\mathrm{i},\mathrm{n}$$\mathcal{G}0,$$\mathcal{G}_{1},$$\mathcal{G}_{0}$$\mathcal{G}2,$$\mathcal{G}_{3}$$A$$\mathit{8}uppose$$\mathcal{G}_{2}$$\mathcal{G}_{3}$$\mathcal{G}_{1}$gauge choice. .‘$?s..uni..q_{\mathrm{f}}’\backslash _{\iota\backslash }\cdot!\wedge\cdot,\backslash \cdot.u.ej.\cdot$$suc_{\mathfrak{i}}.h..a.:conn.e,c.:t‘.i\mathit{0}n_{l},$$M.oreov\wedge\cdot,.\cdot.$$t.heg.ra.ph\backslash .\cdot.,’\backslash$

.

.

‘

Proof The proof of the first assertion is exactly the same as Proposition 5.1. So wehave only to show the uniqueness of irreducible connections corresponding to each vertex
of $K$ . Again we know that there is at least one -unitary connection on the four graphs. So we show that these arewith an initial edge corresponding to each vertex of
the only irreducible -unitary connections up to vertical gauge choice.
In the case of $K=D_{n}$ , we know that there is an irreducible A-K connections wit.h.
. Take one of the. We denote it bywhich correspond theindex. Take a finite$K^{W}K$byand denote it
two non-equivalent fundamental connections on(or ) and decompose them. In this way we get some irreducible A-K
product
connections. We remark that the fusion graph with initial vertex and generator has
the same Perron-Frobenius eigenvalue as the index of the connection . So it must be oneof A-D-E Dynkin diagrams. It is easy to see that irreducible connections corresponding
appears in this procedure. So the graph has
to any choice of the initial vertex of. From the estimates of global index, these arevertices at least as many as that of
all the irreducible A-K connections because of the equality $|A_{2n+1}|=2\cdot|D_{n}|$ . Hereto all the vertices of the graph .
represents the global index
works. So we have only to
In the case of $K=E_{6,7,8}$ , the salne proof as in the case of,, i.e.
show the following estimates of global
represents a connection corresponding to an initial. Hereanddenotes its index.
andthe graphedge connected to the distinguished
These are shown by Wenzl’s index formula [14, Theorem 4.3.3]. Actually the square ofas above are the same as the indices of the corresponding
the indices of connections
and , i.e.
GHJ subfactors, which are exactly the quotient of two global indices of.
equalities
Wenzl’s index forlllula. So the above$|\alpha_{K}|^{2}=|K|/|A|$ for$\mathrm{b}\mathrm{i}$$I\mathrm{c}’([25])$$\mathrm{b}\mathrm{i}$$\sqrt{2}$$\backslash \mathrm{t}\mathrm{o}$$A\alpha_{K}$$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}D_{n}$$D_{n}$$\alpha w\overline{w}\cdots w$$\overline{w}$$w$$\alpha$$w$$I\backslash ^{r}$$D_{\eta}$$i$$|I\mathrm{i}’|$$l\mathrm{i}’$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{0}\mathrm{n}\mathrm{d}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$$D_{n}$$\mathrm{i}_{\mathrm{I})}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{S}$$|E_{8}|=|\alpha_{E_{8}}|^{2}\cdot|A_{29}|$$|E_{6}|=|\alpha_{E_{6}}|^{2}\cdot|A_{11}|,$$|E_{7}|=|\alpha_{E_{7}}|^{2}\cdot|A_{17}|$$\alpha_{I\mathrm{c}’}$$I_{1}’$$\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}*\mathrm{o}\mathrm{f}$$|\alpha_{K}|$$\alpha_{K}$$A$$I\mathrm{t}^{\mathcal{F}}$$K=E_{6,7,8}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{l}$$\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\square$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$Proposition 5.1 and Proposition 5.6 we get the following theorem.

one.-. to-one. ThereTheorem 5.7 Let $K$ be one of the Dynkin diagramsand equivalence classes of irreducible A-K
correspondence between vertices of the graph-unitary connections.
$A_{n},$ $D_{n},$ $E_{6,7,8}$$i\mathit{8}$$a$$I\mathrm{c}^{-}$$bi$

