Classification of Noncommuting Quadrilaterals of Factors
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF
FACTORS
PINHAS GROSSMAN AND MASAKI IZUMI
Abstra
t. A quadrilateral of fa
tors is an irredu
ible in
lusion of fa
tors N ⊂ M
with intermediate subfa
tors P and Q su
h that P and Q generate M and the
interse
tion of P and Q is N . We investigate the stru
ture of a non
ommuting
quadrilateral of fa
tors with all the elementary in
lusions P ⊂ M , Q ⊂ M , N ⊂ P ,
and N ⊂ Q 2-supertransitive. In parti
ular we 
lassify non
ommuting quadrilater-
als with the indi
es of the elementary subfa
tors less than or equal to 4. We also

ompute the angles between P and Q for quadrilaterals 
oming from α-indu
tion
and asymptoti
 in
lusions.
1. Introdu
tion
If M is a fa
tor with a �nite group G a
ting by outer automorphisms, then N =
MG, the algebra of �xed points of the a
tion, is an irredu
ible subfa
tor of M with
index equal to the order of G. It is well known that in this 
ase the intermediate
subalgebras N ⊂ P ⊂ M are pre
isely the �xed point algebras of the subgroups of G
[48℄.
In this spirit one thinks of the 
lassi�
ation of intermediate subfa
tors as a �non-

ommutative Galois theory�. Motivated by von Neumann's study of proje
tion lat-
ti
es in Hilbert spa
e as a �
ontinuous geometry�, Watatani proposed studying latti
es
of intermediate subfa
tors, whi
h are �nite for irredu
ible �nite-index in
lusions, as
a quantization [56℄. Sano and Watatani introdu
ed the notion of angles between
subfa
tors, a numeri
al invariant whi
h measures the degree of non
ommutativity of
pairs of subfa
tors [54℄.
A major step towards the 
lassi�
ation of intermediate subfa
tors was Bis
h's 
har-
a
terization of intermediate subfa
tors as biproje
tions in planar algebras- elements of
the standard invariant whi
h are proje
tions in both of the dual algebrai
 stru
tures
[4℄. Bis
h and Jones then des
ribed a generi
 
onstru
tion of intermediate subfa
-
tors by 
onstru
ting a universal planar algebra generated by a single biproje
tion,
the Fuss-Catalan algebra with parameters 
orresponding to the indi
es [7℄. These
intermediate subfa
tors are supertransitive in the sense that the standard invariants
are minimal.
It was hoped that a similar generi
 planar algebra for multiple intermediate sub-
fa
tors 
ould be des
ribed in terms of indi
es and invariants su
h as angles. Indeed, a
tensor produ
t does yield a generi
 
onstru
tion of 
ommuting pairs of intermediate
subfa
tors, but 
onstru
ting pairs with nontrivial angles has proven more di�
ult.
Work supported by JSPS.
http://arxiv.org/abs/0704.1121v2
2 PINHAS GROSSMAN AND MASAKI IZUMI
In [13℄, Jones and the �rst-named author showed that there are essentially only 2
examples of non
ommuting irredu
ible quadrilaterals of fa
tors su
h that all of the
elementary in
lusions are supertransitive. One is a quadrilateral of subgroups of the
symmetri
 group S3 and the other 
omes from the GHJ family of subfa
tors and all
the elementary in
lusions have index 2 +
It is known that the stru
ture of an intermediate subfa
tor N ⊂ P ⊂ M is deter-
mined by the relationship between the two systems of P -P bimodules arising from
N ⊂ P and P ⊂ M (see [6℄,[26℄). After [13℄ appeared, the se
ond-named author
gave a short proof of the above mentioned result using simple 
omputation of P −P
bimodules (see Remark 4.2). It turns out from the proof that the assumption in [13℄
is too restri
tive from the view point of bimodules. Therefore to 
apture more in-
teresting and general stru
ture, one should relax the assumption of supertransitivity.
Indeed, the �rst named author [12℄ re
ently obtained a 
omplete 
lassi�
ation result
of irredu
ible non-
ommuting quadrilaterals with N ⊂ P and N ⊂ Q isomorphi
 to
the Jones subfa
tor of index less than 4 (without posing any assumption on P ⊂ M
or Q ⊂ M) and showed that a series of su
h quadrilaterals exists.
In this paper we investigate quadrilaterals whose elementary in
lusions P ⊂ M ,
Q ⊂ M , N ⊂ P , and N ⊂ Q are ea
h 2-supertransitive (an in
lusion L ⊂ K is
2-supertransitive if the 
omplement of L in the L − L bimodule de
omposition of
K is irredu
ible.) In this 
ase the systems of P − P bimodules for N ⊂ P and for
P ⊂ M are ea
h generated by a single irredu
ible P −P bimodule, and the question
is how these two bimodules are related.
A key tool in this analysis is the notion of se
ond 
ohomology for subfa
tors,
introdu
ed by the se
ond named author and Kosaki in [27℄, whi
h 
ounts the inner

onjuga
y 
lasses of subfa
tors sharing the same basi
 extension (as a bimodule 
lass).
For a quadrilateral with N ⊂ P and N ⊂ Q 2-supertransitive, non
ommutativity is
equivalent to the existen
e of an N −N bimodule isomorphism from P to Q [13℄. If
N ⊂ P (dual) has trivial se
ond 
ohomology, this N −N bimodule isomorphism 
an
a
tually be realized as an algebra isomorhism. In parti
ular they showed that any
3-supertransitive subfa
tor has trivial se
ond 
ohomology.
Ultimately this leads to a 
lassi�
ation of non
ommuting quadrilaterals whose ele-
mentary subfa
tors are 2-supertransitive with trivial se
ond 
ohomology into two ba-
si
 types: those whi
h are not 
o
ommuting (�
o
ommuting� means that the quadri-
lateral of 
ommutants is a 
ommuting square) and have [M : P ] = [P : N ], like the
2)2 example, and those whi
h are 
o
ommuting and have [M : P ] = [P : N ]−1,
like the S3 example. The latter type is further subdivided a

ording to the Galois
group of N ⊂ M , whi
h must be a subgroup of S3. In the 
ase that the elementary
subfa
tors have index less than or equal to 4 we examine all the possibilities and
arrive at the following result.
Theorem 1.1. There are exa
tly seven non
ommuting irredu
ible quadrilaterals of
hyper�nite II1 fa
tors whose elementary in
lusions all have indi
es less than or equal
to 4, up to 
onjuga
y.
If (GN⊂P , GP⊂M) are the prin
ipal graphs of the elementary subfa
tors, the possible

on�gurations are:
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 3
(A7, A7), (E
7 , E
(A5, A3), (D6, A4), (E
7 , A5), (E
6 , D4)
6 , A3)
The fully supertransitive 
ases are (A7, A7), whi
h is the (2 +
2)2 example, and
(A5, A3), whi
h is the S3 example, and all other 
ases have some extra stru
ture. The
new examples in
lude a non
o
ommuting quadrilateral whose elementary subfa
tors
have index 4 (E
7 , E
7 ), a 
o
ommuting quadrilateral whose indi
es are (3 +
and (5+
5)/2 (D6, A4), and several 
o
ommuting quadrilaterals arising from group
a
tions. Note that any quadrilateral is either non
ommuting, non
o
ommuting (and
hen
e dual to a non
ommuting quadrilateral), or else is both 
ommuting and 
o
om-
muting, in whi
h 
ase there is no obstru
tion to the 
hoi
es of N ⊂ P and N ⊂ Q.
It is also interesting to study su
h quadrilaterals with larger indi
es. In fa
t the
exoti
 Haagerup subfa
tor appears as an elementary subfa
tor with maximal super-
transitivity in both types of quadrilaterals and the Asaeda-Haagerup subfa
tor may
appear as well. It is still unknown how many examples there are of the various types
of quadrilaterals (although there is an in�nite series of 
o
ommuting quadrilaterals
with 3-supertransitive elementary subfa
tors, 
oming from symmetri
 group a
tions.)
Se
ond 
ohomolgy also turns out to be 
losely related to angles. For a non
ommut-
ing quadrilateral with N ⊂ P and N ⊂ Q 2-supertransitive, there are at most two
possibilities for the angle for ea
h 
ohomolgy 
lass of N ⊂ P (dual). In parti
ular,
if N ⊂ P is 3-supertransitive, the angle is always cos−1 1/([P : N ] − 1). We also

ompute the angles in quadrilaterals arising from α-indu
tion (whi
h are identi�ed
with the GHJ pairs of [13℄) and asymptoti
 in
lusions, the former of whi
h in
ludes
the forked Temperley-Lieb quadrilaterals of [12℄.
The paper is organized as follows:
Se
tion 2 sets forth the basi
 notation of quadrilaterals, se
tors and Q-systems
whi
h are used throughout. We often deal with in�nite fa
tors sin
e we are a
tually
studying properties of the standard invariant.
Se
tion 3 studies the relationship between se
ond 
ohomology for subfa
tors and
angles.
Se
tion 4 proves that quadrilaterals whose elementary subfa
tors are 2-supertransitive
with trivial se
ond 
ohomology fall into the two basi
 types dis
ussed above, and that
for the 
o
ommuting type the Galois group of the total in
lusion is a subgroup of S3.
Se
tion 5 analyzes several 
lasses of quadrilaterals a

ording to the results of the
previous se
tion and for ea
h 
lass provides an example with maximal supertransi-
tivity.
Se
tion 6 
lassi�es all non
ommuting irredu
ible quadrilaterals whose elementary
subfa
tors have indi
es less than or equal to 4.
Se
tion 7 
omputes the angles for quadrilaterals 
oming from α-indu
tion and
asymptoti
 in
lusions.
The Appendix 
onstru
ts a Q-system from a se
tor in the Haagerup prin
ipal graph
and shows that it is unique, �lling in a gap for one of the examples in Se
tion 5.
4 PINHAS GROSSMAN AND MASAKI IZUMI
2. Preliminaries
We �rst �x notation used throughout this paper. We always assume that Hilbert
spa
es are separable and von Neumann algebras have separable preduals.
2.1. Quadrilaterals. For an in
lusion of II1 fa
tors N ⊂ M , the Jones index, the
Jones proje
tion, and the tra
e preserving 
onditional expe
tation onto N are de-
noted by [M : N ], eN and EN respe
tively [29℄. We use the same symbols for an
in
lusion of properly in�nite fa
tors N ⊂ M 
onsidering a unique minimum 
ondi-
tional expe
tation [38℄, [15℄. Although the Jones index [M : N ] does not ne
essarily

oin
ides with the minimum index for a II1 in
lusion in general, no 
onfusion arises
as we always 
onsider extremal in
lusions throughout this paper. For a II1 fa
tor, we
denote by tr the unique normalize tra
e. For an in
lusion of general fa
tors N ⊂ M
of �nite index, we use the same symbol tr for the restri
tion of EN to the relative

ommutant M ∩N ′.
Let N ⊂ M be an in
lusion of fa
tors of �nite index with asso
iated tower
N = M−1 ⊂ M = M0 ⊂ M1 ⊂ M2 ⊂ · · · ,
where Mk+1, k ≥ 0 is the von Neumann algebra on the standard spa
e of Mk gener-
ated by Mk and the Jones proje
tion ek+1 = eMk−1. Ea
h ek 
ommutes with N , so
{1, e1, .., ek} generates a *-subalgebra, whi
h we will 
all TLk+1, of the kth relative

ommutant N ′ ∩Mk. The following de�nition �rst appeared in [32℄.
De�nition 2.1. Call a �nite-index subfa
tor N ⊂ M k-supertransitive (for k > 1) if
N ′ ∩Mk−1 = TLk. We will say N ⊂ M is supertransitive if it is k-supertransitive for
all k.
Note that N ⊂ M is k-supertransitive if and only if M ⊂ M1 is k-supertransitive.
Sano and Watatani [54℄ introdu
ed the notion of angles between two subfa
tors.
De�nition 2.2. For two subfa
tors P and Q of a fa
tor M , the set of angles
Ang(P,Q) between P and Q is the spe
trum of the angle operator of the two
Jones proje
tions eP and eQ, that is, the spe
trum of cos
−1√eP eQeP − eP ∧ eQ where
eP eQeP − eP ∧ eQ is regarded as the operator a
ting on its support.
Note that Ang(P,Q) = Ang(Q,P ) always holds.
De�nition 2.3. A quadrilateral of fa
tors Q =
P ⊂ M
N ⊂ Q
is an in
lusion of fa
tors
N ⊂ M of �nite index with two intermediate subfa
tors P 6= Q su
h that P and Q
generate M and P ∩Q = N . The subfa
tors N ⊂ P , N ⊂ Q, P ⊂ M , and Q ⊂ M
are said to be the elementary subfa
tors for Q. When M ∩N ′ = C, the quadrilateral
Q is said to be irredu
ible. When EP 
ommutes with EQ, the quadrilateral Q is
said to be 
ommuting. Let M1 = JMN
′JM , P̂ = JMP
′JM , and Q̂ = JMQ
′JM be the
basi
 extensions of M by N , P , and Q respe
tively. Then P̂ and Q̂ are intermediate
subfa
tors between M ⊂ M1, whi
h form the dual quadrilateral Q̂ =
Q̂ ⊂ M1
N ⊂ P̂
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 5
The quadrilateral Q is said to be 
o
ommuting if Q̂ is 
ommuting. The quadrilateral
Q is said to be (p, q)-supertransitive if P ⊂ M and Q ⊂ M are p-supertransitive and
N ⊂ P and N ⊂ Q are q-supertransitive. Two quadrilaterals Q =
P ⊂ M
N ⊂ Q
P̃ ⊂ M̃
Ñ ⊂ Q̃
are said to be 
onjugate if there exists an isomorphism π from M
onto M̃ su
h that π(P ) = P̃ and π(Q) = Q̃. The two quadrilateral Q and Q̃ are
said to be �ip 
onjugate if either they are 
onjugate or there exists an isomorphism
π from M onto M̃ su
h that π(P ) = Q̃ and π(Q) = P̃ .
We denote by P̂ ⊂ M the in
lusion M ⊂ P̂ and by N̂ ⊂ P the in
lusion P̌ ⊂ N ,
where P̌ ⊂ N ⊂ P is the downward basi
 
onstru
tion. Note that P̌ is uniquely
determined up to inner 
onjuga
y in N .
When Q =
P ⊂ M
N ⊂ Q
is a non-
ommuting quadrilateral of subfa
tors su
h that
N ⊂ P and N ⊂ Q are 2-supertransitive, the two proje
tions eP and eQ are of rank
2 in M1∩N ′ and eP ∧ eQ = eN . Thus Ang(P,Q) 
onsists of at most one point, whi
h
will be denoted by Θ(P,Q) if Q is non-
ommuting.
The following lemma is essentially proved in [13, Proposition 3.2.11, Theorem 3.3.4℄
using Landau's result [41℄:
Lemma 2.4. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible 
o
ommuting quadrilaterals of
subfa
tors su
h that N ⊂ P and N ⊂ Q are 2-supertransitive. If [M : P ] = [P : N ],
the quadrilateral Q is 
ommuting, and if [M : P ] 6= [P : N ],
cos2Θ(P,Q) =
[P : N ]− [M : P ]
[M : P ]([P : N ]− 1) .
We show an easy example of a non
ommuting quadrilateral 
oming from a �nite
group a
tion. For an automorphism group G of a fa
tor R, we denote by RG the
�xed point subalgebra of R under G. The next lemma follows from Lemma 2.4.
Lemma 2.5. Let G be a �nite group and H and K be subgroups of G su
h that the
natural a
tions of G on G/H and G/K are 2-transitive. Assume that [G : H ] = [G :
K], [G : H ] 6= [H : H ∩ K], and G a
ts on a fa
tor R as an outer automorphism
group. We set M = RK∩H , P = RH , Q = RK , and N = RG. Then Q =
P ⊂ M
N ⊂ Q
is an irredu
ible non
ommuting and 
o
ommuting quadrilateral of fa
tors su
h that
N ⊂ P and N ⊂ Q are 2-supertransitive. The angle between P and Q is given by
cos2Θ(P,Q) =
[G : H ]− [H : H ∩K]
[H : H ∩K]([G : H ]− 1) .
6 PINHAS GROSSMAN AND MASAKI IZUMI
Example 2.6. For a �nite set F , we denote by SF the set of permutations of
F and by AF the set of even permutations of F . Let F0 = {1, 2, · · · , n}, F1 =
{1, 2, · · · , n − 1}, and F2 = {2, 3, · · · , n}. Then G = SF0 (G = AF0), H = SF1
(H = AF1), and K = SF2 (K = AF2) with n ≥ 3 (n ≥ 4) satis�es the assumption of
Lemma 2.5 and Θ(P,Q) = cos−1(1/(n− 1)).
In [13℄ Jones and the �rst-named author obtained the following theorem:
Theorem 2.7. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors. If Q is (6,6)-supertransitive, one of the following two o

urs:
(1) All the elementary subfa
tors of Q are the A7 subfa
tors and Θ(P,Q) =
cos−1(
2 − 1). When M is the hyper�nite II1 fa
tor, su
h a quadrilateral
exists and is unique up to 
onjuga
y.
(2) [M : P ] = [M : Q] = 2, [P : N ] = [Q : N ] = 3, Θ(P,Q) = π/3, and the
prin
ipal graphs of N ⊂ P and N ⊂ Q are A5. In this 
ase, there exists an
outer a
tion of the symmetri
 group S3 of degree 3 on M su
h that N is the
�xed point algebra of the a
tion. When M is the hyper�nite II1 fa
tor, su
h
a quadrilateral is unique up to 
onjuga
y.
In Se
tion 4, we give a new proof of the above theorem, ex
ept for the existen
e,
relaxing the assumption. The theorem still holds if we assume that Q is (3,4)-
supertransitive.
2.2. Se
tors. To 
ompute the angle between P and Q, it is more 
onvenient to work
with properly in�nite fa
tors using se
tors as we will see below. Thus we re
all a few
basi
 fa
ts about se
tors here. The reader is referred to [20℄ for details. Note that
the stru
ture of the intermediate subfa
tor latti
e of an in
lusion N ⊂ M , in
luding
information of the angle, only depends on the standard invariant (or paragroup,
plannar algebra) of N ⊂ M . Every standard invariant realized in the type II1 
ase is
also realized in the properly in�nite 
ase and vi
e versa.
For the reason stated above, we always assume in the proofs that the subfa
tors
involved are properly in�nite though we state results for general quadrilaterals of
fa
tors. When we show uniqueness results, we need to deal with quadrilaterals of
hyper�nite II1 fa
tors. Sin
e the 
lassi�
ation theory of subfa
tors for the hyper�nite
II1 fa
tor and the hyper�nite II∞ fa
tor is the same as far as strongly amenable
subfa
tors are 
on
erned [52℄, we may assume in the proofs that the fa
tors involved
are isomorphi
 to the hyper�nite II∞ fa
tors in this 
ase.
Remark 2.8. When one dis
usses se
tors, it is 
ustomary to assume that every fa
tor
involved is of type III. However the whole theory also works for general properly
in�nite fa
tors. This is based on the following fa
ts: (1) for an in
lusion of properly
in�nite fa
tors N ⊂ M , every non-zero proje
tion in the relative 
ommutant N ′ ∩
M is an in�nite proje
tion in M , and (2) a properly in�nite fa
tor has a unique
representation whose 
ommutant is also properly in�nite.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 7
Let M and N be properly in�nite fa
tors. We denote by Mor(N,M) the set of
normal unital homomorphisms from N to M . For two morphisms ρ, σ ∈ Mor(N,M),
we denote by (ρ, σ) the set of intertwiners between ρ and σ
(ρ, σ) = {v ∈ M ; vρ(x) = σ(x)v, ∀x ∈ N}.
When ρ is irredu
ible, that is M ∩ρ(N)′ = C, the spa
e (ρ, σ) is a Hilbert spa
e with
an inner produ
t 〈v1, v2〉 = v∗2v1 for v1, v2 ∈ (ρ, σ). The two morphisms ρ and σ are
said to be equivalent if there exists a unitary u ∈ (ρ, σ). We denote by Sect(M,N) the
quotient of Mor(N,M) by this equivalen
e relation. We denote by [ρ] the equivalen
e

lass of ρ, whi
h we 
all a se
tor. However, we often omit the quare bra
ket when
no 
onfusion arises. When M = N , we use the notation Mor(M,M) = End(M) and
Sect(M,M) = Sect(M).
For M −N bimodule X and ρ ∈ Mor(L,N), we denote by Xρ the M −L bimodule
de�ned by
x · ξ · y = xξρ(y), ξ ∈ X, x ∈ M, y ∈ L.
For σ ∈ Mor(L,M), we denote by σX the L−N bimodule similarly de�ned.
Let L2(M) be the standard Hilbert spa
e of M , whi
h is a M −M bimodule with
xξy = xJMy
∗JMξ, where JM is the modular 
onjugation. Then it is known that for
every M − N bimodule X , there exists a unique se
tor [ρ] ∈ Sect(M,N) su
h that
X is equivalent to L2(M)ρ. With this 
orresponden
e, the relative tensor produ
t of
two bimodules is transformed into the 
omposition of two morphisms. A dire
t sum
of bimodules is easily translated into the se
tor language too (see [20℄).
We denote by Mor0(N,M) the set of ρ ∈ Mor(N,M) whose image has �nite
index. For Mor0(N,M), we denote by d(ρ) the square root of [M : ρ(N)], 
alled
the (statisti
al) dimension of ρ, whi
h is multipli
ative under 
omposition of two
se
tors and is additive under dire
t sum. We use the notation Sect0(M,N) et
. in
an obvious sense.
Corresponding to the 
omplex 
onjugate bimodule of L2(M)ρ, the 
onjugate se
tor
of ρ is de�ned. For ρ ∈ Mor0(N,M), the (equivalen
e 
lass of) 
onjugate morphism
ρ̄ ∈ Mor0(M,N) is 
hara
terized by existen
e of two isometries rρ ∈ (idN , ρ̄ρ) and
r̄ρ ∈ (idM , ρρ̄) satisfying the following equation
r∗ρρ̄(r̄ρ) =
, r̄∗ρρ(rρ) =
Throughout this paper, we �x su
h rρ and r̄ρ for ea
h ρ. When ρ is not self-
onjugate,
we 
an and do assume rρ̄ = r̄ρ. This is not the 
ase for a self-
onjugate ρ. In this

ase r∗ρρ(rρ) = ±1/d(ρ) holds and so rρ = ±r̄ρ [42℄. A self-
onjugate endomorphism
ρ is said to be real if rρ̄ = r̄ρ and said to be pseudo-real if rρ̄ = −r̄ρ.
We set φρ(x) = r
ρρ̄(x)rρ for x ∈ N , whi
h is 
alled the standard left inverse of ρ.
We denote by Eρ the minimum 
onditional expe
tation from M onto the image of ρ.
Then we have Eρ = ρφρ.
For an in
lusion of properly in�nite fa
tors N ⊂ M of �nite index, we denote
by ιM,N the in
lusion map ιM,N : N →֒ M . Then the M − N bimodule X =
2(M)N is nothing but L
2(M)ιM,N . Sin
e ML
2(M1)M is isomorphi
 to X ⊗N X∗,
we have ML
2(M1)M ∼= L2(M)ιM,N ιM,N . The endomorphism ιM,N ιM,N of M is 
alled
8 PINHAS GROSSMAN AND MASAKI IZUMI
the 
anoni
al endomorphism for N ⊂ M . The endomorphism ιM,N ιM,N of N is

alled the dual 
anoni
al endomorphism, whi
h is the 
anoni
al endomorphism for
the in
lusion ιM,N (M) ⊂ N .
We often use diagrams for 
omputation of intertwiners. An intertwiner s ∈ (ρ, στ)
is expressed by the diagram
For t ∈ (σ, µν) and u ∈ (τ, ξη), the produ
ts ts ∈ (ρ, µντ) and σ(u)s ∈ (ρ, σξη) are
expressed as
µ ν τ
, σ(u)s =
For spe
ial operators 1 ∈ (ρ, ρ),
d(ρ)rρ ∈ (id, ρ̄ρ), and
d(ρ)r̄∗ρ ∈ (id, ρρ̄), we use
the following diagrammati
 expressions:
d(ρ)rρ =
d(ρ)r̄∗ρ =
Then the equations d(ρ)r̄∗ρρ(rρ) = 1 and d(ρ)ρ̄(r̄
ρ)rρ = 1 are expressed as
The following lemma holds (see [28, Se
tion 3℄):
Theorem 2.9. Let N ⊂ M be an irredu
ible in
lusion of properly in�nite fa
tors
with �nite index and let
[ιM,N ιM,N ] =
ni[ρi]
be the irredu
ible de
omposition, where ni is the multipli
ity of ρi in ιM,N ιM,N . Then
(1) ni ≤ d(ρi).
(2) Let Hi = (ιM,N , ιM,Nρi). For s1, s2 ∈ Hi,
EN(s1s
〈s1, s2〉
d(ρi)
(3) dimHi = ni.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 9
(4) Let {s(i)j}nij=1 be an orthonormal basis of Hi. Then x ∈ M has the following
unique expansion
s(i)∗jx(i)j
with 
oe�
ients x(i)j ∈ N . The 
oe�
ients x(i)j is given by
x(i)j = d(ρi)EN(s(i)jx).
The fourth statement above says that {
d(ρi)s(i)
j}ij is a Pimsner-Popa basis
[51℄. The advantage of working on properly in�nite fa
tors is that we 
an 
hoose
ea
h element of the basis from an intertwiner spa
e, and so (4) is 
onsidered as a

rossed produ
t type de
omposition.
Let Ai be the linear span of {s∗eN t; s, t ∈ Hi}. Then [28, Se
tion 3℄ shows that
Ai is a simple 
omponent of M ∩N ′ and
M1 ∩N ′ =
We 
an introdu
e a right a
tion of Ai on Hi by
v · (s∗eN t) =
d(ρi)
〈v, s〉t,
whi
h gives a ∗-isomorphism from the opposite algebra of Ai onto B(Hi). Let e(i)jk =
d(ρi)s(i)
jeNs(i)k. Then {e(i)jk}jk is a system of matrix unit of Ai. Let P be an
intermediate subfa
tor and let mi = dim(ιP,N , ιP,Nρi). Then we may and do assume
that {s(i)j}mij=1 is an orthonormal basis for Hi ∩ P = (ιP,N , ιP,Nρi).
Lemma 2.10. Let the notation be as above. Then
(1) The restri
tion EP |Hi of EP toHi is the proje
tion ontoHi∩P = (ιP,N , ιP,Nρi).
(2) The Jones proje
tion eP is expressed as
e(i)jj.
(3) Let zi be the unit of Ai. Then
s · zieP = EP (s), ∀s ∈ Hi.
Proof. (1) Note that EP (Hi) = Hi ∩ P = (ιP,N , ιP,Nρi) holds and EP |Hi is an idem-
potent. Thus it su�
es to show that EP |Hi is self-adjoint. For s, t ∈ Hi, we have
〈EP (s), t〉 = d(ρi)EN(EP (s)t∗) = d(ρi)EN(EP (EP (s)t∗))
= d(ρi)EN(EP (s)EP (t)
∗)) = 〈EP (s), EP (t)〉,
whi
h shows the statement.
(2) Let ϕ be a faithful normal state of M satisfying ϕ · EN = ϕ. Then (1) and
Theorem 2.9,(4) imply the following for x ∈ M and the GNS 
y
li
 and separating
10 PINHAS GROSSMAN AND MASAKI IZUMI
ve
tor Ω for ϕ:
ePxΩ = EP (
t(i)∗jx(i)j)Ω =
EP (t(i)j)
∗x(i)jΩ
t(i)∗jx(i)jΩ =
d(ρi)
t(i)∗jEN(t(i)jx)Ω
d(ρi)
t(i)∗jeN t(i)jxΩ.
This shows (2). (3) follows from (2). �
Remark 2.11. The above lemma gives a powerful method to 
ompute the angle be-
tween subfa
tors. Let Q be another intermediate subfa
tor of N ⊂ M . To 
ompute
the angle between P and Q, it su�
es to 
ompute the angle between two subspa
es
(ιP,N , ιP,Nρi) and (ιQ,N , ιQ,Nρi) of (ιM,N , ιM,Nρi) for those ρi 
ontained in ιP,N ιP,N
and ιQ,N ιQ,N . Equivalently, it su�
es to 
ompute the spe
trum of the restri
tion of
EPEQ to (ιP,N , ιP,Nρi).
Remark 2.12. Even when the in
lusion N ⊂ M is redu
ible, the angle Ang(P,Q) 
an
be 
omputed from EPEQ|(ιP,N ,ιP,Nρi) as long as N ⊂ P and N ⊂ Q are irredu
ible.
Indeed we have M = P ⊕ kerEP and
N(ιP,N , ιP,Nρi).
Sin
e EPEQEP preserves (ιP,N , ιP,Nρi), it su�
e to 
ompute EPEQ|(ιP,N ,ιP,Nρi) to ob-
tain the eigenvalues of EPEQEP .
3. Q-systems and angles
P ⊂ M
N ⊂ Q
is an irredu
ible non
ommuting quadrilateral of subfa
tors and
N ⊂ P and N ⊂ Q are 2-supertransitive, it is observed in [13, Lemma 4.2.1℄ that
L2(P ) and L2(Q) are equivalent as N −N bimodules, or in other words, ιP,N ιP,N is
equivalent to ιQ,NιQ,N . Consequently, [P : N ] = [Q : N ] and so [M : P ] = [M : Q].
This follows from the fa
t that the two proje
tions eP and eQ are equivalent in
M1 ∩ N ′, whi
h is identi�ed with EndNL2(M)N . Let P̌ and Q̌ be subfa
tors of N
su
h that P̌ ⊂ N ⊂ P and Q̌ ⊂ N ⊂ P are towers. It is known that under a

ertain situation, su
h as 3-supertransitivity of N ⊂ P and N ⊂ Q, the equivalen
e
2(P )N ∼= NL2(Q)N implies inner 
onjuga
y of P̌ and Q̌ in N [27℄, [35℄ though it
is not ne
essarily the 
ase in general.
To 
hara
terize the 
anoni
al endomorphism, Longo [43℄ introdu
ed the notion of
Q-systems.
De�nition 3.1. Let M be a properly in�nite fa
tor and let γ ∈ End0(M). We say
that (γ, v, w) is a Q-system if the following three 
onditions are satis�ed:
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 11
(1) v ∈ (idM, γ) and w ∈ (γ, γ2) are isometries.
(2) There exists a positive number d su
h that v∗w = w∗γ(v) = 1/d.
(3) ww = γ(w)w.
Two Q-systems (γ, v, w) and (γ′, v′, w′) are said to be equivalent if there exists a
unitary u ∈ (γ, γ′) su
h that v′ = uv and w′ = uγ(u)wu∗.
It is known that (3) is equivalent to ww∗ = γ(w∗)w under the assumption (1) and
(2) [45℄, [27℄. If γ = σσ̄ with σ ∈ Mor0(N ,M) and v = r̄σ ∈ (idM, γ), w = σ(rσ) ∈
σ((idN , σ̄σ)) ⊂ (γ, γ2), then (γ, v, w) is a Q-system with d = d(σ). The equivalen
e

lass of this Q-system only depends on the inner 
onjuga
y 
lass of σ(N ) in M.
Thus we say that (γ, v, w) arises from σ(N ) ⊂ M. On the other hand, Longo [43℄
showed that any Q-system (γ, v, w) arises from a subfa
tor N ⊂ M, determined by
EN (x) = w
∗γ(x)w, and equivalen
e of two Q-systems exa
tly 
orresponds to inner

onjuga
y of the 
orresponding two subfa
tors.
Let N ⊂ M be an in
lusion of properly in�nite fa
tors of �nite index and let
(γ, v, w) be the Q-system arising from the in
lusion. The se
ond-named author and
Kosaki [27℄ introdu
ed the �se
ond 
ohomology" H2(N ⊂ M), whi
h is always a
�nite set, by the equivalen
e 
lasses of Q-systems (γ1, v1, w2) su
h that γ and γ1
are equivalent. When N is the �xed point algebra of M by an outer a
tion of a
�nite group G, then we 
an identify H2(N ⊂ M) with the se
ond 
ohomology group
H2(G,T).
The following is [27, Lemma 6.1℄.
Lemma 3.2. Let N ⊂ M be an irredu
ible in
lusion of fa
tors of �nite index. If
N ⊂ M is 3-supertransitive, then H2(N ⊂ M) is trivial.
When Q =
P ⊂ M
N ⊂ Q
is a non
ommuting quadrilateral of subfa
tors and N ⊂ P
and N ⊂ Q are 2-supertransitive, the two in
lusions P̌ ⊂ N and Q̌ ⊂ N give
equivalent 
anoni
al endomorphisms as we observed above.
De�nition 3.3. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors with 2-supertransitive N ⊂ P and N ⊂ Q. We denote by c(Q) the 
lass of
the Q-system arising from Q̌ ⊂ N in H2(N̂ ⊂ P ).
The 
lass c(Q) is very mu
h related to the angle Θ(P,Q) and there are at most
two possibilities for Θ(P,Q) for a given 
lass c(Q) as we will see below.
The following lemma is an easy 
onsequen
e of the de�nition above:
Lemma 3.4. For an irredu
ible non
ommuting quadrilateral Q =
P ⊂ M
N ⊂ Q
fa
tors with 2-supertransitive N ⊂ P and N ⊂ Q, the following three 
onditions are
equivalent:
12 PINHAS GROSSMAN AND MASAKI IZUMI
(1) The 
lass c(Q) ∈ H2(N̂ ⊂ P ) is trivial.
(2) The two subfa
tors P̌ and Q̌ are inner 
onjugate in N .
(3) There exists an isomorphism π : P → Q of P onto Q su
h that the restri
tion
π|N of π to N is the identity map.
Sin
e the Q-systems we en
ounter in this paper always have γ de
omposed into two
irredu
ible 
omponents, we give a detailed des
ription of the 
ondition (3) of De�ni-
tion 3.1 in this 
ase. Let σ ∈ End0(M) be an irredu
ible endomorphism satisfying
[idM] 6= [σ] and let γ = idM ⊕ σ. Note that for γ to have a Q-system, the endo-
morphism σ must be self-
onjugate. We �x isometries v ∈ (idM, γ) and v1 ∈ (σ, γ),
whi
h means γ(x) = vxv∗ + v1σ(x)v
1. We give a des
ription of a Q-system of the
form (γ, v, w) in terms of an isometry in (σ, σ2). It is easy to show that an isometry
w ∈ (γ, γ2) satisfying (2) of De�nition 3.1 is of the following form and vi
e versa:
d(σ) + 1
v1σ(v)v
d(σ) + 1
d(σ) + 1
cv1σ(v1)rσv
d(σ)− 1
d(σ) + 1
v1σ(v1)sv
where c is a 
omplex number with |c| = 1 and s ∈ (σ, σ2) is an isometry. Passing from
w to uγ(u)wu∗ with u = vv∗ + c̄v1v
1 (or rede�ning rσ) if ne
essary, we may always
assume c = 1 for a Q-system (γ, v, w). We say that su
h a Q-system is normalized
(with respe
t to (v, v1, rσ)). It is straightforward to show the following:
Lemma 3.5. Let σ, γ, v, and v1 be as above. If s ∈ (σ, σ2) is an isometry satisfying
(3.1) σ(s)rσ = srσ,
(3.2)
d(σ)rσ + (d(σ)− 1)s2 =
d(σ)σ(rσ) + (d(σ)− 1)σ(s)s,
and w is given by
(3.3)
d(σ) + 1
v1σ(v)v
d(σ) + 1
d(σ) + 1
v1σ(v1)rσv
d(σ)− 1
d(σ) + 1
v1σ(v1)sv
then (γ, v, w) is a Q-system. Conversely, every normalized Q-system is of this form
with an isometry s ∈ (σ, σ2) satisfying (3.1) and (3.2). If s and s′ are isometries in
(σ, σ2) satisfyingly (3.1) and (3.2), the 
orresponding Q-systems are equivalent if and
only if s = ±s′.
Remark 3.6. When the Q-system (γ, v, w) as above arises from a 3-supertransitive
subfa
tor, we have dim(σ, σ2) = 1. If s, s′ ∈ (σ, σ2) are isometries satisfying (3.1)
and (3.2), there exists a 
omplex number ω of modulus 1 su
h that s′ = ωs and (3.2)
implies ω2 = 1. This proves Lemma 3.2 again.
We keep using the same notation as above and 
onsider the 
ase with an in
lusion
M ⊂ M1 su
h that γ = ῑι, v = rι, w = ῑ(r̄ι) where ι = ιM1,M. We set
d(ι)√
ι(v∗1)r̄ι,
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 13
whi
h is an isometry in (ι, ισ) thanks to Frobenius re
ipro
ity [20, Proposition 2.2℄.
Then Theorem 2.9,(4) shows that for every x ∈ M1, we have
x = EM(x) + d(σ)EM(xt
In parti
ular,
t2 = EM(t
2) + d(σ)EM(t
2t∗)t,
t∗ = EM(t
∗) + d(σ)EM(t
∗2)t = d(σ)EM(t
∗2)t,
where we use the fa
t EM(t) ∈ (idM, σ) = {0}. Therefore the ∗-algebra stru
ture of
M1 is 
ompletely determined by EM(t2) and EM(t2t∗), whi
h 
an be 
omputed in
terms of the Q-system as follows:
d(ι)2
r∗ι ῑ(ι(v
1)r̄ιι(v
1)r̄ι)rι =
v∗1 ῑ(ι(v
1)r̄ι)rι
σ(v∗1)v
1wv =
2t∗) =
d(ι)3
r∗ι ῑ(ι(v
1)r̄ιι(v
1)r̄ιr̄
ι ι(v1))rι =
v∗1 ῑ(ι(v
1)r̄ι)v1
σ(v∗1)v
1wv1 =
d(σ)− 1
d(σ)3
Lemma 3.7. Let the notation be as above. Then the following hold:
d(σ)− 1
d(σ)r∗σt.
Corollary 3.8. Let M be a properly in�nite fa
tor and let σ ∈ End0(M) be an
irredu
ible self-
onjugate endomorphism. If there exists a Q-system for idM⊕σ, then
[σ] is a real se
tor, that is, rσ = r̄σ.
Proof. Using one of the above formulae, we get
t = (t∗)∗ =
d(σ)(r∗σt)
d(σ)t∗rσ = d(σ)r
σtrσ = d(σ)r
σσ(rσ)t,
whi
h shows r∗σσ(rσ) = 1/d(σ). �
Now we show how the 
lass c(Q) determines the angle between P and Q. Let
P ⊂ M
N ⊂ Q
be an irredu
ible quadrilateral of fa
tors su
h that N ⊂ P and
N ⊂ Q are 2-supertransitive. We assume that NL2(P )N and NL2(Q)N are equivalent.
We apply the above argument to the 
ase where M1 = P , M = N , ι = ιP,N
and γ = ῑι. We set P̌ = ῑ(P ). We 
hoose two isometries v = rι ∈ (id, γ) and
v1 ∈ (σ, γ). Then we may assume that the Q-system (γ, v, wP = ῑ(r̄ι)) is normalized
with respe
t to (v, v1, rσ). By 
hoosing an appropriate representative of the 
onjugate
14 PINHAS GROSSMAN AND MASAKI IZUMI
se
tor of [ιQ,N ], we may also assume γ = ιQ,N ιQ,N , v = rιQ,N , and that the Q-system
(γ, v, wQ = ιQ,N(r̄ιQ,N )) is normalized with respe
t to (v, v1, rσ). By de�nition, the

lass c(Q) is given by (γ, v, wQ) in H
2(P̌ ⊂ N). Let sP and sQ be isometries in (σ, σ2)