5.3 Classification of irreducible K-K -unitary connections.
central projections of the Kthe one-to-one correspondence between$K$ double triangle
and K-K -unitary connections, for every minimal central
projection of the K-K double triangle algebra we can associate an index of the subfactor
generated by the corresponding connections. We call the square root of the index of thecentral projection an index of the projection
subfactor corresponding to a
. Because two equivalent -unitary connections give rise to
and we denote it by$bi$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$\mathrm{b}\mathrm{i}$$\mathrm{a}\mathrm{i}_{\mathrm{g}\mathrm{e}}\mathrm{b}\mathrm{r}\mathrm{a}$

$p$

$p$

$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{l}$$\mathrm{b}\mathrm{i}$$\mathrm{d}(p)$

$p$

128

an isomorphic subfactor, this definition is well-defined. Moreover if twocentralprojections and coincide, they
have the sanue index, i.e.
.
For the case of the Dynkin diagrallls $K$ we havecentral projections
and-which is called chiral projectors (see Section 2.3). We call the subset ofwhich$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$$p$$11\mathrm{l}\mathrm{U}\mathrm{s}\mathrm{t}$$q$$\mathrm{d}(p)=\mathrm{d}(q)$$\mathrm{s}_{1}\supset \mathrm{e}\mathrm{c}\mathrm{i}\mathrm{a}1$$\Psi_{+}$$\Psi$$(Z, \cdot)$

consists of minimal central projections contained in(resp.
) chiral left part (resp.
chiral right part). Because the chiral left part (resp. chiral right part) coincide with theset of minimal central projections appears in the system generated by(resp.
) theyform two fusion rule subalgebras of. The intersection of chiral left part and chiralright part is called ambichiral part of the fusion rule algebra
and it corresponds
to the ambichiral projector. In the following we use the notations
andtorepresent the fusion rule subalgebras of chiral left part, chiral right part and anlbichiralpart respectively.
$\Psi_{-}$$\Psi_{+}$$p_{1}^{+}$$p_{1}^{-}$$(Z, \cdot)$$(Z, \cdot)$$\Psi_{\pm}$$Z_{l},$$Z_{a}$$\mathcal{Z}_{r}$Proposition 5.8 Let $K$ be one of Dynkin diagrams. Suppose we have afusion rule subalgebra of the fusion rule algebras of all irreducible K-K connections
2. Then 2 $decompoSe\mathit{8}$ into left cosets of and right cosets of $B,$ $i.e$ . we have subsets$X,$of irreducible connections (representatives of left and right) such that
if $x\neq x’\in X$ andif.
$A_{n},$ $D_{n},$ $E_{6,7,8}$

$B$

$B$

$\mathrm{Y}\subset Z$$co\mathit{8}ets$$\mathcal{Z}=\bigcup_{x\in x^{X\cdot B}}=\bigcup_{y\in Y}B\cdot y,$$x\cdot B\cap x^{J}\cdot g=\emptyset$$B\cdot y\cap B\cdot y’=\emptyset$$y\neq y’\in \mathrm{Y}$Proof We will give a proof for the left cosets. We have only to show the following;
for irreducible $x’\in Z$ , then $x\cdot B=x’\cdot B$ . Suppose we have$x\cdot B\cap x’\cdot B=\emptyset$
for $x’\in$ Z. Then there are irreducible K-K connections $b’\in B$
and $z\in Z$ , such that $x\cdot b\succ z$ and $x’\cdot b’\succ z$ . Hence $x\cdot B\supset x\cdot b\cdot B\supset z\cdot B$ andholds.
the Frobenius reciprocity, we haveand. So the converse inclusions
andholds. Thus we have $x\cdot B=x’\cdot B=z\cdot B$.
$x\cdot B\cap x’\cdot B=\emptyset$

$x,$

$x,$

$b,$

$x’\cdot B\supset x’\cdot b’\cdot B\supset z\cdot B$$\mathrm{F}\mathrm{r}\mathrm{o}\ln$$z\cdot\overline{b}’\succ x’$$z\cdot\overline{b}\succ x$$z\cdot B\supset z\cdot\overline{b}\cdot B\supset x\cdot B$$z\cdot B\supset z\cdot\overline{b}’\cdot B\supset x’\cdot B\coprod$

Remark 5.9 It is easy to see that this proposition holds true for more general fusion rulealgebras such as those treated in Hiai-Izumi [ . We only need the property of Frobenius
reciprocity. For example any fusion rule algebras of bimodule (or) arising from
have
this
coset decomposition property.
subfactors$\mathit{1}\mathit{5}J$$\mathit{8}ectors$