orresponding to wP and wQ through (3.3) respe
tively. We set
d(σ) + 1
ιP,N(v
1)r̄ιP,N ∈ (ιP,N , ιP,Nσ),
d(σ) + 1
ιQ,N(v
1)r̄ιQ,N ∈ (ιQ,N , ιQ,Nσ),
whi
h are isometries.
Lemma 3.9. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible quadrilateral of fa
tors su
h
that N ⊂ P and N ⊂ Q are 2-supertransitive. If L2(P ) and L2(Q) are equivalent as
N −N bimodules, then
(1) 〈tP , tQ〉 satis�es the following quadrati
 equation:
〈tP , tQ〉2 −
d(σ)− 1
〈sP , sQ〉〈tP , tQ〉 −
(2) 〈sP , sQ〉 and 〈tP , tQ〉 are real numbers.
Proof. (1) Lemma 3.7 implies
〈tP , tQ〉2 = t2Q
t2P =
d(σ)− 1
〈sP , sQ〉〈tP , tQ〉.
(2) Thanks to Lemma 2.10, we have EP (tQ) = 〈tQ, tP 〉tP and EQ(tP ) = 〈tP , tQ〉tQ.
EN(tP tQ) = EN (EP (tP tQ)) = 〈tQ, tP 〉EN(t2P ) =
〈tQ, tP 〉√
EN(tP tQ) = EN (EQ(tP tQ)) = 〈tP , tQ〉EN(t2Q) =
〈tP , tQ〉√
whi
h shows that 〈tP , tQ〉 is real. This and (1) show that 〈sP , sQ〉 is also real. �
Note that dim(ιP,N , ιP,Nσ) = dim(ιQ,N , ιQ,Nσ) = 1 by Frobenius re
ipro
ity. Thus
Remark 2.11 shows cosΘ(P,Q) = |〈tP , tQ〉|.
Theorem 3.10. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible quadrilateral of fa
tors su
h
that N ⊂ P and N ⊂ Q are 2-supertransitive. If L2(P ) and L2(Q) are equivalent as
N −N bimodules, then
(1) Q is not 
ommuting.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 15
(2) The angle Θ(P,Q) is given by
cosΘ(P,Q) =
(d(σ)− 1)2〈sP , sQ〉2 + 4d(σ) + (d(σ)− 1)|〈sP , sQ〉|
2d(σ)
(d(σ)− 1)2〈sP , sQ〉2 + 4d(σ)− (d(σ)− 1)|〈sP , sQ〉|
2d(σ)
Proof. (1) If Q were 
ommuting, we would have 〈tP , tQ〉 = 0, whi
h never o

urs due
to Lemma 3.9,(1). (2) follows from Lemma 3.9 too. �
Note that |〈sP , sQ〉| is a numeri
al invariant of the 
lass c(Q). Therefore the above
theorem says that there are only two possibilities of the angle Θ(P,Q) given c(Q).
Corollary 3.11. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors su
h that N ⊂ P and N ⊂ Q are 2-supertransitive. Then
0 < Θ(P,Q) ≤ cos−1 1
[P : N ]− 1 .
The equality holds if and only if the 
lass c(Q) ∈ H2(P̌ ⊂ N) is trivial. In parti
ular,
if N ⊂ P is 3-supertransitive,
Θ(P,Q) = cos−1
[P : N ]− 1 .
Proof. Note that |〈tP , tQ〉| = 1 never o

urs sin
e |〈tP , tQ〉| = 1 would imply P = Q.
The statement follows from
(d(σ)− 1)2〈sP , sQ〉2 + 4d(σ)− (d(σ)− 1)|〈sP , sQ〉|
2d(σ)
where the quality holds if and only if 〈sP , sQ〉 = ±1. �
Corollary 3.12. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors su
h that N ⊂ P and N ⊂ Q are 2-supertransitive. If Q is 
o
ommuting,
[M : P ] ≤ [P : N ]− 1.
The equality holds if and only if the 
lass c(Q) ∈ H2(P̌ ⊂ N) is trivial. In parti
ular,
if Q is 
o
ommuting and N ⊂ P is 3-supertransitive,
[M : P ] = [P : N ]− 1.
Proof. This follows from Lemma 2.4 and Lemma 3.11. �
Remark 3.13. There is an example of a quadrilateral Q ful�lling the assumption of
Corollary 3.12 with non-trivial c(Q). Let q be a prime power and let F be a �nite
�eld with q elements. We set G = PSL(n, q) with n ≥ 3, whi
h naturally a
ts on the
proje
tive spa
e PF n−1. It is known that this a
tion is 2-transitive. We denote by
16 PINHAS GROSSMAN AND MASAKI IZUMI
F n∗ the dual spa
e of F n, on whi
h G naturally a
ts too. Let {ei}ni=1 be the 
anoni
al
basis of F n and let {e∗i }ni=1 be the dual basis of {ei}ni=1. Let H be the stabilizer of Fe1
and let K be the stabilizer of Fe∗1. Then we have [G : H ] = [G : K] = (q
n−1)/(q−1)
and [H : H ∩K] = [K : H ∩K] = qn−1. Let Q be the quadrilateral 
onstru
ted by
Lemma 2.5 with these G, H , and K. Then [M : P ] is stri
tly smaller than [P : N ]−1.
In this 
ase we have cosΘ(P,Q) = q−n/2.
Lemma 3.14. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors su
h that N ⊂ P and N ⊂ Q are 2-supertransitive and H2(N̂ ⊂ P ) is trivial.
Let [ιP,N ιP,N ] = [idN ] ⊕ [σ]. If dim(ιM,N , ιM,Nσ) = 2 and there is an intermediate
subfa
tor N ⊂ R ⊂ M other than P and Q su
h that L2(R) is equivalent to L2(P )
as N −N bimodules, then [P : N ] = 3.
Proof. Let tP and tQ be as in Lemma 3.9. We may and do assume 〈tP , tQ〉 = 1/d(σ)
by repla
ing tQ with −tQ if ne
essary. Assume that there exists an intermediate
subfa
tor N ⊂ R ⊂ M other than P and Q su
h that L2(R) is equivalent to L2(P )
as N − N bimodules. Then we may 
hoose an isometry t ∈ (ιR,N , ιR,Nσ) su
h that
〈t, tP 〉 = 1/d(σ), 〈t, tQ〉 = ǫ/d(σ) with ǫ ∈ {1,−1}, and EN (t2) = 1/
d(σ)rσ. Sin
e
dim(ιM,N , ιM,Nσ) = 2, the intertwiner spa
e (ιM,N , ιM,Nσ) is spanned by tP and tQ.
Therefore, the �rst two equalities imply
d(σ)− ǫ
d(σ)2 − 1(tP + ǫtQ).
Sin
e t is an isometry, we get
1 = 〈t, t〉 = (d(σ)− ǫ)
(d(σ)2 − 1)2 〈tP + ǫtQ, tP + ǫtQ〉 =
(d(σ)− ǫ)2
(d(σ)2 − 1)2
2(d(σ) + ǫ)
2(d(σ)− ǫ)
d(σ)(d(σ)2 − 1)
If ǫ = 1, we would have d(σ) = 1, whi
h 
ontradi
ts dim(ιM,N , ιM,Nσ) = 2 and
Theorem 2.9,(1). Therefore we get ǫ = −1 and d(σ) = 2, whi
h means [P : N ] =
Remark 3.15. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors. Then there exists σ 
ontained both in ιP,N ιP,N and in ιQ,N ιQ,N so that the
multipli
ity of σ in ιM,N ιM,N is at least 2. Therefore σ is not an automorphism due
to Theorem 2.9,(1). In parti
ular, the subfa
tor N ⊂ P is not the 
rossed produ
t
by a �nite group a
tion.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 17
4. (2,2)-supertransitive quadrilaterals
In what follows, for a non-
ommuting quadrilateral
P ⊂ M
N ⊂ Q
of properly in�nite
fa
tors with 2-supertransitive elementary subfa
tors, we use the following notation:
the symbols ι, κ, ιQ, and κQ denote the in
lusion maps
ι : N →֒ P, κ : P →֒ M, ιQ : N →֒ Q, κQ : Q →֒ M.
Sin
e ῑι, ιῑ, et
 are de
omposed into two irredu
ible se
tors, one of whi
h is the
identity se
tor, we 
hoose representatives ξ, η ∈ End0(P ), ξQ, ηQ ∈ End0(Q), ξ̂, ξ̂Q ∈
End0(N), and η̂, η̂Q ∈ End0(M) satisfying
[ιῑ] = [idP ]⊕ [ξ], [κ̄κ] = [idP ]⊕ [η],
[ιQῑQ] = [idQ]⊕ [ξQ], [κ̄QκQ] = [idQ]⊕ [ηQ],
[ῑι] = [ῑQιQ] = [idN ]⊕ [ξ̂],
[κκ̄] = [idM ]⊕ [η̂], [κQκ̄Q] = [idM ]⊕ [η̂Q].
Note that neither ξ nor ξ̂ is an automorphism (see Remark 3.15). When N ⊂ P
and N ⊂ Q are 3-supertransitive, the se
tors ξι and ξQιQ are de
omposed into two
irredu
ible 
omponents and we use the following notation:
[ξι] = [ι]⊕ [ι′], [ξQιQ] = [ιQ]⊕ [ι′Q].
In this 
ase, we also have
[ιξ̂] = [ι]⊕ [ι′], [ιQξ̂Q] = [ιQ]⊕ [ι′Q].
In the same way, when P ⊂ M and Q ⊂ M are 3-supertransitive, we use the following
notation:
[κη] = [κ]⊕ [κ′], [κQηQ] = [κQ]⊕ [κ′Q],
[η̂κ] = [κ]⊕ [κ′], [η̂QκQ] = [κQ]⊕ [κ′Q].
In the sequel we often use the te
hniques developed in [17℄. For example, let ρ, σ
be irredu
ible endomorphisms of P asso
iated with the in
lusion N ⊂ P . Then ρξ

ontains σ if and only if either [ρ] = [σ] or the distan
e between ρ and σ in the dual
prin
ipal graph of N ⊂ P is two. When [ρ] 6= [σ], the multipli
ity of σ in ρξ is the
number of the paths from ρ to σ. The multipli
ity of ρ in ρξ is
i=1 n
i − 1 where
[ρι] = ⊕mi ni[τi], is the irredu
ible de
omposition of ρι. In parti
ular, the in
lusion
N ⊂ P is 3-supertransitive if and only if the multipli
ity of ξ in ξ2 is one.
Lemma 4.1. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors. We assume that Q is (2,2)-supertransitive. Then
(1) [ξ] 6= [η].
(2) ξ2 
ontains η. Consequently, the depth of N ⊂ P is at least 4.
18 PINHAS GROSSMAN AND MASAKI IZUMI
Proof. (1) Sin
e N ⊂ M is irredu
ible, we have dim(κι, κι) = 1. Thus Frobenius
re
ipro
ity implies
1 = dim(κι, κι) = (κ̄κ, ιῑ) = (idP ⊕ η, idP ⊕ ξ),
whi
h shows [ξ] 6= [η].
(2) Sin
e Q is non
ommuting and
L2(P ) ∼= L2(Q) ∼= L2(N)⊕ L2(N)ξ̂,
as N −N bimodules, the N −N bimodule L2(M) 
ontains L2(N)ξ̂ with multipli
ity
at least two. Thus
2 ≤ dim(ῑκ̄κι, ξ̂) = dim(κ̄κ, ιξ̂ῑ) = dim(κ̄κ, ιῑιῑ)− dim(κ̄κ, ιῑ)
= dim(idP ⊕ η, ξ ⊕ ξ2) = 1 + dim(η, ξ2),
whi
h shows the statement. �
Remark 4.2. Using the above lemma, we give a short proof of Theorem 2.7 only
assuming (4,4)-supertransitivity of Q (see Corollary 4.9 and Corollary 5.18 below for
sharper results in this dire
tion). Sin
e N ⊂ P is 4-supertransitive, Lemma 4.1 shows
[ξ2] = [idP ]⊕ [ξ]⊕ [η].
If the prin
ipal graph is A5, η is an automorphism and M is the 
rossed produ
t
P ⋊η Z/2Z. Therefore the proof of [18, Theorem 3.1℄ implies that there exists an
outer a
tion of S3 on M su
h that N is the �xed point algebra of the a
tion. Assume
now that the prin
ipal graph of N ⊂ M is not A5. Then we have dim(ηι, ηι) ≥ 2.
Frobenius re
ipro
ity implies
dim(η2, ξ) = dim(η, ηξ) = dim(η, ηιῑ)− dim(η, η) = dim(ηι, ηι)− 1
Sin
e P ⊂ M is 4-supertransitive, we get [η2] = [idP ]⊕[η]⊕[ξ]. Therefore d(ξ) = d(η)
and d(ξ)2 = 1+2d(ξ), whi
h shows that [M : P ] = [P : N ] = 2+
2 = 4 cos2(π/8) and
all the elementary subfa
tors are the A7 subfa
tor. Uniqueness of su
h a quadrilateral
follows from Lemma 3.2 and Lemma 3.14 as η is uniquely determined by N ⊂ P (see
Theorem 4.7 below for details).
The following is one of the main results in this paper, whi
h says that irre-
du
ible non
ommuting (2,2)-supertransitive quadrilaterals with trivial 
ohomologi
al
obstru
tion are 
lassi�ed into four 
lasses and one ex
eptional 
ase.
Theorem 4.3. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors. We assume that Q is (2,2)-supertransitive and the 
lass c(Q) ∈ H2(N̂ ⊂ P )
is trivial. Then
Θ(M,N) = cos−1
[P : N ]− 1 .
Moreover,
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 19
(1) If Q is not 
o
ommuting and the 
lass c(Q̂) in H2(P ⊂ M) is trivial, then
[M : P ] = [P : N ]. In this 
ase, there exists an outer automorphism α of
P su
h that [η] = [αξ] = [ξα−1] and ξ2 
ontains η. (In 
onsequen
e, the
automorphism α is 
ontained in ξ3.)
(2) If Q is 
o
ommuting, then [M : P ] = [P : N ] − 1 and [ξ] = [κ̄κQπ], where
π is an isomorphism from P to Q su
h that the restri
tion of π to N is the
identity map. In this 
ase, the Galois group
Gal(M/N) = {θ ∈ Aut(M); θ(x) = x, ∀x ∈ N}
is either trivial or isomorphi
 to Z/2Z,Z/3Z, or the symmetri
 group S3 of
degree 3.
If Gal(M/N) = {idM , θ} ∼= Z/2Z, then θ swit
hes P and Q and [ξ] = [κ̄θκ].
If Gal(M/N) = {idM , θ, θ2} ∼= Z/3Z, then either θ(P ) = Q or θ(Q) = P
holds and [ξ] = [κ̄θκ] = [κ̄θ2κ].
If Gal(M/N) ∼= S3, then N is the �xed point algebra of an outer a
tion of
S3 on M and P and Q are the �xed point algebras of two di�erent order two
elements in S3.
Proof. The angle between P and Q was already 
omputed in Theorem 3.10. Lemma
3.4 shows that sin
e c(Q) is trivial there exists an isomorphism π from P onto Q
su
h that the restri
tion of π to N is trivial. This means ιQ = πι. Thus κι = κQιQ
implies κι = κQπι and
1 = dim(κι, κQπι) = dim(κιῑ, κQπ) = dim(κ, κQπ) + dim(κξ, κQπ)
= dim(κ, κQπ) + dim(ξ, κ̄κQπ).
We 
laim [κ] 6= [κQπ], and in 
onsequen
e, 
laim that ξ is 
ontained in κ̄κQπ. Indeed,
if [κ] = [κQπ] were the 
ase, there would exist a unitary u inM su
h that x = uπ(x)u
holds for all x ∈ P . In parti
ular, the unitary u 
ommutes with all x ∈ N and u is
a s
alar. However, this means P = Q, whi
h is a 
ontradi
tion. Therefore the 
laim
holds.
(1) Assume that Q is not 
o
ommuting and the 
lass c(Q̂) in H2(P ⊂ M) is
trivial. Then Lemma 3.4 applied to the dual quadrilateral Q̂ implies that P and Q
are inner 
onjugate in M and there exists a unitary v ∈ M satisfying vQv∗ = P .
We introdu
e an automorphism α of P by setting α(x) = vπ(x)v∗ for x ∈ P . This
implies [κα] = [κQπ] and so
[κ̄κQπ] = [κ̄κα] = [α]⊕ [ηα].
Sin
e ξ is 
ontained in κ̄κQπ, either [ξ] = [α] or [ξ] = [αη] holds. Note that [ξ] = [α]
never o

urs thanks to Remark 3.15. Thus we have [ξ] = [ηα] and so [η] = [ξα−1],
whi
h is equal to [αξ] for η is self-
onjugate. Sin
e ξ2 
ontains η and idP , we 
on
lude
that ξ3 
ontains α.
(2) Assume that Q is 
o
ommuting. Then Theorem 3.10,(1) applied to the dual
quadrilateral Q̂ implies that [κκ̄] 6= [κQκ̄Q] and so
dim(κ̄κQπ, κ̄κQπ) = dim(κκ̄, κQκ̄Q) = 1.
20 PINHAS GROSSMAN AND MASAKI IZUMI
This shows that κ̄κQπ is irredu
ible and [ξ] = [κ̄κQπ], whi
h implies
[P : N ]− 1 = d(ξ) = d(κ)d(κQ) = [M : P ].
(This is also shown in Corollary 3.12 with help of Lemma 2.4 though we don't need
to use it here.)
Note that the Galois group of the in
lusion N ⊂ M is isomorphi
 to the set of
1-dimensional se
tors G 
ontained in κιῑκ̄. (Note that sin
e N ⊂ M is irredu
ible,
the multipli
ity of any automorphism in κιῑκ̄ is at most one.) We assume that G is
non-trivial now. Using [ξ] = [κ̄κQπ], we get
[κιῑκ̄] = [κκ̄]⊕ [κξκ̄] = [idM ]⊕ [η̂]⊕ [κκ̄κQπκ̄]
= [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄].
Assume that η̂ is an automorphism. Then [M : P ] = 2 and so [P : N ] = 3.
Sin
e N ⊂ P is 2-supertransitive, it is the A5 subfa
tor. Therefore, the proof of [18,
Theorem 3.1℄ shows that there exists a unique outer a
tion of S3 on M su
h that N
is the �xed point algebra of the a
tion.
Assume that η̂ is not an automorphism. We 
laim that κQπκ̄ 
ontains an auto-
morphism. By the assumption of non-triviality of Gal(M/N), either κQπκ̄ or η̂κQπκ̄

ontains an automorphism. Assume that the latter 
ontains an automorphism, say
α. Then Frobenius re
ipro
ity implies
1 = dim(η̂κQπκ̄, α) = dim(κQπκ̄, η̂α),
and κQπκ̄ 
ontains η̂α. Sin
e d(κQπκ̄) = d(η̂α) + 1, we 
on
lude that κQπκ̄ 
ontains
an automorphism.
Let θ be an automorphism 
ontained in κQπκ̄. Then Frobenius re
ipro
ity implies
[θκ] = [κQπ]. We may assume that θκ = κQπ holds by 
hoosing an appropriate
representative of [θ]. Then θ(P ) = Q and θκι = κQπι = κQιQ = κι, whi
h shows
θ ∈ Gal(M,N). Using [θκ] = [κQπ], we get
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ [θκκ̄]⊕ [η̂θκκ̄]
= [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂].
If η̂θη̂ does not 
ontain any automorphism, then Gal(M,N) = {idM , θ} ∼= Z/2Z.
Assume that η̂θη̂ does 
ontain an automorphism, say, β. Then Frobenius re
i-
pro
ity again implies [η̂θ] = [βη̂] (and [θη̂] = [η̂β]), and
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [βη̂2].
Therefore
G = {[idM ], [θ]} ∪ {[βγ]}γ∈Gal(M/η̂(M)).
Sin
e [β] ∈ G, the group G1 = {[γ]}γ∈Gal(M/η̂(M)) is a subgroup of G su
h that
#G = 2 + #G1. Sin
e #G1 divides #G, either G1 is trivial or #G1 = 2. If G1 is
trivial, we get Gal(M/N) ∼= {[idM ], [θ], [β]} ∼= Z/3Z.
Suppose that G1 = {[idM ], [γ]}. Then G = {[idM ], [θ], [β], [βγ]} and [θ] = [γ].
This would imply [θη̂] = [γη̂] = [η̂] and [η̂θη̂] = [η̂2] would 
ontain [idM ]. However,
this means that κιῑκ̄ 
ontain idM with multipli
ity two, whi
h is a 
ontradi
tion.
Therefore G1 is trivial. �
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 21
Corollary 4.4. Let Q =
P ⊂ M
N ⊂ Q
be a (3,3)-supertransitive irredu
ible quadrilat-
eral of fa
tors su
h that Q is neither 
ommuting nor 
o
ommuting. Then Θ(P,Q) =
cos−1 1/([P : N ] − 1), [M : P ] = [P : N ] and there exists an outer automorphism α
of P su
h that [η] = [αξ] = [ξα−1]. Moreover,
(1) The automorphism α satis�es [αι′] = [ι′].
(2) dim(κι, κιξ̂) = 2.
(3) The Galois group Gal(M/N) is either trivial or isomorphi
 to Z/2Z. It is
isomorphi
 to Z/2Z if and only if the order of [α] is two. When Gal(M/N) =
{id, θ}, the automorphism θ swit
hes P and Q.
Proof. The �rst part follows from Lemma 3.2 and Theorem 4.3.
(1) Sin
e αξ is 
ontained in ι′ῑ, by Frobenius re
ipro
ity ι′ is 
ontained in αξι =
αι⊕αι′. Sin
e d(ι′) = d(ι)(d(ι)2−2), the equality [ι′] = [αι] would imply that N ⊂ P
is the A5 subfa
tor, whi
h does not allow ξ
to 
ontain αξ, and so [αι′] = [ι′].
(2) Thanks to Lemma 4.1, we have
[ῑκ̄κι] = [ῑι]⊕ [ῑηι] = [idN ]⊕ [ξ̂]⊕ [ῑαξι] = [idN ]⊕ [ξ̂]⊕ [ῑαι]⊕ [ῑαι′]
= [idN ]⊕ [ξ̂]⊕ [ῑαι]⊕ [ῑι′].
Frobenius re
ipro
ity implies
dim(ῑαι, ξ̂) = dim(αι, ιξ̂) = dim(αι, ι) + dim(αι, ι′) = dim(αι, ι),
where we use [αι] 6= [ι′] again. Sin
e the Galois group of N ⊂ P is trivial, we
have dim(αι, ι) = 0 and dim(ῑαι, ξ̂) = 0. Sin
e N ⊂ P is 3-supertransitive, the
endomorphism ῑι′ 
ontains ξ̂ with multipli
ity one and we get the statement.
(3) The Galois group Gal(M/N) is isomorphi
 to the set of 1 dimensional se
tors

ontained in κιῑκ̄. By a similar 
omputation as above,
[κιῑκ̄] = [κκ̄]⊕ [κξκ̄] = [idM ]⊕ [η̂]⊕ [κηακ̄] = [idM ]⊕ [η̂]⊕ [κακ̄]⊕ [κ′ακ̄].
Sin
e the depth of P ⊂ M is at least 4 due to Lemma 4.1, the endomorphism η̂ is
not an automorphism. If κ′ακ̄ 
ontained an automorphism θ, Frobenius re
ipro
ity
would imply [κ′α] = [θκ] and d(κ) = d(κ′), whi
h is a 
ontradi
tion.
Assume that Gal(M/N) is not trivial. Then κακ̄ 
ontains an automorphism, say,
θ. Then Frobenius re
ipro
ity implies [θκ] = [κα] and
[idP ]⊕ [αξ] = [κ̄κ] = [α−1κ̄κα] = [idP ]⊕ [α−1αξα] = [idP ]⊕ [ξα].
Sin
e [αξ] = [ξα−1], this shows that ξ2 
ontains α2. If α2 were outer, we would have
[α2ι] = [ι′] as ι′ῑ would 
ontain α2. However this 
ontradi
ts d(ι) 6= d(ι′). Thus
the order of [α] is two. This implies [θ2κ] = [κα2] = [κ]. Sin
e the Galois group of
P ⊂ M is trivial, we get [θ2] = [idM ]. Let β be another automorphism 
ontained
in κακ̄. Then the above argument shows that [θκ] = [βκ] and [β−1θκ] = [κ], whi
h
implies [β−1θ] = [idM ]. Therefore Gal(M/N) ∼= Z/2Z.
Assume now that the order of [α] is two. Then
[(κα)κα] = [α−1(idP ⊕ η)α] = [idP ]⊕ [α−1ηα] = [idP ]⊕ [η] = [κ̄κ].
22 PINHAS GROSSMAN AND MASAKI IZUMI
Sin
e H2(P̂ ⊂ M) is trivial, the subfa
tors α−1κ̄(M) and κ̄(M) are inner 
onjugate
in P and so there exists an automorphism θ of M su
h that [θκ] = [κα]. Sin
e the
Galois group of κ̄(M) ⊂ P is trivial, we have [κα] 6= [κ] and θ is outer. The same

omputation as above implies that [θ] is 
ontained in [κιῑκ̄] and so Gal(M/N) is not
trivial.
The above argument shows that when Gal(M/N) is isomorphi
 to Z/2Z, we have
[θκ] = [κα] = [κQπ] and we 
an 
hoose a representative θ of [θ] so that θκ = κQπ
holds. Then θ(P ) = Q and θκι = κQπι = κQιQ = κι, and so Gal(M/N) = {id, θ}.
Corollary 4.5. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilat-
eral of fa
tors su
h that Q is (2,2)-supertransitive and the 
lass c(Q) is trivial.
Assume that Q is 
o
ommuting and the Galois group Gal(M/N) is isomorphi
 to
Z/3Z. Then [M : P ] ≥ (5+
13)/2 and Q is neither (4,2)-supertransitive nor (2,3)-
supertransitive. If Q is (3,2)-supertransitive, the dual prin
ipal graph (to be more
pre
ise, the indu
tion-redu
tion graph for M − M and M − P bimodules) 
ontains
one of the graphs of the Haagerup subfa
tor.
Proof. Let Gal(M/N) = {idM , θ, θ2 = θ−1}. The proof of Theorem 4.3 shows that
[θη̂] = [η̂θ−1], the endomorphism η̂2 does not 
ontain any non-trivial automorphism
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [θ2η̂]⊕ [θ2η̂2].
Note that [η̂], [θη̂], [θ2η̂] are distin
t se
tors. Sin
e η̂2 
ontains η̂, the endomorphism
κιῑκ̄ 
ontains θ2η̂ with multipli
ity at least 2. Sin
e [θκι] = [κι], the multipli
ities of
η̂ and θη̂ in κιῑκ̄ are also at least 2, whi
h implies η̂2 
ontains idM ⊕ η̂ ⊕ θη̂ ⊕ θ2η̂.
This shows that P ⊂ M is not 4-supertransitive and d(η̂)2 ≥ 1+3d(η̂), whi
h implies
[M : P ] = 1 + d(η̂) ≥ (5 +
13)/2. Note that N ⊂ P is 3-supertransitive if and only
if dim(ξ, ξ2) = 1. Using [ξ] = [κ̄θκ], we get
dim(ξ, ξ2) = dim(κ̄θκ, κ̄θκκ̄θκ) = dim(κκ̄θκκ̄, θκκ̄θ)
= dim((idM ⊕ η̂)θ(idM ⊕ η̂), θ(idM ⊕ η̂)θ)
= dim(θ ⊕ θη̂ ⊕ θ2η̂ ⊕ θ2η̂2, θ2 ⊕ η̂)
= dim(θ2η̂2, θ2 ⊕ η̂) ≥ 2,
whi
h shows that N ⊂ P is not 3-supertransitive.
Assume that Q is (3,2)-supertransitive. Then κ′κ̄ 
ontains η̂ ⊕ θη̂ ⊕ θ2η̂. and the
dual prin
ipal graph of P ⊂ M 
ontains the following graph,
M − P
θη̂ θ
whi
h is one the prin
ipal graphs of the Haagerup subfa
tor [1℄. �
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 23
In Se
tion 5 and in the Appendix, we will show that there exists a (3,2)-supertransitive
quadrilateral of fa
tors in the above 
lass su
h that P ⊂ M is the Haagerup subfa
tor
and that su
h a quadrilateral is unique up to �ip 
onjuga
y.
Corollary 4.6. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors. Assume that Q is 
o
ommuting and (2,3)-supertransitive. Then Θ(P,Q) =
cos−1 1/([P : N ] − 1), [M : P ] = [P : N ] − 1, and the Galois group Gal(M/N) is
either trivial or isomorphi
 to Z/2Z or S3. Moreover,
(1) Two equalities [ηι] = [ι′] and [ῑκ̄κι] = [idM ]⊕ [ξ̂]⊕ [ῑι′] hold. In 
onsequen
e,
the endomorphism ι′ῑ 
ontains η with multipli
ity one and [ι′ῑ] = [η]⊕ [ηξ].
(2) dim(κι, κιξ̂) = 2.
(3) Assume that Gal(M/N) is isomorphi
 to Z/2Z and Gal(M/N) = {idM , θ}.
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂],
and dim(κιῑκ̄, κιῑκ̄) ≥ 7. The se
tors [η̂], [θη̂], [η̂θ], and [θη̂θ] are distin
t.
The endomorphism η̂θη̂ 
ontains θη̂θ with multipli
ity one and it does not

ontain either any automorphism or any of η̂, θη̂, η̂θ. The endomorphism η̂2
does not 
ontain any of θη̂, η̂θ, θη̂θ.
(4) The endomorphism ξ2 
ontains π̄ηQπ and [π̄ηQπι] = [ι
′] holds. Assume that
H2(P̂ ⊂ M) is trivial. Then the Galois group Gal(M/N) is trivial if and only
if [η] 6= [π̄ηQπ]. Assume further that Gal(M/N) is trivial. Then the dual
prin
ipal graph of N ⊂ P 
ontains D6 in su
h a way that every endpoint of
D6 is also an endpoint of the dual prin
ipal graph.
(5) Assume that P ⊂ M is 3-supertransitive and Gal(M/N) is trivial. Then
Q ⊂ M is 3-supertransitive as well and
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ 2[κQπκ̄]⊕ [κ′Qκ̄Q] = [idM ]⊕ [η̂Q]⊕ 2[κQπκ̄]⊕ [κ′κ̄].
The se
tor [κQπκ̄] is irredu
ible and self-
onjugate and [κ
′] = [κ′Qπ] holds.
Unless the prin
ipal graph of P ⊂ M is A4, the inequality dim(κιῑκ̄, κιῑκ̄) ≥ 8
holds.
Proof. The �rst part follows from Lemma 3.4, Theorem 4.3, and Corollary 4.5.
(1) Let d = d(ι) = [P : N ]− 1. Then we have d(ξ) = d2 − 1 and d(ι′) = d(d2 − 2).
On the other hand, we have d(η) = [M : P ] − 1 = [P : N ] − 2 = d2 − 2. Sin
e ηι

ontains ι′ and d(ηι) = d(ι′), we get [ηι] = [ι′].
(2) Using (1), we get
[ῑκ̄κι] = [ῑι]⊕ [ῑηι] = [idN ]⊕ [ξ̂]⊕ [ῑι′].
Sin
eN ⊂ P is 3-supertransitive, ῑι′ 
ontains ξ̂ with multipli
ity one and dim(κι, κιξ̂) =
dim(ῑκ̄κι, ξ̂) = 2.
(3) The proof of Theorem 4.3 shows that η̂θη̂ 
ontain no automorphism, [θκ] =
[κQπ], [ξ] = [κ̄θκ], and
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂].
24 PINHAS GROSSMAN AND MASAKI IZUMI
If [θη̂] = [η̂], we would have [η̂θη̂] = [η̂2], whi
h 
ontains idM . Thus [θη̂] 6= [η̂]. In the
same way, we get [η̂θ] 6= [η̂] and [θη̂] 6= [θη̂θ] 6= [η̂θ]. If [η̂] = [θη̂θ], we would have
[κκ̄] = [idM ]⊕ [η̂] = [idM ]⊕ [θη̂θ] = [θκκ̄θ] = [κQππ̄κ̄Q] = [κQκ̄Q],
whi
h 
ontradi
ts the assumption thatQ is 
o
ommuting thanks to Theorem 3.10,(1).
Thus [η̂] 6= [θη̂θ] and [θη̂] 6= [η̂θ]. Sin
e N ⊂ P is 3-supertransitive, we have
dim(ξ, ξ2) = 1, and
1 = dim(ξ, ξ2) = dim(κ̄θκ, κ̄θκκ̄θκ) = dim(κκ̄θκκ̄, θκκ̄θ)
= dim((idM ⊕ η̂)θ(idM ⊕ η̂), θ(idM ⊕ η̂)θ)
= dim(θ ⊕ θη̂ ⊕ η̂θ ⊕ η̂θη̂, idM ⊕ θη̂θ)
= dim(η̂θη̂, θη̂θ),
whi
h means that η̂θη̂ 
ontains θη̂θ with multipli
ity one. Sin
e [θκι] = [κι], the
multipli
ities of η̂, θη̂, η̂θ and θη̂θ in κιῑκ̄ should be the same, whi
h is the same as
η̂θη̂ 
ontains θη̂θ with multipli
ity one. Thus η̂θη̂ does not 
ontain any of η̂, θη̂, η̂θ.
Thanks to Frobenius re
ipro
ity we 
on
lude that η̂2 does not 
ontain any of θη̂, η̂θ,
θη̂θ.
(4) Sin
e [ξ] = [κ̄κQπ] = [π̄κ̄Qκ], we have
[ξ2] = [π̄κ̄Qκκ̄κQπ] = [π̄κ̄QκQπ]⊕ [π̄κ̄Qη̂κQπ],
whi
h 
ontains π̄ηQπ. Sin
e N ⊂ P is 3-supertransitive and π̄ηQπ (whi
h is not
equivalent to id) is 
ontained in ξ2, the morphism ι′ῑ 
ontains π̄ηQπ and so π̄ηQπι

ontains ι′. Sin
e d(ι′) = d(π̄ηQπι), we 
on
lude [π̄ηQπι] = [ι
The proof of Theorem 4.3 shows that Gal(M/N) is non-trivial if and only if there
exists and automorphism θ of M su
h that [θκ] = [κQπ]. Assume �rst that su
h
an automorphism θ exists. Then [κ̄κ] = [π̄κ̄QκQπ] and [η] = [π̄ηQπ] holds. Assume

onversely that [η] = [π̄ηQπ] holds. Then
[κ̄κ] = [idP ]⊕ [η] = [π̄(idQ ⊕ ηQ)π] = [π̄κ̄QκQπ].
Sin
e H2(P̂ ⊂ M) is trivial, the subfa
tors κ̄(M) and π̄κ̄Q(M) are inner 
onjugate in
P and there exists an automorphism θ of M su
h that [θκ] = [κQπ]. Therefore the
Galois group Gal(M/N) is trivial if and only if [η] 6= [π̄ηQπ].
Now we assume that Gal(M/N) is trivial. Then the dual prin
ipal graph of N ⊂ P
is as follows:
η π̄ηQπP − P
whi
h 
ontains D6.
(5) Sin
e [η] 6= [π̄ηQπ],
dim(κQπκ̄, κQπκ̄) = dim(π̄κ̄QκQπ, κ̄κ) = dim(idP ⊕ π̄ηQπ, idP ⊕ η) = 1,
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 25
and κQπκ̄ is irredu
ible. Sin
e [ξ] = [κ̄κQπ] is self-
onjugate,
1 = dim(κ̄κQπ, π̄κ̄Qκ) = dim(κQπκ̄, κπ̄κ̄Q),
and κQπκ̄ is self-
onjugate. This and the proof of Theorem 4.3 imply
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κπ̄κ̄Q]
= [idM ]⊕ [η̂]⊕ 2[κQπκ̄]⊕ [κ′π̄κ̄Q].
On the other hand, repla
ing P by Q, we get
[κιῑκ̄] = [idM ]⊕ [η̂Q]⊕ [κQπκ̄]⊕ [η̂QκQπκ̄].
Sin
e [η̂] 6= [η̂Q], the endomorphism η̂QκQπκ̄ 
ontains η̂ and
1 ≤ dim(η̂QκQπκ̄, η̂) = dim(η̂QκQπ, η̂κ) = dim(η̂QκQπ, κ⊕ κ′).
If η̂QκQπ 
ontained κ, we would have [η̂Q] = [κπ̄κ̄Q], whi
h 
ontradi
ts d(η̂Q) 6=
d(κπ̄κ̄Q). Therefore η̂QκQπ 
ontains κQπ and κ
, and so [η̂QκQπ] = [κQπ]⊕ [κ′] as we
have d(η̂QκQπ) = d(κQπ) + d(κ
′). This shows that Q ⊂ M is 3-supertransitive and
[κ′Qπ] = [κ
′], whi
h �nishes the proof. �
About uniqueness, we have the following:
Theorem 4.7. Let Q =
P ⊂ M
N ⊂ Q
and Q̃ =
P̃ ⊂ M̃
Ñ ⊂ Q̃
be irredu
ible non
om-
muting and non
o
ommuting quadrilaterals of fa
tors su
h that Q and Q̃ are (3,3)-
supertransitive. Assume that there exists an isomorphism Φ from P onto P̃ su
h that
Φ(N) = Ñ . If there exists no se
tor σ 
ontained in ξ2 su
h that [ξ] 6= [σ], [η] 6= [σ],
and d(σ) = d(η), then Q and Q̃ are 
onjugate.
Proof. Sin
e P ⊂ M and P̃ ⊂ M̃ are 3-supertransitive, H2(P̂ ⊂ M) and H2( ˜̂P ⊂ M̃)
are trivial thanks to Lemma 3.2. Note that we have [ιP̃ ,Ñ ιP̃ ,Ñ ] = [ΦιP,N ιP,NΦ
Therefore Lemma 4.1 implies [ιM̃,P̃ ιM̃,P̃ ] = [ΦιM,P ιM,PΦ
−1], whi
h shows that Φ ex-
tends to an isomorphism Ψ from M onto M̃ as H2(P̂ ⊂ M) and H2( ˜̂P ⊂ M̃) are
trivial. Lemma 3.14 and Corollary 4.4 show Ψ(Q) = Q̃. �
Theorem 4.8. Let Q =
P ⊂ M
N ⊂ Q
and Q̃ =
P̃ ⊂ M̃
Ñ ⊂ Q̃
be irredu
ible non-

ommuting and 
o
ommuting quadrilaterals of fa
tors su
h that H2(P̂ ⊂ M) and
H2( ˜̂P ⊂ M̃) are trivial and Q and Q̃ are (2,3)-supertransitive. Assume that there
exists an isomorphism Φ from P onto P̃ su
h that Φ(N) = Ñ . Then
(1) If Gal(M/N) ∼= Z/2Z and η is the only se
tor σ 
ontained in ξ2 su
h that
[σι] = [ι′], then Q and Q̃ are 
onjugate.
(2) If Gal(M/N) and Gal(M̃/Ñ) are trivial and there exists only two se
tors σ

ontained in ξ2 su
h that [σι] = [ι′], then Q and Q̃ are �ip 
onjugate.
26 PINHAS GROSSMAN AND MASAKI IZUMI
Proof. The proof of (1) is the same as that of Theorem 4.7. In (2),
[ιM̃,P̃ ιM̃,P̃ ] = [Φ(idP ⊕ π̄ηQπ)Φ−1] = [Φπ−1ιM,QιM,QπΦ−1]
may o

ur instead of [ιM̃ ,P̃ ιM̃ ,P̃ ] = [ΦιM,P ιM,PΦ
−1]. In this 
ase, the two quadrilater-
als are �ip 
onjugate. �
We 
an improve the assumption of Theorem 2.7.
Corollary 4.9. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors. Then
(1) If Q is not 
o
ommuting and (3,4)-supertransitive, then all the elementary
subfa
tors are isomorphi
 to A7 subfa
tors. There exists a unique su
h quadri-
lateral up to isomorphism when the fa
tors are hyper�nite II1 fa
tors.
(2) If Q is 
o
ommuting and (2,4)-supertransitive, then there exists an outer a
-
tion of S3 on M su
h that N is the �xed point algebra of the a
tion. There
exists a unique su
h quadrilateral up to isomorphism when the fa
tors are
hyper�nite II1 fa
tors.
Proof. Sin
e N ⊂ P is 4-supertransitive, we have [ξ2] = [idP ]⊕ [ξ]⊕ [η].
(1) Sin
e Q is not 
o
ommuting, there exists an automorphism θ of P su
h that
[η] = [θξ] and so we get d(ξ)2 = 1 + 2d(ξ). This shows that the prin
ipal graphs of
N ⊂ P and P ⊂ Q are A7. The uniqueness follows from Theorem 4.7.
(2) Sin
e Q is 
o
ommuting, we have [ηι] = [ι′] and so the prin
ipal graph of
N ⊂ P is A5. The rest of the proof has already been stated in Remark 4.2 ex
ept
for uniqueness, whi
h follows from the uniqueness of outer a
tions of �nite groups by
Jones [29℄ and the Galois 
orresponden
e. �
5. Classifi
ation I
A