The following can be easily shown from Proposition 5.8 and the estimates of global
indices (Theorem 5.5).
Corollary 5.10 If the chiral left partdoes not coincide with the chiral right part ,then the principal fusion graphs of minimal central projections
andcannot be the
Dynkin diagram of type A. Conversely if one of the principal fusionof andis the Dynkin diagram of type , then we have $Z_{l}=Z_{r}=Z_{a}=Z$ .
$\mathcal{Z}_{l}$$Z_{r}$$p_{1}^{+}$$p_{1}^{-}$$graph_{\mathit{8}}$$p_{1}^{+}$$p_{1}^{-}$$A$

5.3.1 The case of$A_{n}$

This is done in the previous subsection 5.1. There iscorrespondence between vertices of the Dynkin diagramand irreducible-unitary connections.
In this case the two minimal central projections
andcoincide. The fusion rule graphfor the generator $[1]=p_{1}^{+}=p_{1}^{-}$ is given in Figure 40.
$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}}$$A_{n}$$A_{n^{-}}A_{n}\mathrm{b}\mathrm{i}$$p_{1}^{+}$

5.3.2 The case of $D_{2n+1}$$p_{1}^{-}$

129
In this case we have gap $(D_{2}n+1)=\infty$ and corresponding recoupling system is $A_{4n-1}$ .
So there are two series of mutually orthogonal minimal central projections
coincide and it corresponds to the identity conand. Hereandby taking . product, this means that the subfactor
arise fromnection. Because all(which is one of the two non-equivalentarising from the connection corresponding to-unitary connections as in Corollary 4.4) have at least $4n-1$ vertices in its principal
this is possible only when the principal graph is thegraph. From the index value
Dynkin diagram $A_{4n-1}$ . So we luave $Z_{l}=Z_{r}=Z_{a}=\mathcal{Z}$ by Corollary 5.10. Hence theseandthe facts thatare all the minimal central projections.
must coincide, the lninimal central projection
the fusion rule. The fusion graph of the two. Hence we getwithgenerator $[1]=p_{1}^{+}$ and $[4n-3]=p_{1}^{-}$ are given as in Figure 41.
Note that we did not use the fact of non-existence of $D_{2n+1}$ subfactors in the above
argument. So it gives another proof of the non-existence of $D_{2n+1}$ subfactors. It alsosquares are $A_{4n-1}$ .
shows that the flat part of$\{p_{k}^{+}\}_{k=0},1,2,\ldots,4n-2$$p_{0}^{+}$$\{p_{k}^{-}\}_{k=}0,1,2,\ldots,4n-2$$p_{0}^{-}$$p_{1}^{+}$$p_{k}^{+}$$p_{1}^{+}$$\mathrm{b}\mathrm{i}$$\mathrm{d}(p_{1}^{+})$$p_{1}^{-}\neq p_{1}^{+},$$i^{\mathrm{F}}\mathrm{r}\mathrm{o}\ln$$p_{1}^{-}\neq p_{4n-3}^{-}$$p_{1}^{-}$$\mathrm{d}(p_{1}^{+})=\mathrm{d}(p_{1}^{-})=\mathrm{d}(p_{4n-3}^{+})=\mathrm{d}(p_{4n}^{-}-3)$$p_{k}^{-}=p_{4n-}^{+}2-k\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{n}\mathrm{l}$$p_{4n-3}^{+}$$D_{2n+1}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$5.3.3 The case of
In this case there are two non-equivalent connections with index 1. One is trivial(we denoteconnection and the other comes fronl the flip of the two tails of the graphit by ). We denote the minimal central projections corresponding to the connection by.
, the fusion graph of chiral left part as well as chiral right part can
Becauseexcept the case $n=5,8$ . But
from Corollary 5.10. So they must be
not be. Hence there is a
we have gap $(D_{10})=16$ andandin the case ofin the casecentral projections
series of mutually orthogonal. These shows that the fusion graph ofin the case of,andofand they lllust beorchiral left part (hence chiral right part as well) can not bethemselves.
andby comparing the indicesnordoes not appear in either
It is easy to see thatis thealgebra of even vertices of, the. In the case of
except the casein this case, either. So we get the coset decolnposition
cyclic group . Hence$D_{2n}$$D_{2n}$$\epsilon$$\epsilon$$p_{\epsilon}$$p_{1}^{+}\neq p_{1}^{-}$$D_{2n}$$A_{4n-3}$$D_{16}$$D_{10}$$\mathrm{g}\mathrm{a}\mathrm{p}(D_{16}++)=28$$p_{0},p_{1},$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}_{1}\mathrm{n}\mathrm{a}1$$p_{0}^{+},p^{+}1$$D_{10}$$\ldots,p_{7}+\in Z_{l}$$D_{16}$$\ldots,p^{+}13\in Z_{l}$$E_{6}$$E_{8}$$D_{16}$$D_{10}$$\mathcal{Z}_{l}$$\mathcal{Z}_{r}$$p_{\epsilon}$$D_{4}$$D_{4}$$D_{4}$$\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{r}\perp \mathrm{r}\mathrm{u}\mathrm{l}\mathrm{e}$$p_{\epsilon}\not\in Z_{l}\cup Z_{r}$$\mathrm{Z}_{3}$. But the estimates of the global indices of the both sets andshows that these are all the irreducible K-K collnectiolls.
,
we haveandthe equalitiesof
rulewhich shows that the even vertices of chiral left and right part coincide from fusion
does not coincide again from theand. It is easy to see that the odd vertices offusion rule. So we obtain the fusion rule graph for two generators $[1]=p_{1}^{+}$ andas in Figure 42.
$\mathcal{Z}$$Z_{l}\cup Z_{l}\cdot p\epsilon$$\mathcal{Z}\supset Z_{l}\cup Z_{l}\cdot p_{\in}$$p_{2}^{+}=p_{2}^{-}$$p_{\epsilon}^{2}=p_{\epsilon}\cdot p_{\epsilon}=id$$p_{1}^{-}=p_{\xi j}\cdot p_{1}+=p_{1}^{+}\cdot p_{\mathcal{E}}$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$Z_{r}$$\mathcal{Z}_{l}$$D_{2n}$$[1^{\sim}]=p_{1}^{-}$