ording to our dis
ussions in the last se
tion, we 
onsider the following four

lasses of quadrilaterals Q of fa
tors. We assume that every quadrilateral Q =
P ⊂ M
N ⊂ Q
of fa
tors appearing in this se
tion is irredu
ible and non
ommuting.
Class I: Q is non
o
ommuting and (3,3)-supertransitive.
Class II: Q is 
o
ommuting and (2,3)-supertransitive. The Galois group Gal(M/N)
is trivial.
Class III: Q is 
o
ommuting and (2,3)-supertransitive. The Galois group Gal(M/N)
is isomorphi
 to Z/2Z.
Class IV: Q is 
o
ommuting and (3,2)-supertransitive. The Class c(Q) is trivial. The
Galois group Gal(M/N) is isomorphi
 to Z/3Z.
For ea
h 
lass, we show that there exists an example of a quadrilateral and we seek
an example with maximal supertransitivity.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 27
We keep using the notation of se
tors su
h as ι, κ, et
. as in the previous se
tion,
We often use expli
it formulae of fusion rules of subfa
tors and the reader is refered
to [17℄ for them.
5.1. Class I. The stru
ture of quadrilaterals in Class I is relatively easy to des
ribe.
Theorem 5.1. Let N ⊂ P be an irredu
ible 3-supertransitive in
lusion of fa
tors.
Let ι = ιP,N and let [ιP,N ιP,N ] = [idP ] ⊕ [ξ] and [ξι] = [ι] ⊕ [ι′] be the irredu
ible
de
omposition. Assume that there exists an outer automorphism α ∈ Aut(P ) su
h
that [ξ] 6= [αξ] = [ξα−1] and ξ2 
ontains αξ. If idP ⊕ αξ has a Q-system, then there
exists a unique irredu
ible non
ommuting quadrilateral Q =
P ⊂ M
N ⊂ Q
in Class I
su
h that [η] = [αξ].
Proof. Assume that there exists a Q-system for idP ⊕αξ. We 
laim that a Q-system
with idP⊕αξ is unique up to equivalen
e. For this, it su�
es to show dim(αξ, (αξ)2) =
1 thanks to Lemma 3.5. As in the proof of Corollary 4.5, we have [αι′] = [ι′], [αι] 6= [ι],
and d(ι) 6= d(ι′). Thus
dim(αξ, (αξ)2) = dim(αξ, ξ2) = dim(αξ, ξιῑ)− dim(αξ, ξ) = dim(αξι, ξι)
= dim(αι⊕ αι′, ι⊕ ι′) = 1,
whi
h shows the 
laim. The 
laim implies that there exists a unique fa
tor M 
on-
taining N su
h that [κ̄κ] = [idP ]⊕ [αξ] where κ = ιM,P . Sin
e [ξ] 6= [αξ], the in
lusion
N ⊂ M is irredu
ible.
We next 
laim that και is irredu
ible and [και] = [κι]. Indeed, we have
dim(και, και) = dim(κ̄κα, αιῑ) = dim(α⊕ αξα, α⊕ αξ) = 1,
dim(και, κι) = dim(κ̄κα, ιῑ) = dim(α⊕ αξα, idP ⊕ ξ) = 1,
whi
h show the 
laim. The 
laim shows that there exists a unitary v ∈ M su
h that
vα(x)v∗ = x holds for every x ∈ N . We set π(y) = vα(y)v∗ for y ∈ P and set
Q = π(P ). By 
onstru
tion Q is an intermediate subfa
tor between N and M su
h
that ιQ,N = πι, [ιQ,N ιQ,N ] = [ῑι], and [M : P ] = [M : Q].
Suppose P = Q. Then π would be an automorphism of P , whi
h satis�es [κπ] =
[κα]. Frobenius re
ipro
ity implies that πα−1 is 
ontained in κ̄κ = idP ⊕ αξ and so
[π] = [α]. Thus there exists a unitary u in P su
h that vα(y)v∗ = uα(y)u∗ for all
y ∈ P and so u∗v ∈ P ′ ∩M = C. This shows that v ∈ P and [αι] = [ι], whi
h is a

ontradi
tion. Therefore P 6= Q.
Sin
e N ⊂ P is 3-supertransitive, it has no intermediate subfa
tor and P ∩Q = N .
Sin
e dim(αξ, (αξ)2) = 1, the in
lusion P ⊂ M is 3-supertransitive too and M is
generated by P and Q. Therefore Q =
P ⊂ M
N ⊂ Q
is an irredu
ible quadrilateral of
fa
tors. Theorem 3.10,(1) shows that Q is non
ommuting. Sin
e P and Q are inner

onjugate in M , Theorem 3.10,(1) applied to the dual quadrilateral Q̂ shows that Q
is non
o
ommuting. �
28 PINHAS GROSSMAN AND MASAKI IZUMI
The above theorem shows that quadrilaterals in Class I is 
ompletely determined
by the subfa
tor N ⊂ P and α.
Jones and the �rst-named author [13℄ showed that there exists a unique quadrilat-
eral of the hyper�nite II1 fa
tors su
h that all the elementary subfa
tors are the the
A7 subfa
tor. It is easy to show that this is the only quadrilateral in Class I satis-
fying [P : N ] < 4. Other than this example, we know two subfa
tors satisfying the
assumption of Theorem 5.1, namely, the E
7 subfa
tor and the Haagerup subfa
tors.
Let N ⊂ P be the E(1)7 subfa
tor. Then the dual prin
ipal graph is as follows:
P − P
Note that the 
ategory of P − P bimodules for the E(1)7 subfa
tor is isomorphi
 to
the 
ategory Â4 of the unitary representations of the alternating group A4. It is
observed in [24, Corollary 4.2℄ that there exists an automorphism of the 
ategory Â4
that �ips the two representations 
orresponding to ξ and αξ. Thus idP ⊕αξ has a Q-
system and gives rise to an irredu
ible non
ommuting quadrilateral in Class I whose
elementary subfa
tors are the E
7 subfa
tor. Note that this quadrilateral satis�es
the assumption of Theorem 4.7 and su
h a quadrilateral is unique.
Sin
e it is easy to see that no other subfa
tors of index less than or equal to 4 �t
into the statement of Theorem 4.3,(1), we get the following:
Theorem 5.2. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral of fa
tors in Class I su
h
that [M : P ] ≤ 4. Then one of the following two 
ases o

urs:
(1) The prin
ipal graphs of all the elementary subfa
tors are A7.
(2) The prin
ipal graphs of all the elementary subfa
tors are E
In ea
h 
ase, su
h a quadrilateral of hyper�nite II1 fa
tors exists and is unique up to

onjuga
y.
Let N ⊂ P be the Haagerup subfa
tor [1℄. Sin
e the Haagerup subfa
tor is not
self-dual, we spe
ify the dual prin
ipal graph (the indu
tion-redu
tion graph of the
P − P and P −N bimodules) as below.
P − P
It is known that α3 = idP and αξ = ξα
holds. Sin
e idP ⊕ ξ has a Q-system,
so does α−1(idP ⊕ ξ)α = idP ⊕ αξ. Thus there exists an irredu
ible non
ommuting
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 29
quadrilateral of fa
tors in Class I whose elementary subfa
tors are the Haagerup
subfa
tor. Although the quadrilateral arising from N ⊂ P does not satisfy the
assumption of Theorem 4.7, we have [κα] = [κQπ] and so [π̄ηQπ] = [α
−1αξα] = [α2ξ].
Thus the same proof of Theorem 4.8,(2) works and we 
an show uniqueness of su
h
a quadrilateral up to �ip 
onjuga
y.
We don't know if there are in�nitely many mutually non-
onjugate quadrilaterals
of the hyper�nite II1 fa
tors in Class I.
5.2. Class II. The following two are the main theorems of this subse
tion:
Theorem 5.3. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral of fa
tors in Class II su
h
that Q is (5,3)-supertransitive. Then
(1) If Q is (6,3)-supertransitive, the prin
ipal graphs of P ⊂ M and Q ⊂ M are
A4 and those of N ⊂ P and N ⊂ Q are D6.
(2) If Q is not (6,3)-supertransitive, the prin
ipal graphs of P ⊂ M and Q ⊂ M
are E
In ea
h 
ase, su
h a quadrilateral of the hyper�nite II1 fa
tors exists and is unique
up to 
onjuga
y.
Theorem 5.4. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral of fa
tors in Class II su
h
that [M : P ] ≤ 4. Then one of the following holds:
(1) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and
N ⊂ Q are D6. Su
h a quadrilateral of the hyper�nite II1 fa
tors exists and
is unique up to 
onjuga
y.
(2) The prin
ipal graphs of P ⊂ M and Q ⊂ M are E6 and the dual prin
ipal
graphs of N ⊂ P and N ⊂ Q are as below.
η π̄ηQπ αξ
αP − P
Su
h a quadrilateral of the hyper�nite II1 fa
tors exists.
(3) The prin
ipal graphs of P ⊂ M and Q ⊂ M are E(1)8 . Su
h a quadrilateral of
the hyper�nite II1 fa
tors exists and is unique up to 
onjuga
y.
In the sequel, the dimension of the spa
e (κιῑκ̄, κιῑκ̄) often gives important infor-
mation about a quadrilateral and we give useful formulae about it �rst.
Lemma 5.5. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors su
h that Q is 
o
ommuting and (2,3)-supertransitive. Then ι′ῑ 
ontains ξ
30 PINHAS GROSSMAN AND MASAKI IZUMI
and η with multipli
ity one, the equality [ι′ῑ] = [η]⊕ [ηξ] holds, and
dim(κιῑκ̄, κιῑκ̄) = 5 + dim(ηξ, ηξ).
If moreover P ⊂ M is 3-supertransitive, then [ηξ] = [ξ]⊕ [κ̄′κQπ] and
dim(κιῑκ̄, κιῑκ̄) = 6 + dim(κ̄′κQπ, κ̄
′κQπ) = 6 + dim(κ
′κ̄′, κQκ̄Q).
Proof. By Corollary 4.6,(1) and Frobenius re
ipro
ity, we have
dim(κιῑκ̄, κιῑκ̄) = dim(ῑκ̄κι, ῑκ̄κι) = dim(idN ⊕ ξ̂ ⊕ ῑι′, idN ⊕ ξ̂ ⊕ ῑι′)
= 2 + 2 dim(ιξ̂, ι′) + dim(ῑι′, ῑι′) = 4 + dim(ῑι′, ῑι′).
Sin
e [ῑιῑι] = 2[ῑι] ⊕ [ῑι′], the se
tor [ῑι′] is self-
onjugate, and so is [ι′ῑ] for a similar
reason. Thus we get
dim(ῑι′, ῑι′) = dim(ῑι′, ῑ′ι) = dim(ι′ῑ, ιῑ′) = dim(ι′ῑ, ι′ῑ).
The statement follows from Corollary 4.6,(1) now. �
In view of Corollary 4.6,(4), a 
anoni
al 
andidate of a quadrilateral in Class II
with the smallest index is that with the D6 subfa
tor for N ⊂ P .
Proposition 5.6. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral in Class II or Class III
su
h that the prin
ipal graph of P ⊂ M is A4. Then Q is in Class II and the prin
ipal
graph of N ⊂ P is D6. Su
h a quadrilateral of the hyper�nite II1 fa
tors exists and
is unique up to 
onjuga
y.
Proof. From Theorem 4.3, we see [P : N ] = [M : P ]+1 = (5+
5)/2 = 4 cos2(π/10),
whi
h shows that the prin
ipal graph of N ⊂ P is either A9 or D6. Thanks to
Theorem 2.7, the former never o

urs. If Q were in Class III, Corollary 4.6,(3) would
imply that η̂θη̂ does not 
ontain any automorphism and it 
ontains θη̂θ. However,
this 
ontradi
ts d(η̂θη̂)− d(θη̂θ) = d(η̂)2 − d(η̂) = 1 and Q is in Class II.
In the 
ase of the hyper�nite II1 fa
tors, uniqueness up to �ip 
onjuga
y follows
from Theorem 4.8,(2) and the uniqueness of the D6 subfa
tor. Let [ξ
2] = [idP ] ⊕
[ξ] ⊕ [ξ′] ⊕ [ξ′′] be the irredu
ible de
omposition of ξ2. To prove uniqueness up to

onjuga
y, it su�
es to show that there exists an automorphism α of P su
h that
α(N) = N and [αξ′α−1] = [ξ′′] (see the proof of Theorem 4.8,(2)). It is shown in [34℄
that there exists an automorphism β of period two on the A9 subfa
tor Ñ ⊂ P̃ su
h
(N ⊂ P ) ∼= (Ñ ⋊β Z/2Z ⊂ P̃ ⋊β Z/2Z).
The dual a
tion β̂ of β a
ts on the higher relative 
ommutants of N ⊂ P non-trivially.
It is routine work to show that β̂ does the right job.
Finally we show existen
e. Let N ⊂ M be an in
lusion of fa
tors whose prin
ipal
graph is D6. We use the following parameterization of se
tors asso
iated with N ⊂
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 31
idM ρ ρ1 ρ2
idN σ σ1 σ2
Re
all the fusion rule for the D6 subfa
tors [17, Proposition 3.10℄:
[ρ2] = [idM]⊕ [ρ]⊕ [ρ1]⊕ [ρ2], [ρ1ρ2] = [ρ2ρ1] = [ρ],
[ρ21] = [idM]⊕ [ρ1], [ρ22] = [idM]⊕ [ρ2],
[µ] = [ρ1ν] = [ρ2ν] = [νσ1] = [νσ2],
where ν = ιM,N . We 
an take representatives ρ1 and ρ2 satisfying µ = ρ1ν = ρ2ν.
We set M = M, P = ρ1(M), Q = ρ2(M), and N = µ(N ) = ρ1(N ) = ρ2(N ).
Then d(ρ1) = d(ρ2) = (1 +
5)/2, whi
h means [M : P ] = [M : Q] = 4 cos2 π/5,
and so the prin
ipal graphs of P ⊂ M and Q ⊂ M are A4. By 
onstru
tion, the
prin
ipal graphs of N ⊂ P and N ⊂ Q are D6. Sin
e µ is irredu
ible, the in
lusion
N ⊂ M is irredu
ible and Q =
P ⊂ M
N ⊂ Q
is an irredu
ible quadrilateral. Sin
e
[ρ21] = [idM] ⊕ [ρ1] and [ρ22] = [idM] ⊕ [ρ2], we have [ιM,P ιM,P ] = [idM ] ⊕ [ρ1] and
[ιM,QιM,Q] = [idM ]⊕ [ρ2], whi
h implies that Q is 
o
ommuting. Therefore [M : P ] 6=
[P : N ] shows that the quadrilateral Q is not 
ommuting. �
Remark 5.7. The above 
onstru
tion shows that the prin
ipal graph and the dual
prin
ipal graph of N ⊂ M are the same, whi
h 
an be easily 
omputed from
[µµ] = [ρ1νν̄ρ1] = [ρ1(idM ⊕ ρ)ρ1] = [idM ]⊕ [ρ1]⊕ [ρ2]⊕ 2[ρ],
[ρ1µ] = [ρ
1ν] = [ν]⊕ [ρ1ν] = [ν]⊕ [µ],
[ρ2µ] = [ρ
2ν] = [ν]⊕ [ρ2ν] = [ν]⊕ [µ],
[ρµ] = [ρρ1ν] = [ρν]⊕ [ρ2ν] = [ν]⊕ 2[µ],
[νµ̄] = [νν̄ρ1] = [ρ1]⊕ [ρρ1] = [ρ1]⊕ [ρ2]⊕ [ρ].
By symmetry of µ and µ̄, there exist two more intermediate subfa
tors L and R
between N and M su
h that the prin
ipal graphs of L ⊂ M and R ⊂ M are D6 and
those of N ⊂ L and N ⊂ R are A4. Sin
e ιM,LιM,L and ιM,RιM,R are 
ontained in
µµ̄, we see that they are equivalent to idM ⊕ ρ. There is no intermediate subfa
tor
other than P,Q, L,R. Indeed, if S is an intermediate subfa
tor, the endomorphism
ιM,SιM,S is 
ontained in µµ̄. Computation of the indi
es [M : S] and [S : N ] shows
that all the possibilities of ιM,SιM,S are exhausted by P,Q, L,R. Thus Lemma 3.14
applied to the dual quadrilateral implies the 
laim.
32 PINHAS GROSSMAN AND MASAKI IZUMI
Proposition 5.8. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral in Class II or Class III
su
h that the prin
ipal graph of P ⊂ M is A5. Then Q is in Class III and the
prin
ipal graph of N ⊂ P is E(1)7 . When M is the hyper�nite II1 fa
tor, su
h a
quadrilateral exists and is unique up to 
onjuga
y.
Proof. Sin
e the prin
ipal graph of P ⊂ M is A5, there exists an outer automorphism
of period two α ∈ Aut(P ) su
h that [κη] = [κ]⊕ [κα]. Sin
e [ξ] = [κ̄κQπ], we get
[ηξ] = [ηκ̄κQπ] = [κ̄κQπ]⊕ [ακ̄κQπ] = [ξ]⊕ [αξ].
Therefore Corollary 4.6,(1) shows that the dual prin
ipal graph of N ⊂ P is
αηP − P
whi
h is E
7 . Existen
e of su
h a quadrilateral follows from Example 2.6 with G =
S{1,2,3,4} and uniqueness follows from Theorem 4.8. Corollary 4.6,(4) shows that Q
is in Class III. �
Proposition 5.9. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral in Class II or Class III
su
h that the prin
ipal graph of P ⊂ M is E6. Then Q is in Class II and the dual
prin
ipal graph of N ⊂ P is as in Theorem 5.4 and [P : N ] = 3+
3. There exists a
quadrilateral of the hyper�nite II1 fa
tors satisfying the above property, whi
h arises
from the GHJ pair [13℄ (see Example 7.10) for E6 with ∗ given by the vertex with the
smallest entry of the Perron-Frobenius eigenve
tor.
Proof. From Theorem 4.3, we see [P : N ] = [M : P ] + 1 = 3 +
3. We re
all the
fusion rule for the E6 subfa
tors [17, p.968℄:
[η2] = [idP ]⊕ [α]⊕ 2[η], [αη] = [ηα] = [η], [α2] = [idP ],
[κη] = [κ]⊕ [κ′]⊕ [κα], [κ̄κ′] = [η],
[η̂2] = [idM ]⊕ [β]⊕ 2[η̂], [η̂κ] = [κ]⊕ [κ′]⊕ [βκ],
with [βκ] = [κα], d(κ) =
3, d(κ′) =
2, d(α) = d(β) = 1, d(η) = d(η̂) =
3, d(ξ) = 2 +
From [ξ] = [κ̄κQπ], we get [ηξ] = [ηκ̄κQπ] = [ξ]⊕ [κ̄′κQπ]⊕ [αξ]. Sin
e
dim(κ̄′κQπ, κ̄
′κQπ) = dim(κ
′κ̄′, κQκ̄Q) = dim(idM ⊕ β, idM ⊕ η̂Q) = 1,
the endomorphism κ̄′κQπ is irredu
ible, and Lemma 5.5 implies dim(κιῑκ̄, κιῑκ̄) = 8.
Suppose that Q is in Class III. Then thanks to Corollary 4.6,(3), we have
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [η̂θη̂],
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 33
and η̂θη̂ does not 
ontain any other irredu
ible 
omponents above. Therefore we
must have dim(η̂θη̂, η̂θη̂) = 3. However,
dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = dim(idM ⊕ θβθ ⊕ 2θη̂θ, idM ⊕ β ⊕ 2η̂)
= 1 + dim(θβθ, β),
whi
h is a 
ontradi
tion. Therefore Q is in Class II. By Corollary 4.6,(4), we get
[π̄ηQπ] = [κ̄
′κQπ] and the prin
ipal graph of N ⊂ P is as stated above. �
We ex
lude A7, D6, and E8 for the prin
ipal graph of P ⊂ M .
Lemma 5.10. There exists no quadrilateral Q =
P ⊂ M
N ⊂ Q
in Class II or Class
III su
h that the prin
ipal graph of P ⊂ M is A7.
Proof. Suppose there exists su
h a quadrilateralQ. Sin
e P ⊂ M is the A7 subfa
tor,
there exist outer automorphisms α of P and β of M and su
h that
[η̂2] = [idM ]⊕ [η̂]⊕ [βη̂],
[κ̄′κ] = [η]⊕ [αη].
Lemma 5.5 shows [ηξ] = [ξ]⊕ [κ̄′κQπ].
Suppose that κ̄′κQπ is irredu
ible. Then Lemma 5.5 shows that dim(κικ̄ῑ, κικ̄ῑ) =
7. Sin
e κ′κ̄ is redu
ible, Corollary 4.6,(3),(5) shows that the quadrilateral Q should
be in Class III and η̂θη̂ is de
omposed into two mutually inequivalent irredu
ibles,
whi
h implies
2 = dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = (idM ⊕ θη̂θ ⊕ θβη̂θ, idM ⊕ η̂ ⊕ βη̂).
Thus either [η̂] = [θβη̂θ], [βη̂] = [θη̂θ], or [βη̂] = [θβη̂θ] should hold. However, if
[η̂] = [θβη̂θ], we would get [η̂2] = [η̂η̂] = [θη̂2θ], whi
h is a 
ontradi
tion. The same
reasoning shows that the other two do not o

ur as well and we 
on
lude that κ̄′κQπ
is redu
ible.
Suppose that κ̄′κQπ is redu
ible. Sin
e any irredu
ible 
omponent ζ of κ̄
′κQπ is

ontained in ι′ῑ, Frobenius re
ipro
ity shows that ζι 
ontains ι′. On the other hand,
[κ̄′κQπι] = [κ̄
′κQιQ] = [κ̄
′κι] = [ηι]⊕ [αηι] = [ι′]⊕ [αι′].
This implies [ι′] = [αι′] and that κ̄′κQπ is de
omposed into two irredu
ibles, say, ζ1
and ζ2 su
h that [ζ1ι] = [ζ2ι] = [ι
′]. This shows [ι′ῑ] = [ξ]⊕[η]⊕[ζ1]⊕[ζ2] and the set of
P −P se
tors appearing the dual prin
ipal graph of N ⊂ P is {[idP ], [ξ], [η], [ζ1], [ζ2]}.
However, sin
e [αι′] = [ι′], the automorphism α appears in ι′ῑ′, whi
h is a 
ontradi
-
tion. Therefore the statement is shown. �
Lemma 5.11. There exists no quadrilateral Q =
P ⊂ M
N ⊂ Q
in Class II or Class
III su
h that the prin
ipal graph of P ⊂ M is D6.
34 PINHAS GROSSMAN AND MASAKI IZUMI
Proof. Suppose that Q is su
h a quadrilateral. We �rst re
all the fusion rule for the
D6 subfa
tor:
[κ′κ̄] = [η̂]⊕ [η̂1]⊕ [η̂2], [κ′κ̄′] = [idM ]⊕ 2[η̂]⊕ [η̂1]⊕ [η̂2],
[η̂2] = [idM ]⊕ [η̂]⊕ [η̂1]⊕ [η̂2],
[η̂21] = [idM ]⊕ [η̂1], [η̂2]2 = [idM ]⊕ [η̂2], [η̂1η̂2] = [η̂2η̂1] = [η̂],
[η̂1η̂] = [η̂η̂1] = [η̂]⊕ [η̂2], [η̂2η̂] = [η̂η̂2] = [η̂]⊕ [η̂1],
d(η̂) = (3 +
5)/2, d(η̂1) = d(η̂2) = (1 +
5)/2.
Lemma 5.5 shows [ηξ] = [ξ]⊕ [κ̄′κQπ]. We 
laim that κ̄′κQπ is irredu
ible. Indeed,
dim(κ̄′κQπ, κ̄
′κQπ) = dim(κ
′κ̄′, κQκ̄Q) = dim(idM ⊕ 2η̂ ⊕ η̂1 ⊕ η̂2, idM ⊕ η̂Q).
Sin
e Q is 
o
ommuting, we have [η̂] 6= [η̂Q] and we get the 
laim. Lemma 5.5 and
the 
laim implies that dim(κιῑκ̄, κιῑκ̄) = 7. Corollary 4.6,(3),(5) shows that Q is in
Class III and dim(η̂θη̂, η̂θη̂) = 2, and so
2 = dim(θη̂2θ, η̂2) = dim(idM ⊕ θη̂θ ⊕ θη̂1θ ⊕ θη̂2θ, idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2)
= 1 + dim(θη̂1θ ⊕ θη̂2θ, η̂1 ⊕ η̂2).
If [θη̂1θ] = [η̂2], we would have [θη̂2θ] = [η̂1] too as the period of θ is two, whi
h is a

ontradi
tion. Thus we may assume [θη̂1θ] = [η̂1] and [θη̂2θ] 6= [η̂1], [η̂2].
Sin
e η̂θη̂ 
ontains θη̂θ with multipli
ity one, there exists an irredu
ible ζ ∈
End0(M) inequivalent to θη̂θ su
h that [η̂θη̂] = [θη̂θ]⊕ [ζ ]. We 
laim that [ζ ] = [η̂θη̂2]
holds. Indeed, we have
dim(η̂θη̂2, η̂θη̂2) = dim(η̂
2, θη̂22θ) = dim(idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2, idM ⊕ θη̂2θ) = 1,
whi
h shows that η̂θη̂2 is irredu
ible. Sin
e [θη̂θ] 6= [η̂θη̂2] and
dim(η̂θη̂, η̂θη̂2) = dim(η̂
2, θη̂2η̂θ) = dim(idM ⊕ η̂ ⊕ η̂1 ⊕ η̂2, θη̂1θ ⊕ θη̂θ) = 1,
we get the 
laim. Sin
e
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [θη̂θ]⊕ [ζ ],
and [θκι] = [κι], we get [θζ ] = [ζ ]. However,
dim(η̂θη̂2, θη̂θη̂2) = dim(η̂θη̂, θη̂
2θ) = dim(θη̂θ ⊕ ζ, idM ⊕ θη̂2θ) = 0,
whi
h is a 
ontradi
tion. Thus the statement is proven. �
Lemma 5.12. There exists no quadrilateral Q =
P ⊂ M
N ⊂ Q
in Class II or Class
III su
h that the prin
ipal graph of P ⊂ M is E8.
Proof. Suppose thatQ is su
h a quadrilateral. Sin
e the E8 fa
tor is 4-supertransitive,
there exists an irredu
ible η̂′ ∈ End0(M) su
h that [η̂2] = [idM ]⊕ [η̂]⊕ [η̂′]. Note that
we have d(η̂) 6= d(η̂′) and η̂ is the only irredu
ible se
tor σ 
ontained in η̂′2 (also in
κ′κ̄′) su
h that d(η̂) = d(σ) (see [17, Se
tion 3.3℄).
As before, we have [ξη] = [ξ]⊕ [κ̄′κQπ] and
dim(κ̄′κQπ, κ̄
′κQπ) = dim(κ
′κ̄′, κQκ̄Q) = 1 + dim(κ
′κ̄′, η̂Q) = 1.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 35
Thus Q is in Class III and
2 = dim(η̂θη̂, η̂θη̂) = dim(θη̂2θ, η̂2) = dim(idM ⊕ θη̂θ ⊕ θη̂′θ, idM ⊕ η̂ ⊕ η̂′),
whi
h shows [θη̂′θ] = [η̂′]. However, this implies [θη̂′2θ] = [η̂′2], whi
h 
ontradi
ts
[θη̂θ] 6= [η̂] as η̂′2 
ontains η̂. �
We are ready to prove Theorem 5.3 now.
Proof of Theorem 5.3. Let Q be in Class II and (5,3)-supertransitive su
h that the
prin
ipal graph of P ⊂ M is not A4. Corollary 4.6,(5) shows dim(κιῑκ̄, κιῑκ̄) ≥ 8 and
Lemma 5.5 shows dim(κ̄′κQπ, κ̄
′κQπ) ≥ 2. Sin
e the prin
ipal graph of P ⊂ M is not
A5, the depth of P ⊂ M is at least 5. Thus we use the following parameterization of
M −M , M − P , and P − P se
tors arising from P ⊂ M .
κ κ′ κ′′
idM η̂ η̂
idP η η
M − P
P − P
Corollary 4.6,(5) shows
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ [η̂Q]⊕ [η̂′]⊕ 2[κQπκ̄].
Sin
e Q is 
o
ommuting, [η̂] 6= [η̂Q]. Sin
e the prin
ipal graph of P ⊂ M is neither
A5 nor A7, we have d(η̂Q) 6= d(η̂′), d(κQπ) 6= d(κ′) and d(κQπ) 6= d(κ′′). Sin
e
dim(κQπκ̄, η̂
′) = dim(κQπ, η̂
′κ) = dim(κQπ, κ
′ ⊕ κ′′) = 0,
we get [κQπκ̄] 6= [η̂′]. Thus dim(κιῑκ̄, κιῑκ̄) = 8 and dim(κ̄′κQπ, κ̄′κQπ) = 2. Thanks
to Corollary 4.6,(4),(5), the endomorphism κ̄′κQπ is de
omposed into π̄ηQπ and an
irredu
ible endomorphism of P , say ξ′. Sin
e
[κ̄′κQπι] = [κ̄
′κι] = [ηι]⊕ [η′ι] = [ι′]⊕ [η′ι],
and [π̄ηQπι] = [ι
′], we get [ξ′ι] = [η′ι], whi
h 
ontains ι′ as ξ′ is 
ontained in ι′ῑ. Thus
Frobenius re
ipro
ity implies
1 = dim(ι′, η′ι) = dim(ι′ῑ, η′) = dim(ξ ⊕ η ⊕ π̄ηQπ ⊕ ξ′, η′).
36 PINHAS GROSSMAN AND MASAKI IZUMI
If [ξ] = [η′], we would have [ξ′ι] = [ι] ⊕ [ι′] and by Frobenius re
ipro
ity [ξ′] = [ξ], a

ontradi
tion. Thus [ξ′] = [η′].
idP η π̄ηQπξ η′
P − P
We 
laim [ηπ̄ηQπ] = [ξ]⊕ [η′]. Thanks to Corollary 4.6(5), we know that κQπκ̄ is
self-
onjugate. Thus
[ξ2] = [κ̄κQπκ̄κQπ] = [κ̄κπ̄κ̄QκQπ]
= [(idP ⊕ η)(idP ⊕ π̄ηQπ)] = [idP ]⊕ [η]⊕ [π̄ηQπ]⊕ [ηπ̄ηQπ],
whi
h shows the 
laim.
Let [η′ι] = [ι′] ⊕
i=1mi[ιi], and [ιiῑ] = mi[η
j=1mij [ξj ] be the irredu
ible
de
ompositions. We 
ompute [ηξ] in two ways:
[ηξ][ξ] = [ξ2]⊕ [π̄ηQπξ]⊕ [η′ξ]
= [idP ]⊕ [ξ]⊕ [η]⊕ [π̄ηQπ]⊕ [η′]⊕ [ξ]⊕ [η]⊕ [η′]
⊕ [ξ]⊕ [η]⊕ [π̄ηQπ]⊕
m2i [η
mimij [ξj]
= [idP ]⊕ 3[ξ]⊕ 3[η]⊕ 2[π̄ηQπ]⊕ (2 +
m2i )[η
mimij [ξj],
[η][ξ2] = [η]⊕ [ηξ]⊕ [η2]⊕ [ηπ̄ηQπ]⊕ [ηη′]
= [η]⊕ [ξ]⊕ [π̄ηQπ]⊕ [η′]⊕ [idP ]⊕ [η]⊕ [η′]⊕ [ξ]⊕ [η′]⊕ [ηη′]
= [idP ]⊕ 2[ξ]⊕ 2[η]⊕ [π̄ηQπ]⊕ 3[η′]⊕ [ηη′].
This shows
[ηη′] = [ξ]⊕ [η]⊕ [π̄ηQπ]⊕ (
m2i − 1)[η′]⊕
mimij [ξj].
Sin
e P ⊂ M is 5-supertransitive, the multipli
ity of η′ in ηη′ is 1, whi
h implies
k = 2 and m1 = m2 = 1. This shows
[κ̄κ′′] = [η′]⊕ [π̄ηQπ]⊕ [ξ]⊕
(m1j +m2j)[ξj]
and P ⊂ M 
annot be 6-supertransitive. Frobenius re
ipro
ity implies that κπ̄ηQπ
and κξ 
ontain κ′′ with multipli
ity one and
[κξ] = [κκ̄κQπ] = [κQπ]⊕ [η̂κQπ],
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 37
On the other hand, the endomorphisms κπ̄ηQπ and η̂κQπ are irredu
ible as we have
dim(κπ̄ηQπ, κπ̄ηQπ) = dim(κ̄κπ̄ηQπ, π̄ηQπ) = dim(π̄ηQπ ⊕ ηπ̄ηQπ, π̄ηQπ)
= dim(π̄ηQπ ⊕ ξ ⊕ η′, π̄ηQπ) = 1,
dim(η̂κQπ, η̂κQπ) = dim(η̂
2, κQκ̄Q) = dim(idM ⊕ η̂ ⊕ η̂′, idM ⊕ η̂Q) = 1,
and so [κ′′] = [κπ̄η̂Qπ] = [η̂κQπ]. By indu
tion, we 
an get d(η) = d(κ)
2 − 1,
d(κ′) = d(κ)(d(κ)2− 2), d(η′) = d(κ)4− 3d(κ2)+ 1, d(κ′′) = d(κ)(d(κ)4− 4d(κ)2+3).
Therefore d(κ′′) = d(κπ̄η̂Qπ) implies d(κ)
4−5d(κ)2+4 = 0 and [M : P ] = d(κ2) = 4.
This shows that the prin
ipal graph of P ⊂ M is E(1)8 .
η′ π̄ηQπ ξ
κπ̄ηQπ κQπM − P
P − P
The above 
omputation shows that ξ is determined solely by the in
lusion P ⊂ M .
Sin
e N ⊂ P is 3-supertransitive, a Q-system for idP ⊕ ξ is unique up to equivalen
e
and so M is uniquely determined by P ⊂ M up to inner 
onjuga
y in P . Therefore
uniqueness of su
h a quadrilateral of the hyper�nite II1 fa
tors up to 
onjuga
y follows
from Theorem 3.14.
We show existen
e now. Let R be a fa
tor and α be an outer a
tion of the al-
ternating group A5. We regard A4 as a subgroup of A5 and �x it. We 
hoose two
mutually inequivalent 2-dimensional irredu
ible proje
tive unitary representations σ1
and σ2. Sin
e σ1 and σ2 
arry a unique non-trivial element in H
2(A5,T), we may
assume that they have the same 
o
y
le ω. Sin
e A4 has a unique 2-dimensional
irredu
ible proje
tive unitary representation up to equivalen
e, we may assume that
the restri
tion of σ1 and that of σ2 to A4 
oin
ide. Let M2(C) be the 2 by 2 matrix
algebra and let βg = αg⊗Adσ1(g) for g ∈ A5, whi
h is an a
tion of A5 on R⊗M2(C).
We set
N = (R⊗ C)⋊β A4 ⊂ P = (R⊗ C)⋊β A5 ⊂ M = (R⊗M2(C))⋊β A5.
Then the prin
ipal graph of P ⊂ M is E(1)8 [11, p.224℄, the in
lusion N ⊂ P is
3-supertransitive [39℄, and it is routine work to show that N ⊂ M is irredu
ible.
Let ug = 1⊗σ2(g)σ1(g)∗, whi
h is a β-
o
y
le, and let {λg}g∈A5 be the implementing
unitary inM = (R⊗M2(C))⋊βA5. Sin
e ug 
ommutes with R⊗C, we 
an introdu
e
a homomorphism π : P → M by setting π(x) = x for x ∈ R ⊗ C and π(λg) = ugλg
for g ∈ A5. We set Q = π(P ). The restri
tion of π to N is the identity map be
ause
ug = 1 for g ∈ A4. Sin
e σ1 and σ2 are not equivalent, there exists g ∈ A5 \ A4
su
h that ug is not a s
alar, and so P 6= Q. Sin
e A4 is maximal in A5 there is
no intermediate subfa
tor between N ⊂ P . Sin
e the prin
ipal graph of P ⊂ M
8 , there is no intermediate subfa
tor between P ⊂ M . Therefore we 
on
lude
38 PINHAS GROSSMAN AND MASAKI IZUMI
that Q =
P ⊂ M
N ⊂ Q
is a quadrilateral, whi
h is non
ommuting thanks to Theorem
3.10,(1). Theorem 4.3 shows thatQ is 
o
ommuting as we have [M : P ] = [P : N ]−1.
To �nish the proof, we show that the prin
ipal graph of Q ⊂ M is also E(1)8 . Sin
e
(N ⊂ Q) ∼= (R ⋊α A4 ⊂ R ⋊α A5), the 
ategory of Q − Q bimodules arising from
N ⊂ Q is equivalent to the unitary dual of A5. Lemma 4.1 shows that this 
ategory