5.3.4 The case of$E_{6}$

at the vertical edges of the. By
we know thatIn the case of. , we can see that this connection is irrecomposite connection corresponding toducible from the criterion [1, Corollary 2, Section 3] and the Frobenius reciprocity. This
. So we have coset deconlposition
is not in the chiral left part, i.e.
means that
and(see Proposition 5.8). The (coset) principal fusion graphs for
Corollary
fronl
have
or . We cannot
are one of the Dynkin diagrams$|A_{11}|=(3+\sqrt{3})|E6|=(1+|p_{1}^{-}|^{2})|E_{6}|$ .
that
show
5.10. The estimates of global indicesthis together with the following inequality $|E_{6}|<|D_{7}|<|A_{11}|$ the both (coset). This also shows the fiatness oflllust beandprincipal fusion graphs for$p_{1}^{+}\neq p_{1}^{-}$$E_{6}$$p_{1}^{+}$$\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{k}\mathrm{i}\mathrm{l}$$p_{1}^{-}$$p_{1}^{-}\in Z_{l}$$p_{1}^{-}$$\mathcal{Z}_{l}$$\mathcal{Z}\supset Z_{l}\cup \mathcal{Z}_{l}\cdot p_{1}^{-}$$D_{7}$$\mathcal{Z}_{l}\cdot p_{1}^{-}$$A_{11}$$E_{6}$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$Z_{l}$$Z_{l}\cdot p_{1}^{-}$$E_{6}$

130

Table 1: Multiplication table for$\cdot\copyright_{N}$. of the fusion rule algebra of$E_{6}$

$N-N$

$N-M$
$\Lambda I-N$

$M-\Lambda I$

Figure 37: The (dual) principal graph$E_{6}$

is generated by theconnections (see Proposition 3.21) and that the whole systemeasy.
connections
and
Now
it
is
to see the fusion rule
two irreducible fundamental
is given in Figure 43 from the index
graph for the two generators $[1]=p_{1}^{+}$ andfusion rule algebra in Table
values. Here we also give the multiplication table of theshown in Figure 37.
correspond to the vertices of
1, where$\mathcal{Z}$$E_{6}$$p_{1}^{+}$$p_{1}^{-}$$[1^{-}]=p_{1}^{-}$$E_{6}$$E_{6}$$\rho_{i}’ \mathrm{s}$5.3.5 The case of$E_{7}$and the irreducibility of the composite connection
can beWe havecase. So we have coset decomposition
andshown in the same way as
andare one of the Dynkin diagramsthe (coset) principal fusion graphs foror . (Again we cannot havefrom Corollary 5.10.)
In this case the global indices satisfies $|E_{7}|=2\beta(3\beta^{2}-15\beta+18)<|D_{10}|=\beta|E_{7}|<$$|A_{17}|=2\beta|E_{7}|$ , where $\beta=4\cos^{2}(\pi/18)$ .
So we have $|A_{17}|=2\beta|E_{7}|>|G_{1}|+\beta|G_{2}|$ ,principal(coset)respectively.
thefusionforanddenoteandwhere
Hence the equality only happens when $G_{1}=D_{10}$ and $G_{2}=E_{7}$ . This shows that the fiat
and that the whole system $Z$ is generated by the twoconnections ispart of the, from the fusion. If
and . Here we claim
fundamental.
. But this is inlpossible because
rule we haveby Frobenius reciprocity, which contradicts the irreducibility. Then by looking at the indices and the fusion. Hence we must haveofrules, weas in Figure
the fusion graph for the two generators $[1]=p_{1}^{+}$ and (0)43.
$p_{1}^{+}\neq p_{1}^{-}$$p_{1}^{+}\cdot p_{1}^{-}$$E_{6}$$Z\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}$$\mathcal{Z}_{l}$$D_{10}$$Z_{l}\cdot p_{1}^{-}$$A_{18}$$E_{7}$$G_{2}$$G_{1}$$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}1_{1}$$Z_{l}$$Z_{l}\cdot p_{1}^{-}$$D_{10}$$E_{7}$$p_{1}^{-}$$p_{2}^{+}\neq p_{2}^{-}$$p_{2}^{+}--p_{2}^{-}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}++\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}p_{1}+$$p_{1}\cdot p_{1}\cdot p_{1}^{-}=2p_{1}^{-}+p_{3}^{-}$$\dim \mathrm{E}\mathrm{n}\mathrm{d}(p_{1}+$$p_{1}^{-)}=$$\dim \mathrm{H}\mathrm{o}\mathrm{m}(p^{+}1^{\cdot}p_{1}^{+}\cdot p^{-}1 , p_{1}^{-})=2$$p_{2}^{+}\neq p_{2}^{-}$$p_{1}^{+}\cdot p_{1}^{-}$$=p_{1}^{-}$$\mathrm{g}.\mathrm{e}\mathrm{t}$5.3.6 The case of$E_{8}$In this case we also have the coset decomposition$\mathcal{Z}\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}$

which is shown131$(\beta^{2}-2\beta+2)|E_{8}|<|A_{17}|=(2\beta^{2}\neg 4\beta+4)|E_{8}|$, where $\beta=4\cos^{2}(\pi/30)$ .