ontains the 
ategory of Q − Q bimodules arising from Q ⊂ M . Sin
e [M : Q] = 4,
we 
on
lude that the prin
ipal graph of Q ⊂ M is E(1)8 . �
To prove Theorem 5.4, we have to ex
lude two more 
ases.
Proposition 5.13. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral in Class II or III su
h
that the prin
ipal graph of P ⊂ M is E(1)7 . Then Q is in Class III. There exists a
fa
tor R and an outer a
tion of G = S5 on R with two maximal subgroups H 6= K
of G su
h that M = RH∩K , P = RH , Q = RK, and N = RG. Su
h a quadrilateral
of the hyper�nite II1 fa
tors exists and is unique up to 
onjuga
y.
Proof. We use the following parameterization of M −M , M − P , and P − P se
tors
arising from P ⊂ M :
M − P
P − P
idM η̂ η̂
′ βη̂ β
κ κ′ βκ = κα
η η′ αη α
The 
ategory of M − M bimodules and P − P bimodules arising from P ⊂ M are
isomorphi
 to the unitary dual of S4, whi
h gives the fusion rule of the above se
tors.
We have
[η2] = [idP ]⊕ [η]⊕ [η′]⊕ [αη], [η′2] = [idP ]⊕ [α]⊕ [η′],
[αη] = [ηα], [αη′] = [η′α] = [η′], [α2] = [idP ],
[η̂2] = [idM ]⊕ [η̂]⊕ [η̂′]⊕ [βη̂], [η̂′2] = [idM ]⊕ [β]⊕ [η̂′],
[βη̂] = [η̂β], [βη̂′] = [η̂β ′] = [η̂′], [β2] = [idM ],
d(η) = d(η̂) = 3, d(η′) = d(η̂′) = 2, d(α) = d(β) = 1.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 39
First we 
laim that κ̄′κQπ is irredu
ible, whi
h shows that Q is in Class III thanks
to Corollary 4.6,(4) and Lemma 5.5. Indeed, we have
dim(κ̄′κQπ, κ̄
′κQπ) = dim(κ
′κ̄′, κQκ̄Q)
= dim(idM ⊕ β ⊕ 2η̂ ⊕ 2βη̂ ⊕ η̂′, idM ⊕ η̂Q)
= 1 + 2 dim(βη̂, η̂Q),
whi
h shows that either [βη̂] 6= [η̂Q] and κ̄′κQπ is irredu
ible or [βη̂] = [η̂Q] and κ̄′κQπ
is a dire
t sum of three irredu
ible distin
t se
tors. Suppose that the latter holds
and [κ̄′κQπ] = [ξ1]⊕ [ξ2]⊕ [ξ3]. Then ξiι would 
ontain ι′ for ea
h i = 1, 2, 3. On the
other hand, we have
[κ̄′κQπι] = [κ̄
′κι] = [ηι]⊕ [η′ι]⊕ [αηι] = [ι′]⊕ [η′ι]⊕ [αι′].
Sin
e d(η′) = 2, d(ι) =
5, and d(ι′) = 3
5, this is impossible and we get the 
laim.
Lemma 5.5 and Corollary 4.6,(3),(5) imply
2 = dim(η̂θη̂, η̂θη̂) = dim(η̂2, θη̂2θ)
= dim(idM ⊕ η̂ ⊕ η̂′ ⊕ βη̂, idM ⊕ θη̂θ ⊕ θη̂′θ ⊕ θβη̂θ)
= 1 + dim(η̂, θβη̂θ) + dim(βη̂, θη̂θ) + dim(βη̂, θβη̂θ) + dim(η̂′, θη̂′θ).
If [η̂] = [θβη̂θ], we would have [η̂2] = [η̂η̂] = [θη̂2θ], whi
h is impossible. For the same
reason, we get [βη̂] 6= [θη̂θ], [βη̂] 6= [θβη̂θ] and so [η̂′] = [θη̂′θ].
In [21, Corollary 4.8℄, the se
ond-named author showed that there exists a fa
tor
R and an outer a
tion µ of S4 on R su
h that M = R
and P = RS4. Let ν = ιR,M .
Then we have [ν̄ν] = [idM ]⊕ 2[σ]⊕ [ϕ] where d(σ) = 2, d(ϕ) = 1 and
[σ2] = [idM ]⊕ [ϕ]⊕ [σ], [ϕσ] = [σϕ] = [σ], [ϕ2] = [idM ].
We 
laim that [η̂′] = [σ] and [β] = [ϕ] hold. Indeed, sin
e d(η̂) = 3 and η̂2 does
not 
ontain any non-trivial automorphism, [idM ], [ϕ], [σ], [η̂], and [ϕη̂] are all distin
t
se
tors. Sin
e P ⊂ R is of depth 2 and [R : P ] = 24, we get
24 = dim(νκκ̄ν̄, νκκ̄ν̄) = dim(ν̄νκκ̄, κκ̄ν̄ν)
= dim((idM ⊕ ϕ⊕ 2σ)(idM ⊕ η̂), (idM ⊕ η̂)(idM ⊕ ϕ⊕ 2σ))
= dim(idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ ϕη̂, idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ η̂ϕ)
+ 2 dim(idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ ϕη̂, η̂σ) + 2 dim(ση̂, idM ⊕ ϕ⊕ 2σ ⊕ η̂ ⊕ η̂ϕ)
+ 4 dim(ση̂, η̂σ)
= 7 + dim(ϕη̂, η̂ϕ) + 2 dim(σ ⊕ ϕσ ⊕ 2σ2, η̂) + 2 dim(η̂2, σ) + 2 dim(ϕη̂, η̂σ)
+ 2 dim(η̂, σ ⊕ σϕ⊕ 2σ2) + 2 dim(σ, η̂2) + 2 dim(ση̂, η̂ϕ) + 4 dim(ση̂, η̂σ),
whi
h shows [ϕη̂] = [η̂ϕ] and
4 = 2 dim(σ2, η̂) + dim(η̂2, σ) + dim(η̂2ϕ, σ) + dim(ση̂, η̂σ)
= 2 dim(η̂2, σ) + dim(ση̂, η̂σ) = 2 dim(idM ⊕ η̂ ⊕ η̂′ ⊕ βη̂, σ) + dim(ση̂, η̂σ)
= 2 dim(η̂′, σ) + dim(ση̂, η̂σ).
If [η̂′] 6= [σ], we would have,
dim(ση̂, ση̂) = dim(σ2, η̂2) = dim(id⊕ ϕ⊕ σ, id⊕ η̂ ⊕ η̂′ ⊕ βη̂) = 1,
40 PINHAS GROSSMAN AND MASAKI IZUMI
whi
h means that ση̂ is irredu
ible. A similar argument shows that η̂σ is irredu
ible
too and dim(ση̂, η̂σ) is at most 1, whi
h is a 
ontradi
tion. Thus we get [σ] = [η̂′]
and in 
onsequen
e, the equality [ϕ] = [β] holds.
Sin
e Q is in Class III, the prin
ipal graph of Q ⊂ M is E(1)7 too and we 
an apply
the same argument to Q ⊂ M . Then there exists a fa
tor R̃ and an outer a
tion of
S4 on R̃ su
h that M = R̃
and Q = R̃S4 . Note that we have [θκ] = [κQπ], whi
h
implies [η̂Q] = [θη̂θ] and
[η̂2Q] = [idM ]⊕ [θη̂θ]⊕ [θη̂′θ]⊕ [θβη̂θ] = [idM ]⊕ [η̂Q]⊕ [η̂′]⊕ [θβη̂θ].
Thus the above argument shows [ιR̃,M ιR̃,M ] = [ιR,M ιR,M ]. The se
ond-named author
and Kosaki [26, Theorem 3.3℄ showed that the 
ohomology H2(M ⊂ R) is identi�ed
with H2(Ŝ3,T) introdu
ed by Wassermann [55℄. Sin
e the order of S3 is square free,
there is only one ergodi
 a
tion of S3 on a 6-dimensional C
-algebra and the latter
is trivial. Therefor we may assume R = R̃. We denote by µQ the a
tion of S4 on R
whose �xed point algebra is Q. We set H = µS4 and K = µ
We 
laim that N ⊂ R is irredu
ible. Indeed, we known that η̂θη̂ is a dire
t sum of
θη̂θ and an irredu
ible, say ζ , with d(ζ) = 6. Thus we have
dim(νκι, νκι) = dim(ν̄ν, κιῑκ̄)
= dim(idM ⊕ β ⊕ 2η̂′, idM ⊕ θ ⊕ η̂ ⊕ θη̂ ⊕ η̂θ ⊕ θη̂θ ⊕ ζ)
= 1 + dim(β, θ).
Sin
e [βη̂] = [η̂β], if [β] = [θ], the endomorphism η̂θη̂ would 
ontain an automorphism,
whi
h 
ontradi
ts Corollary 4.6,(3). Therefore the 
laim is shown.
Let G be the group generated by H and K in Aut(R). Then the �xed point algebra
for RG is P ∩ Q = N and the a
tion of G on R is outer for N ⊂ R is irredu
ible.
Sin
e N ⊂ P is 3-supertransitive, the a
tion of G on G/H is 3-transitive. Therefore
from [G : H ] = [P : N ] = [M : P ] + 1 = 5, we 
on
lude that G is isomorphi
 to S5
be
ause any 3-transitive transforation group of the �ve point set is either A5 or S5.
Uniqueness in the 
ase of the hyper�nite II1 fa
tors follows from Theorem 4.8. �
Proposition 5.14. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral in Class II or III su
h
that the prin
ipal graph of P ⊂ M is E(1)6 . Then Q is in Class III. There exists a
fa
tor R and an outer a
tion of G = A5 on R with two maximal subgroups H 6= K
of G su
h that M = RH∩K , P = RH , Q = RK, and N = RG. Su
h a quadrilateral
of the hyper�nite II1 fa
tors exists and is unique up to 
onjuga
y.
Proof. We employ the same strategy as in the proof of the previous proposition using
A4 instead of S4. Sin
e the prin
ipal graph of P ⊂ M is E(1)6 , there exists outer
automorphisms of period 3 α ∈ Aut(P ) and β ∈ Aut(M) su
h that
[η2] = [idP ]⊕ [α]⊕ [α2]⊕ 2[η], [αη] = [ηα] = [η],
[η̂2] = [idM ]⊕ [β]⊕ [β2]⊕ 2[η̂], [βη̂] = [η̂β] = [η̂].
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 41
idM η̂ β β2
κ βκ = κα β2κ = κα2
η α α2
M − P
P − P
Sin
e [M : Q] = 4, Theorem 5.3 and Proposition 5.13 show that the prin
ipal graph
of Q ⊂ M is also E(1)6 . Therefore there exists outer automorphism βQ ∈ Aut(M) of
period 3 su
h that
[η̂2Q] = [idM ]⊕ [βQ]⊕ [β2Q]⊕ 2[η̂Q], [βQη̂Q] = [η̂QβQ] = [η̂Q].
Sin
e [κη] = [κ] ⊕ [κα] ⊕ [κα2], we have [ηξ] = [ηκ̄κQπ] = [ξ] ⊕ [αξ] ⊕ [α2ξ], and
Lemma 5.5 shows dim(κιῑκ̄, κιῑκ̄) = 8. If Q is in Class III, Corollary 4.6,(3) implies
3 = dim(η̂θη̂, η̂θη̂) = dim(η̂2, θη̂2θ)
= dim(idM ⊕ β ⊕ β2 ⊕ 2η̂, idM ⊕ θβθ ⊕ θβ2θ ⊕ 2θη̂θ),
whi
h shows either [β] = [βQ] or [β] = [β
Q]. Sin
e the se
ond 
ohomology of the

y
li
 group {idM , β, β2} ∼= A3 is trivial, a similar proof, using either [16℄ or [21℄, as
in the 
ase of E
7 works .
Suppose now that Q is in Class II. (Note that sin
e the 
ohomology of the E
subfa
tor is not trivial, we 
annot apply Corollary 4.6,(4) as before.) Corollary 4.6,(4)
shows that π̄ηQπ is 
ontained in ξ
, whi
h is
[ξ2] = [idP ]⊕ [ξ]⊕ [αξ]⊕ [α2ξ]⊕ [η],
and so [η] = [π̄ηQπ]. The proof of Theorem 4.3 shows
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ [κQπκ̄]⊕ [η̂κQπκ̄].
Sin
e dim(κQπκ̄, κQπκ̄) = dim(π̄κ̄QκQπ, κ̄κ) = 2, the endomorphism κπ̄κ̄Q is de
om-
posed two irredu
ible endomorphism, say ρ and σ, neither of whi
h is an automor-
phism as we assumed that Q is in Class II. By Frobenius re
ipro
ity, the morphisms
ρκQπ and σκQπ 
ontain κ with multipli
ity one. Sin
e
[κπ̄κ̄QκQπ] = [κ(id⊕ η)] = 2[κ]⊕ [βκ]⊕ [β2κ],
we may assume [ρκQπ] = [κ]⊕[βκ] and [σκQπ] = [κ]⊕[β2κ]. By Frobenius re
ipro
ity
again, the morphism βκπ̄κ̄Q 
ontains ρ and the morphism β
2κπ̄κ̄Q 
ontains σ, and so
κπ̄κ̄Q 
ontains β
2ρ and βσ. If [βσ] = [σ], we also have [β2ρ] = [ρ] and so [βκπ̄κ̄Q] =
[κπ̄κ̄Q]. However, this would imply
2 = dim(βκπ̄κ̄Q, κπ̄κ̄Q) = dim(καπ̄κ̄Q, κπ̄κ̄Q) = dim(κ̄κα, π̄κ̄QκQπ)
= dim(α⊕ η, idP ⊕ η),
42 PINHAS GROSSMAN AND MASAKI IZUMI
whi
h is a 
ontradi
tion. Thus we get [ρ] = [βσ] and d(σ) = 2 and
[κιῑκ̄] = [idM ]⊕ [η̂]⊕ [σ]⊕ [βσ]⊕ [η̂σ]⊕ [η̂βσ]
= [idM ]⊕ [η̂]⊕ [σ]⊕ [βσ]⊕ 2[η̂σ].
Sin
e d(σ) = 2, the endomorphism η̂2 does not 
ontain σ and by Frobenius re
ipro
ity
η̂σ does not 
ontain η̂. Therefore the multipli
ity of η̂ in κιῑκ is one, and so the
multipli
ity of η̂Q is also one by symmetry. However, sin
e d(η̂Q) = 3 and [η̂] 6= [η̂Q],
this is impossible. Therefore Q is in Class III. �
Now the proof of Theorem 5.4 follows from the above propositions, lemmas, and
Theorem 5.3.
We don't know if there are in�nitely many mutually non-
onjugate quadrilaterals
of the hyper�nite II1 fa
tors in Class II.
5.3. Class III.
Theorem 5.15. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral of fa
tors in Class III su
h
that Q is (5,3)-supertransitive. Then
(1) If Q is (6,3)-supertransitive, the prin
ipal graphs of P ⊂ M and Q ⊂ M are
A5 and those of N ⊂ P and N ⊂ Q are E(1)7 .
(2) If Q is not (6,3)-supertransitive, the dual prin
ipal graph of P ⊂ M 
on-
tains one of the prin
ipal graphs of the Asaeda-Haagerup subfa
tor [1℄, and
in 
onsequen
e [M : P ] ≥ (5 +
17)/2.
Proof. Assume that the prin
ipal graph of P ⊂ M is not A5. Sin
e it is neither A3
nor A4, the depth of P ⊂ M is at least 5. We use the same symbols η̂′, κ′, κ′′, and
η′ for M − M , M − P , and P − P se
tors arising from P ⊂ M as in the proof of
Theorem 5.3. Corollary 4.6,(3) implies
2 ≤ dim(η̂θη̂, η̂θη̂) = dim(idM ⊕ η̂ ⊕ η̂′, idM ⊕ θη̂θ ⊕ θη̂′θ) = 1 + dim(η̂′, θη̂′θ),
whi
h shows that [η̂′] = [θη̂′θ] and the endomorphism η̂θη̂ is a dire
t sum of θη̂θ and
an irredu
ible endomorphism, whi
h we 
all ζ̂. Therefore we have
[κιῑκ̄] = [idM ]⊕ [θ]⊕ [η̂]⊕ [θη̂]⊕ [η̂θ]⊕ [θη̂θ]⊕ [ζ̂].
Sin
e θκι = κι, we have [θζ̂] = [ζ̂θ] = [ζ̂], whi
h is also self-
onjugate.
We 
laim [θη̂′] 6= [η̂′]. Suppose, on the 
ontrary, that [θη̂′] = [η̂′] holds. Then
[κ′]⊕ [κ′′] = [η̂′κ] = [θη̂′κ] = [θκ′]⊕ [θκ′′].
If [θκ′] = [κ′′], then the prin
ipal graph of P ⊂ M would be A9 and θ would 
ommute
with η̂, whi
h is a 
ontradi
tion. Thus [θκ′] = [κ′] holds. However, this implies
[η̂]⊕ [η̂′] = [κ′κ̄] = [θκ′κ̄] = [θη̂]⊕ [θη̂′] = [θη̂]⊕ [η̂′],
and [θη̂] = [η̂], whi
h is a 
ontradi
tion again. Therefore the 
laim is proven.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 43
Let [κ′′κ̄] = [η̂′]⊕
i=1mi[η̂i] be the irredu
ible de
omposition. We 
ompute η̂θη̂
in two ways:
[η̂θη̂][η̂] = [θη̂θη̂]⊕ [ζ̂ η̂] = [η̂θ]⊕ [θζ̂ ]⊕ [ζ̂ η̂] = [η̂θ]⊕ [ζ̂]⊕ [ζ̂ η̂],
[η̂θ][η̂2] = [η̂θ]⊕ [η̂θη̂]⊕ [η̂θη̂′] = [η̂θ]⊕ [θη̂θ]⊕ [ζ̂]⊕ [η̂η̂′θ]
= 2[η̂θ]⊕ [θη̂θ]⊕ [η̂′θ]⊕ [ζ̂]⊕
mi[η̂iθ],
whi
h implies
[ζ̂ η̂θ] = [η̂]⊕ [θη̂]⊕ [η̂′]⊕
mi[η̂i].
Sin
e [θ][ζ̂ η̂θ] = [ζ̂ η̂θ] and [θ][η̂′] 6= [η̂′], we may assume that m1 = 1, [θη̂′] = [η̂1] and
mi[θη̂i] =
mi[η̂i].
By Frobenius re
ipro
ity, the endomorphism η̂′θη̂ 
ontains ζ̂ with multipli
ity one.
On the other hand, we have
[η̂′θη̂] = [θη̂′η̂] = [θη̂]⊕ [θη̂′]⊕ [η̂′]⊕
mi[θη̂i] = [θη̂]⊕ [θη̂′]⊕ [η̂′]⊕
mi[η̂i].
Corollary 4.6,(3) shows [ζ̂] 6= [θη̂], and [θη̂′] 6= [η̂′] implies [ζ̂] 6= [η̂′], [θη̂′] as [θζ̂] = [ζ̂]
holds. Thus we may assume that m2 = 1 and [ζ̂] = [η̂2]. This in parti
ular shows
that Q 
annot be (6, 3)-supertransitive.
Now we have
[ζ̂ η̂] = [η̂θ]⊕ [θη̂θ]⊕ [η̂′]⊕ [θη̂′]⊕ [ζ̂]⊕
mi[η̂iθ],
and [ζ̂] 6= [η̂iθ] for i ≥ 3. Sin
e the multipli
ity of ζ̂ in ζ̂ η̂ is one, there are exa
tly
two verti
es in the dual prin
ipal graph of P ⊂ M 
onne
ted to ζ̂, one of whi
h is κ′′.
Therefore there exists an irredu
ible morphism κ′′′ su
h that [ζ̂κ] = [κ′′]⊕ [κ′′′] with
[κ′′′] 6= [κ′′]. We 
laim [κ′′′] = [θη̂θκ] = [η̂θκ]. Indeed, 
omputing dim(θη̂θκ, θη̂θκ),
dim(η̂θκ, η̂θκ), and dim(η̂θκ, θη̂θκ), we 
an show that θη̂θκ and η̂θκ are irredu
ible
and [θη̂θκ] = [η̂θκ]. Sin
e θη̂θκκ̄ 
ontains ζ̂ with multipli
ity one, Frobenius re
i-
pro
ity implies that ζ̂κ 
ontains θη̂θκ. If [θη̂θκ] = [κ′′] we would have d(η̂)d(κ) =
d(κ′′), whi
h implies [M : P ] = 4. Sin
e P ⊂ M is 5-supertransitive, the prin
ipal
graph of P ⊂ M should be E(1)8 . However, sin
e the M − M se
tors arising from
P ⊂ M is non
ommutative, we get a 
ontradi
tion. Therefore the 
laim is proven.
44 PINHAS GROSSMAN AND MASAKI IZUMI
The above 
omputation shows that the dual prin
ipal graph is as follows.
idM κ η̂ κ
′ η̂′ κ′′ θη̂′ θκ′ θη̂ θκ θ
θη̂θκ = η̂θκ
θη̂θ η̂θ
whi
h 
ontains one of the prin
ipal graphs of the Asaeda-Haagerup subfa
tor [1℄. �
Remark 5.16. We do not known whether there exists a (5,3)-supertransitive quadri-
lateral of fa
tors in Class III with [M : P ] = (5 +
17)/2. If su
h a quadrilateral of
hyper�nite II1 fa
tors exists, it must be unique up to 
onjuga
y. Indeed, the above
argument shows that the in
lusion P ⊂ M must be Asaeda-Haagerup subfa
tor and
ξ = κ̄θκ is determined by P ⊂ M . Sin
e N ⊂ P is 3-supsertransitive, a Q-system for
idP ⊕ ξ is unique up to equivalen
e and M is uniquely determined by P ⊂ M up to
inner 
onjuga
y in P . Sin
e Q = θ(P ), where the representative θ of the se
tor [θ] is

hosen so that the restri
tion θ|N is trivial, the quadrilateral is uniquely determined
by P ⊂ M . For existen
e, the same 
riterion using a Q-system as in the 
ase of
the Haagerup subfa
tor below (Theorem 5.19) should work in prin
iple, though it
involves heavy 
omputation in pra
ti
e.
Theorem 5.15 and the propositions and lemmas in the previous subse
tion imply
the following theorem:
Theorem 5.17. Let Q =
P ⊂ M
N ⊂ Q
be a quadrilateral of fa
tors in Class III su
h
that [M : P ] ≤ 4. Then one of the following holds:
(1) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and
N ⊂ Q are E(1)7 . The quadrilateral is given by the 
onstru
tion in Example
2.6 with S{1,2,3,4}.
(2) The prin
ipal graphs of P ⊂ M and Q ⊂ M are E(1)6 . The quadrilateral is
given by the 
onstru
tion in Example 2.6 with A{1,2,3,4,5}.
(3) The prin
ipal graphs of P ⊂ M and Q ⊂ M are E(1)7 . The quadrilateral is
given by the 
onstru
tion in Example 2.6 with S{1,2,3,4,5}.
In ea
h 
ase, su
h a quadrilateral of the hyper�nite II1 fa
tors exists and is unique
up to 
onjuga
y.
Corollary 5.18. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting (6,3)-supertransitive
quadrilateral of fa
tors. Then one of the following holds:
(1) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and
N ⊂ Q are A5.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 45
(2) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and
N ⊂ Q are D6.
(3) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and
N ⊂ Q are E(1)7 .
(4) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A7 and those of N ⊂ P and
N ⊂ Q are A7.
In ea
h 
ase, su
h a quadrilateral of the hyper�nite II1 fa
tors exists and is unique
up to 
onjuga
y.
Thanks to Example 2.6, there are 
ountably many mutually non-
onjugate (3,3)-
supertransitive quadrilaterals of the hyper�nite II1 fa
tors in Class III.
5.4. Class IV.
Theorem 5.19. There exists a quadrilateral of fa
tors Q =
P ⊂ M
N ⊂ Q
in Class
IV with [M : P ] = (3 +
13)/2. Su
h a quadrilateral of the hyper�nite II1 fa
tors is
unique up to �ip 
onjuga
y.
Proof. Let P ⊂ M be the Haagerup subfa
tor whose prin
ipal graph and the dual
prin
ipal graph are as follows.
M − P
θη̂ θ
idP κ̄θκη η
P − P
It is known that θ and η̂ satis�es the following relation [1℄:
[θ3] = [idM ], [θ
−1η̂] = [η̂θ], [η̂2] = [idM ]⊕ [η̂]⊕ [θη̂]⊕ [θ2η̂].
In the Appendix, we will show that there exists a unique Q-system for idP ⊕κ̄θκ up to
equivalen
e. Therefore there exists a unique subfa
tor N of P up to inner 
onjuga
y
su
h that [ιῑ] = [idP ]⊕ [κ̄θκ], where ι : N →֒ P is the in
lusion map. Sin
e
dim(κι, κι) = dim(κ̄κ, ιῑ) = dim(idP ⊕ η, idP ⊕ κ̄θκ) = 1,
the in
lusion N ⊂ M is irredu
ible. Sin
e
[κιῑκ̄] = [κκ̄]⊕ [κκ̄θκκ̄] = [idM ]⊕ [η̂]⊕ [(idM ⊕ η̂)θ(idM ⊕ η̂)]
= [idM ]⊕ [η̂]⊕ [θ]⊕ [θη̂]⊕ [θ2η̂]⊕ [θ2η̂2]
= [idM ]⊕ [θ]⊕ [θ2]⊕ 2[η̂]⊕ 2[θη̂]⊕ 2[θ2η̂],
46 PINHAS GROSSMAN AND MASAKI IZUMI
we 
an take representative θ of the se
tor [θ] su
h that θκι = κι. Then θ3 is an inner
automorphism satisfying θ3κι = κι and so θ3 = idM as N ⊂ M is irredu
ible. We
set Q = θ(P ). Using a similar argument as in the proof of Theorem 5.1, we 
an
show that
P ⊂ M
N ⊂ Q
is the desired quadrilateral. Uniqueness up to �ip 
onjuga
y
follows from uniqueness of the Q-system for id⊕ κ̄θκ. �
We do not know if there exists a quadrilateral of the hyper�nite II1 fa
tors in Class
IV di�erent from the above example.
6. Classifi
ation II
Theorem 6.1. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral
of fa
tors su
h that the indi
es of all the elementary subfa
tors are less than or equal
to 4. Then one of the following o