the above coset decompois .
First we show the principal fusion graph of$|A_{17}|=(2\beta^{2}-4,\theta+4)|E_{8}|>|G_{1}|+\beta|G_{2}|$. Heresition, we have the following inequality,. If
anddenote the (coset) principal fusion graph forandholds, tlle even
. Ifor
$G_{1}=D_{16}$ , there are two possibilities, i.e.
from thepart
coinciderightchiraland those of thevertices of the chiral left part
fusion rule. And we have the following coset decomposition,. But the smallest possible value of the global indices isbecause$|D_{16}|+\beta|E8|+\beta(\beta-2)2|E_{8}|=(\beta^{3}-3\beta 2+3\beta+2)|E_{8}|>|A_{29}|$ and this is impossible. If
by comparing the indices. So we have the coset
holds, then
. The smallest possible value of the global indicesdecomposition.
is $|D_{16}|+\beta|E8|+(\beta-1)^{2}|E_{8}|=(2\beta 2-3\beta+3)|E_{8}|>|A_{29}|$ . So again this is. And this shows the fiatness ofmust beThus the principal fusion graph ofleft
and right part as in Figure 38.
connections. We label the vertices of the chiral
Next we show that the ambichiral part $Z_{a}=Z_{l}\cap Z_{r}$ is, we have two series of mutually orthogonal minimal central projections$E_{8}$$p_{1}^{+}$$Z_{l}$$G_{2}$$G_{1}$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$p_{2}^{+}=p_{2}^{-}$$Z_{l}\cdot p_{1}^{-}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{y}$$p_{2}^{+}=p_{2}^{-}$$p_{2}^{+}\neq p_{2}^{rightarrow}$$Z_{r}$$\mathcal{Z}_{l}$$Z\supset Z_{l}\cup Z_{l}\cdot p_{1}^{-}\cup Z_{l}\cdot p_{3}^{-}$$p_{3}^{-}\not\in \mathcal{Z}_{l}\cup Z_{l}\cdot p_{1}^{-}$$p_{2}^{-}\not\in \mathcal{Z}_{l}\cup Z_{l}\cdot p_{1}^{-}$$p_{2}^{+}\neq p_{2}^{-}$$Z\supset Z_{l}\cup z_{lp_{1}^{-}}.\cup \mathcal{Z}_{l}\cdot p_{2}^{-}$$\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{p}_{0}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}$$E_{8}$$E_{8}$$p_{1}^{+}$$\{\rho_{0}, \rho_{6^{--}}\beta 6\}\sim i^{\mathrm{F}\mathrm{g}\mathrm{a}_{\mathrm{P}(}}\mathrm{r}\mathrm{o}\mathrm{n}1E8)$$=0-\mathrm{g}\mathrm{a}\mathrm{p}(E8)=10$$\mathrm{t}\mathrm{h}\sim \mathrm{d}\beta_{k}\mathrm{a}\mathrm{n}pk\rho_{k}=\mathrm{T}+\sim.\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}$$\mathrm{a}\mathrm{r}\mathrm{e}\{p_{k}^{+}\}_{k0,1}\mathrm{t}\mathrm{W}\mathrm{o}\mathrm{p}\mathrm{o}=,2,3,4\mathrm{a}\mathrm{n}_{\mathrm{i}}\mathrm{d}\{pk\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{i}1\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{S},-\}_{k}.=0,1,2,3,4\cdot \mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}_{2}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}1\mathrm{s}\mathrm{o}1\mathrm{a}_{\mathrm{O}}\mathrm{b}\mathrm{e}\mathrm{l}1\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}_{1}p_{k}^{+}=\mathrm{e}.\rho 2=\rho 2\mathrm{o}\mathrm{r}\sim\rho\neq\rho 2\mathrm{f}\mathrm{r}\mathrm{m}\mathrm{t}11\mathrm{e}\mathrm{i}_{1}\mathrm{d}\mathrm{e}\mathrm{x}\mathrm{V}\mathrm{a}1\mathrm{u}\mathrm{e}\mathrm{s}.\mathrm{I}\mathrm{f}\rho_{2}=\rho_{2}-,$. But this isfrom the fusion rule (Table 3) we haveby Frobenius reciprocity
impossible because
do not coincide. Then again
and. So
and it contradicts the irreducibility of. Now ifcontradicts
whichfrom the fusion rule we cannot havethen we can check the following coset
by comparing indices. The slnallest possible value of the global index of these
cosets is $(2\beta^{2}-3\beta+3)|E_{8}|>|A_{29}|$ , which contradicts the estimates of global indices. It is easy to see that the odd vertices of(Theorem 5.5). Hence we must have.
cannot coincide because of the fusion rule (Table 3) and the fact
and$\beta 1^{\cdot}\rho 1^{\cdot}\rho 1^{-}=(p_{0}+\rho_{2})\cdot\rho_{1}^{\sim}=2\beta_{1}^{\sim}+\rho 3\sim$$\dim \mathrm{E}\mathrm{n}\mathrm{d}(\rho_{1\rho_{1}^{\sim})}.=\dim \mathrm{H}\mathrm{o}\mathrm{n}\mathrm{u}(\beta_{1}.(\rho_{1}\cdot\rho_{1}^{\sim}), \rho_{1})$$\rho_{1}\cdot\rho_{1}-$$\rho_{2}^{\sim}$$\rho_{2}$$\rho_{6}\neq\rho_{6^{-}}$$\rho_{2}\neq\rho_{2}^{\sim}$$\rho_{4}=\rho_{4}^{\sim}$$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{u}\mathrm{p}_{\mathrm{o}\mathrm{s}}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},$$\mathcal{Z}\supset Z_{l^{\cup}}\mathcal{Z}l\rho^{-}1\cup \mathcal{Z}_{l}\cdot\rho_{2}^{\sim}\cup \mathcal{Z}_{l}\cdot\rho^{\sim}5$$\cup Z_{l}\cdot\rho_{6}^{\sim}$$\rho_{6}=\rho_{6^{-}}$$\rho_{1}=\rho_{1}^{\sim}$$Z_{r}$$\mathcal{Z}_{l}$and 3 are for the system
Remark 5.11 Here we remark that the multiplication
we have different
systems
connection
K-Kof bimodules or sectors. But in the case of
the two fundamental connections of Agradings. Actually we have equality
corresponds
D-E Dynkin diagrams. This is impossible in the case of bimodules
sense,
makes
sectorself-conjugate
to an $N-M$ bimodule. In the case of sectors
but these have different meaning. (; From the fact that the fundamental connection ,in the multiplication tables, is self-conjugate in our sense as awhich $corre\mathit{8}ponds$ to. for odd andfor even in the casepair of connections, we can easily get .
. for even does not make sense even inof and . Again one will notice thatwhile the right hand side ininthe case of sectors because the left hand side
. But in our setting of K-K connection systems such things can happen. Henceasand
we can read the multiplication
3.
and
2
Table
in
examplesrespectively forand$table\mathit{8}\mathit{1},\mathit{2}$

$u$ )

$u$ )

$=\overline{w}f\mathit{0}r$$becau\mathit{8}ew$$\rho_{1}=\overline{\rho}_{1}$$w$$\rho_{1}$$\rho_{k}$$E_{6}$$E_{8}$$k$$=\overline{\rho}_{k}$$\rho_{k}’=\rho_{k}$

$k$

$k$

$\rho_{k^{\backslash }}’=\rho_{k}$$i\mathit{8}$$?S$$\mathrm{s}_{\mathrm{e}\mathrm{c}\mathrm{t}}(\Lambda I)$$\mathrm{s}_{\mathrm{e}\mathrm{c}\mathrm{t}}(N)$$\overline{\rho}_{1}\cdot\rho_{2}=\overline{\rho}_{1}+\overline{\rho}_{3}$$\overline{\rho}_{1}\cdot\rho_{3}=\rho_{2}’+\rho_{4}’$$\rho_{1}\cdot\rho_{2}=\rho_{1}+\rho_{3}$$\rho_{1}\cdot\rho_{3}=\rho_{2}+\rho_{4}$

. So
and
Now from the fusion rule we know
. And the estimates
we have the coset decomposition$|\mathrm{A}_{29}|=(2\beta^{2}-4\beta+4)|E_{8}|$and $|E_{8}|+\beta|E_{8}|$of the global indices of both hand side is$+(\beta-1)^{2}|E_{8}|+\beta(-\beta^{3}+7\beta^{2}-13\beta+5)^{2}|E_{8}|=(2\beta^{2}-4\beta+4)|E_{8}|$ , where we used theequality $\beta^{4}-7\beta^{3}+14\beta^{2}-8\beta+1=0$ to colnpute the right hand side. So we get the equality$\rho_{7^{-}}=\rho 6^{\cdot}\rho 1\sim,$$\rho_{4^{-}}=\rho 6^{\cdot}\rho^{\sim}2$$\mathcal{Z}\supset \mathcal{Z}_{l}\cup Z_{l}\cdot\rho_{1^{-}}\cup \mathcal{Z}_{l}\cdot\rho_{2}\sim\cup Z_{l}\cdot\rho 5^{-}$$\rho_{3}=\rho_{6\rho_{5}}\sim\cdot\sim$

132
even
The chiral left part
odd

even
The chiral right part
odd

Figure 38: The chiral left and right part of the fusion rule algebra $Z$ of

Table 2: Multiplication table for$\cdot\otimes_{N}\cdot$$E_{8}$.

of the fusion rule algebra of $E_{8}(1)$in the above coset decomposition. This shows that the whole system $Z$ is generated by
the two elements
and. Finally by looking at the index values and the fusion rule,
we obtain the fusion rule graph for the two generators $[1]=p_{1}^{+}$ andas in inFigure 45.
$p_{1}^{+}$$p_{1}^{-}$$[1^{\sim}]=p_{1}^{-}$

Remark 5.12 We will explain the meaning of Figures 40 to 45. The white vertices
and black vertices represents even and odd vertices. The large double circled
vertices denote the ambichiral part. The thick edges and thin edges represent the chiral left
graphs and the left coset graphs, which are obtained as Cayley graphs for multiplicationby the generator
representfrom the left. The thick dotted edges and thin dotted
the chiral right graphs and the right coset, which are obtained as Cayley graphs for$re\mathit{8}pectively$$p_{1}^{+}$$edge\mathit{8}$$graph_{\mathit{8}}$

$N-N$
$N-M$
$M-N$
$M-M$

Figure 39: The (dual) principal graph$E_{8}$

133

Table 3: Multiplication table formultiplication by the generator$p_{1}^{-}$$\cdot\otimes_{N}\cdot$

of. the fusion rule algebra of$E_{8}(2)$from the right.

In the procedure to get the complete classification of irreducible K-K -unitary connections. We also obtained the complete classification of fiat connections and flat part ofnon-flat connections on the Dynkin diagrams. We state this as the following corollary.
$\mathrm{b}\mathrm{i}$Corollary 5.13 The (fundamental)

ure 35 are flat in the case ofand . The flat part of-unitary connections on the four graphs as in Figand . They are not flat in the case of $D_{2n+1}$
respectively.
connections are $A_{4n-1}$ and$A_{n},$ $D_{2n},$ $E_{6}$$D_{2n+1}$$E_{7}$$bi$and$E_{\overline{\prime}}$$E_{8}$$D_{10}$Hence it provides another proof of the colnplete classification of subfactors of thefactor with index less than 4.
hyperfinite$\mathrm{I}\mathrm{I}_{1}$Corollary 5.14 There is only one subfactor with principal graphfor each $n\geq 2$ .
There is only one subfactor with principal graphfor each $n\geq 2$ . There are two andrespectively. These
andonly two $non- i_{\mathit{8}}omorphiC$ subfactors with principal graph
less
index
than
are all the subfactors of the hyperfinite
4.
factor with
$A_{n}$$D_{2n}$$E_{8}$$E_{6}$$II_{1}$

By examining each case we obtain the following structural result on the fusion rulealgebra of all K-K -unitary connections.
$\mathrm{b}\mathrm{i}$Theorem 5.15 Let $K$ be one of the A-D-E Dynkin diagrams. The fusion rule algebrasof all K-K -unitary connections are generated by the two minimal central projections. Moreover the chiral left part and the chiral right part commutes.
$bi$$p_{1}^{\pm}$Remark 5.16 The first assertion of the above theorem can be shown directly by taking. lVIinimal central projections appear in thisthe . product of two chiral projectors
and . And it is not
generated bysubalgebra
rule
the
in
product is contained
difficult to show that the product contains the identity element of the K-K double triangle
A. This result can
algebra by using the non-degenerate braiding on the recoupling
be generalized in more abstractof double triangle algebra as in [5].
$\Psi_{+}\cdot\Psi_{-}$$p_{1}^{+}$$fu\mathit{8}i_{on}$$p_{1}^{-}$$\mathit{8}ystem$$\cdot$$\mathit{8}etting$Remark 5.17 Though it is not written in detail in [24], Ocneanu showed stronger result
than the commutativity of the chiral left and nght part. He showed that the chiral leftpart and chiral right part of the fusion rule algebra of K-K -unitary connections has
$bi$

134

$[0][egg0]$$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$

Figure 40: Chiral symlnetry for the Coxeter graph$A_{n}$

non-degenerated braiding. (See the explanation of the picture $‘(Quantum$ Symmetry for
Coxeter graphs” in [24].) He defined the choice of intertwiner ingraphically andtheof the non-degenerate braiding. This shows that thefusion rule algebra of the ambichiral part has non-degenerate braiding. And it implies
the existence of non-degenerate braiding on the ystern of bimodule corresponding to evenvertices of the Dynkin diagram $D_{2n}u$ $hich$ was shown by D. E. Evans-Y. Kawahigashi in[12]. .
$\mathrm{H}\mathrm{o}\mathrm{m}(p_{i}\cdot p_{j},p_{j}\cdot p_{1})$$\mathit{8}howed$$exi\mathit{8}tence$$s$)

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Figure 42: Chiral$D_{6}$$\mathrm{s}\mathrm{y}_{11\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}$$D_{8}$

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Figure 43: Chiral symnuetry for the Coxeter graph$E_{6}$[24] A. Ocneanu, Paths on Coxeter Diagrams: From Platonic Solids and Singularities
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138

Figure 44: Chiral$\mathrm{s}\mathrm{y}_{\mathrm{l}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{t}\Gamma}\mathrm{y}$for the Coxeter graph$E_{7}$[32] V. G. Turaev&O. Y. Viro, State sum invariants of 3-manifolds and quantum 6jsymbols, Topology, 31 (1992), 865-902.

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139

$\lceil 01$

Figure 45: Chiral symlnetry for the Coxeter graph$E_{8}$