urs:
(1) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and
N ⊂ Q are A5. In this 
ase N is the �xed point algebra of an outer a
tion of
the symmetri
 group S3 on M .
(2) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A4 and those of N ⊂ P and
N ⊂ Q are D6.
(3) The prin
ipal graphs of all the elementary subfa
tors are A7.
(4) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A3 and those of N ⊂ P and
N ⊂ Q are D(1)6 . In this 
ase N is the �xed point algebra of an outer a
tion
of the dihedral group D8 of order 8 on M .
(5) The prin
ipal graphs of P ⊂ M and Q ⊂ M are D4 and those of N ⊂ P and
N ⊂ Q are E(1)6 . In this 
ase N is the �xed point algebra of an outer a
tion
of the alternating group A4 on M .
(6) The prin
ipal graphs of P ⊂ M and Q ⊂ M are A5 and those of N ⊂ P and
N ⊂ Q are E(1)7 . In this 
ase the quadrilateral Q is given by the 
onstru
tions
in Example 2.6 with S{1,2,3,4}.
(7) The prin
ipal graphs of all the elementary subfa
tors are E
In ea
h 
ase, su
h a quadrilateral of the hyper�nite II1 fa
tors is unique up to 
onju-
ga
y.
We use the symbols ι = ιP,N , ιQ = ιQ,N , κ = ιM,P , κQ = ιM,Q. When N ⊂ P is
2-supertransitive, we also use the symbols ξ and ξ̂ as in Se
tion 4.
Lemma 6.2. Let Q =
P ⊂ M
N ⊂ Q
be an irredu
ible non
ommuting quadrilateral of
fa
tors su
h that the indi
es of all the elementary subfa
tors are less than or equal to
4. Then
(1) The prin
ipal graph of N ⊂ P is neither A3, D4 nor D(1)4 .
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 47
(2) If the prin
ipal graph of N ⊂ P is E(1)6 , then that of P ⊂ M is D4. There
exists an outer a
tion a
tion of the alternating group A4 on M su
h that N
is the �xed point algebra of the a
tion.
(3) If the prin
ipal graph of N ⊂ P is D(1)n , then n = 6 and the prin
ipal graph of
P ⊂ M is A3. There exists an outer a
tion a
tion of the dihedral group D8
of order 8 on M su
h that N is the �xed point algebra of the a
tion.
(4) If N ⊂ P and N ⊂ Q are 2-supertransitive, then the prin
ipal graph of P ⊂ M
is not D
Proof. (1) follows from Remark 3.15.
(2) Sin
e the prin
ipal graph of N ⊂ P is E(1)6 , there exists an outer automorphism
α ∈ Aut(P ) of order 3 su
h that
[ξ2] = [idP ]⊕ [α]⊕ [α2]⊕ 2[ξ].
The proof of Lemma 4.1 shows that dim(κ̄κ, ξ) = 0 and
2 ≤ dim(κ̄κ, ξ2) = dim(κ̄κ, idP ⊕ α⊕ α2).
Therefore κ̄κ 
ontains α and so the prin
ipal graph of P ⊂ M is D4. The rest of the
statement follows from [16℄, [21℄.
(3) Thanks to (1), the number n is at least 5 and we have irredu
ible de
omposition
[ιῑ] = [idP ]⊕ [α]⊕ [ξ], [ῑι]⊕ [τ ]⊕ [ξ̂],
where d(ξ) = d(ξ̂) = 2 and α ∈ Aut(P ) and τ ∈ Aut(N) are outer automorphisms
of order 2 satisfying [αξ] = [ξ], [τ ξ̂] = [ξ̂]. Sin
e [κιτ ] = [κι], the 
rossed produ
t
N ⋊τ Z/2Z is identi�ed with the intermediate subfa
tor of N ⊂ P generated by N
and a unitary u ∈ M satisfying uxu∗ = τ(x) for all x ∈ N . If ῑQιQ 
ontained τ
as well, the two in
lusions N ⊂ P and N ⊂ Q would have a 
ommon intermediate
subfa
tor N ⋊ τZ/2Z, whi
h is a 
ontradi
tion as we have P ∩ Q = N . Sin
e Q is
non
ommuting ῑQιQ must 
ontain ξ̂ and the prin
ipal graph of N ⊂ Q is either A5
m with m ≥ 5.
Assume that the prin
ipal graph of N ⊂ Q is D(1)m . Then we have the irredu
ible
de
omposition [ῑQιQ] = [idN ] ⊕ [τQ] ⊕ [ξ̂] su
h that [τQ] 6= [τ ] and τQ is 
ontained
in ξ̂2. If n were not equal to 6, the endomorphism ξ̂2 would 
ontain exa
tly two
automorphisms idP and τ , whi
h would imply [τ ] = [τQ], a 
ontradi
tion. Thus we
get n = m = 6 and [ξ̂2] = [idN ]⊕ [τ ]⊕ [τQ]⊕ [τ ′], where τ ′ is an automorphisms of N .
It is known that there are 4 di�erent subfa
tors with the prin
ipal graph D
6 [25℄.
We show that the only one 
ase among them is possible.
Sin
e the dual prin
ipal graph of N ⊂ P is also D(1)6 , we have [ξ2] = [id] ⊕ [α] ⊕
[α1]⊕ [α2] where α1 and α2 are automorphisms of P . The proof of Lemma 4.1 shows
2 ≤ dim(ῑκ̄κι, ξ̂) = 1 + dim(κ̄κ, α1 ⊕ α2).
If κ̄κ 
ontained both α1 and α2, it would 
ontain α too, whi
h 
ontradi
ts irredu
ibil-
ity of N ⊂ M . Thus we may assume that κ̄κ 
ontains only α1, whose period must be
two in 
onsequen
e. Therefore either [M : P ] = 2 or the prin
ipal graph of P ⊂ M is
48 PINHAS GROSSMAN AND MASAKI IZUMI
k . Let L = P ⋊α1 Z/2Z, whi
h is regarded as an intermediate subfa
tor of P and
M . Then it is known [26, Chapter 8℄ that N ⊂ L is given by the �xed point algebra
of an outer a
tion of either D8 or the Ka
-Paljutkin algebra on L and
[ῑσ̄σι] = [idN ]⊕ [τ ]⊕ [τQ]⊕ [τ ′]⊕ 2[ξ̂],
where we set σ = ιL,P . We 
laim that Q is an intermediate subfa
tor of N ⊂ L.
Sin
e d(ξ̂) = 2, Theorem 2.9,(1) shows that the multipli
ity of ξ̂ in ῑκ̄κι is two and
ῑκ̄κι 
ontains automorphisms with multipli
ity at most one. On the other hand, the
endomorphism ῑQιQ is 
ontained in ῑσ̄σι, and so [28, Theorem 3.9℄ shows the 
laim.
Sin
e Q is a quadrilateral we get M = L. Sin
e [M : P ] = [M : Q] = 2, there exist
period two automorphisms β, β1 ∈ Aut(M) su
h that P = Mβ and Q = Mβ1 . If
N ⊂ M were given by the Ka
-Paljutkin algebra, we would have
[κιῑκ̄] = [idM ]⊕ [β1]⊕ [β2]⊕ [β1β2]⊕ 2[ρ],
with d(ρ) = 2, and the two automorphisms β1 and β2 would generate a group G of
order 4 in Aut(M). However, this means P ∩Q = MG 6= N , whi
h is a 
ontradi
tion.
Therefore N is the �xed point algebra of an outer a
tion of D8.
Now we assume that the prin
ipal graph of N ⊂ P is A5 and we show that Q
would be 
ommuting, whi
h 
ontradi
ts the assumption. Sin
e [M : P ][P : N ] =
[M : Q][Q : N ] we have [M : P ] = 3 and [M : Q] = 4. In this 
ase, we have
[ῑQιQ] = [idN ]⊕ [ξ̂], [ιQῑQ] = [idQ]⊕ [ξQ],
[ξ̂2] = [idN ]⊕ [τ ]⊕ [ξ̂], [τ 2] = [idM ], [τ ξ̂] = [ξ̂τ ] = [ξ̂],
[ξ2Q] = [idQ]⊕ [αQ]⊕ [ξQ], [α2Q] = [idQ], [αQξQ] = [ξQαQ] = [ξQ].
As in [18, Lemma 3.2℄, we 
an 
hoose representatives of [αQ] and [τ ] su
h that α
idQ, τ
2 = idN , and αQιQ = ιQτ hold.
We 
laim [κQαQ] = [κQ]. Indeed, sin
e κι = κQιQ and [ιτ ] = [ι], we have
[κQαQιQ] = [κQιQτ ] = [κQιQ] and
1 = dim(κQαQιQ, κQιQ) = dim(κQαQ, κQιQῑQ) = dim(κQαQ, κQ ⊕ κQξQ).
Sin
e we have
dim(κQαQ, κQξQ) = dim(κQ, κQξQαQ) = dim(κQ, κQξQ) = dim(κ̄QκQ, ξQ) = 0,
we get the 
laim.
The 
laim implies that we 
an regard L := Q⋊αQZ/2Z as an intermediate subfa
tor
of Q ⊂ M . On the other hand, we 
an regard R := N ⋊τ Z/2Z as an intermediate
subfa
tor of N ⊂ P . Sin
e we have αQιQ = ιQτ , we have the in
lusion R ⊂ L.
Moreover [18, Theorem 3.1℄ shows that N is the �xed point algebra of an outer
a
tion of S3 on L. In parti
ular
R ⊂ L
N ⊂ Q
is a 
ommuting square as we have
[L : R] = 3 and [L : Q] = 2. By Galois 
orresponden
e, the subfa
tor R is the �xed
point subalgebra of L under a subgroup of S3 isomorphi
 to Z/3Z, and by duality,
there exists an order three outer automorphism β ∈ Aut(R) su
h that L = R⋊βZ/3Z.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 49
Sin
e [P : R] = 2, there exits an order two automorphism ϕ ∈ Aut(R) su
h that
P = R ⋊ϕ Z/2Z. Sin
e ιM,RιM,R 
ontains ϕ and β and [M : R] = 6, we have
M = N ⋊ Γ, where Γ is the group generated by [ϕ] and [β], whose order must be
6. In parti
ular
P ⊂ M
R ⊂ L
is a 
ommuting square. Therefore we have EQEP =
EQELEP = EQER = EN , whi
h shows that Q is 
ommuting, a 
ontradi
tion. Thus
the statement is proven.
(4) Suppose that N ⊂ P and N ⊂ Q are 2-supertransitive and the prin
ipal
graph of P ⊂ M is D(1)n . If Q were 
o
ommuting, Corollary 3.12 would imply
4 = [M : P ] ≤ [P : N ] − 1 whi
h is a 
ontradi
tion. If Q were non
o
ommuting,
the statement (3) applied to the dual quadrilateral Q̂ would imply M = N ⋊ D8.
However, this shows that Q is 
ommuting and we get a 
ontradi
tion. Therefore the
lemma is proven. �
Proof of Theorem 6.1. Assume �rst that N ⊂ P and N ⊂ Q are 2-supertransitive.
Then Lemma 6.2,(4) shows that P ⊂ M is either 2-supertransitive or M = P ⋊ G
where G is a �nite abelian group of order 2,3, or 4. If the latter o

urs, then the
proof of Lemma 4.1 implies that ξ2 
ontains a non-trivial automorphism, and so
the prin
ipal graph is either A5, E6, or E
6 . Applying Lemma 6.2,(1) to the dual
quadrilateral Q̂, we see that Q is 
o
ommuting. Sin
e the E6 subfa
tor has trivial
se
ond 
ohomology, Corollary 3.12 implies that E6 never o

urs. For the same reason,
we have [M : P ] = [P : N ]− 1 = 2 for A5 and we get 
ase (1) from Theorem 2.7. For
6 , we get 
ase (5) from Lemma 6.2,(2). If P ⊂ M andQ ⊂ M are 2-supertransitive,
either Theorem 4.3 or Lemma 6.2,(2) is applied and we get 
ases (1),(2),(3),(6), and
(7) from Theorem 5.2, 5.4, and 5.17.
Assume now that N ⊂ P is not 2-supertransitive. Then Lemma 6.2,(1) implies
that the prin
ipal graph of N ⊂ P is D(1)n with n ≥ 5, and so Lemma 6.2,(4) implies
the 
ase (4). �
Remark 6.3. Ex
ept for the 
ase (4), we have Θ(P,Q) = cos−1 1/([P : Q]−1) thanks
to Lemma 2.5 and Corollary 3.11. In the 
ase (4), we have Θ(P,Q) = π/4 thanks to
[54, Theorem 6.1℄.
7. α-indu
tion and GHJ pairs
7.1. α-indu
tion and angles. Let N be a properly in�nite fa
tor and {λi}i∈I be a
�nite system of irredu
ible endomorphisms in End0(N ). We assume:
(1) there exists 0 ∈ I su
h that λ0 = idN ,
(2) for i 6= j ∈ I, we have [λi] 6= [λj ],
(3) for ea
h i ∈ I, there exists ī ∈ I su
h that [λi] = [λī],
(4) there exist non-negative integers Nkij su
h that
[λiλj] =
Nkij [λk],
50 PINHAS GROSSMAN AND MASAKI IZUMI
(5) the system {λi}i∈I has a braiding {ε(λi, λj)}i,j∈I (see [3, De�nition 2.2℄ for the
de�nition),
(6) N ⊂ M is an irredu
ible in
lusion of fa
tors of �nite index su
h that
[ιM,N ιM,N ] =
nj [λj],
where J is a subset of I and nj is a positive integer.
We naturally extend the braiding to the 
ategory generated by {λi}i∈I and use
the same symbol ε(ρ, σ) for the extension. We set ε+(ρ, σ) = ε(ρ, σ) and ε−(ρ, σ) =
ε(σ, ρ)∗. For simpli
ity, we use the following notation: ν = ιM,N , γ = νν̄ ∈ End0(M),
γ̂ = ν̄ν ∈ End0(N ). Then α-indu
tion α±λi ∈ End0(M) is de�ned by
α±λi = ν̄
−1 · Adε±(λi, γ̂) · λi · ν̄.
(See [44℄, [58℄.) Note that for x ∈ N we have ν̄(ν(x)) = γ̂(x) and α±λi(x) = λi(x),
whi
h means α±λiν = νλi.
Let Hj = (ν, νλj). Then thanks to Theorem 2.9, we have
as a linear spa
e. We 
hoose an orthonormal basis {t(j)k}njk=1 of Hj .
Lemma 7.1. Let the notation be as above. Then for t ∈ Hj we have
α±λi(t) = ε
±(λi, λj)
Proof. Sin
e ν̄(t) ∈ (γ̂, γ̂λj), the statement follows from the equation
ε(λi, γ̂)λi(ν̄(t))ε(λi, γ̂)
∗ = ν̄(t)
γ̂ λi
γ̂ λi λj
= ν̄(t)
γ̂ λi
γ̂ λi λj
= γ̂(ε(λi, λj)
∗)ν̄(t).
From the above lemma, it is easy to show (ρ, σ) ⊂ (α±ρ , α±σ ).
We �x i0 ∈ I and set λ = λi0 , M = M, P = α+λ (M), Q = α−λ (M), N = P ∩ Q,
and R = λ(N ). Sin
e α±λ ν = νλ, we have R ⊂ N . In general these two fa
tors do
not 
oin
ide and we give a des
ription of N now.
We set
Ji0 = {j ∈ J ; ε(λj, λ)ε(λ, λj) is a s
alar},
and set mj = ε(λj, λ)ε(λ, λj) for j ∈ Ji0. Re
all that φρ = r∗ρρ̄(·)rρ is the standard
left inverse of ρ. Sin
e ε(λj, λ)ε(λ, λj) ∈ (λλj, λλj), we have φλ(ε(λj, λ)ε(λ, λj)) ∈
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 51
(λj , λj), whi
h is a s
alar. It is easy to show that j ∈ Ji0 if and only if
|φλ(ε(λj, λ)ε(λ, λj))| = 1.
Lemma 7.2. Let the notation be as above. Then
N = α+λ (
j∈Ji0
NHj) = α−λ (
j∈Ji0
NHj).
In parti
ular, the von Neumann algebra N is a fa
tor.
Proof. Let x, y ∈ M satisfy α+λ (x) = α−λ (y). Then Theorem 2.9 shows that we have
expansion of x and y as
a(j)kt(j)k,
b(j)kt(j)k,
where a(j)k and b(j)k are elements in N uniquely determined by x and y respe
tively.
Thus we get
λ(a(j)k)ε(λ, λj)
∗t(j)k = α
λ (x) = α
λ (y) =
λ(b(j)k)ε(λj, λ)t(j)k,
whi
h implies λ(a(j)k)ε(λ, λj)
∗ = λ(b(j)k)ε(λj, λ) for all j ∈ J and 1 ≤ k ≤ nj . Ap-
plying φλ to both λ(a(j)k) = λ(b(j)k)ε(λj, λ)ε(λ, λj) and λ(a(j)k)(ε(λj, λ)ε(λ, λj))
b(j)k, we get
a(j)k = b(j)kφλ(ε(λj, λ)ε(λ, λj)),
a(j)kφλ(ε(λj, λ)ε(λ, λj)) = b(j)k,
whi
h shows that a(k)j 6= 0 only if |φλj (ε(λ, λj)ε(λj, λ))| = 1, whi
h is equivalent to
j ∈ Ji0 . On the other hand, when j ∈ Ji0 we have ε(λj, λ) = mjε(λ, λj)∗ and
α+λ (
j∈Ji0
a(j)kt(j)k) = α
j∈Ji0
m−1j a(j)kt(j)k), ∀a(j)k ∈ N ,
whi
h shows the �rst statement. Sin
e
j∈Ji0
NHj is an intermediate von Neumann
algebra between N and M, it must be a fa
tor, and so is N . �
Assume that P ⊂ M has no non-trivial intermediate subfa
tor. Then Q =
P ⊂ M
N ⊂ Q
is a quadrilateral though it is not irredu
ible in general.
Theorem 7.3. Let the notation be as above. Then
Ang(P,Q) = {cos−1 |φλ(ε(λj, λ)ε(λ, λj))|; j ∈ J} \ {0,
52 PINHAS GROSSMAN AND MASAKI IZUMI
In parti
ular, when the braiding is non-degenerate, we have
Ang(P,Q) = {cos−1 |S00Si0j ||S0i0S0j |
; j ∈ J} \ {0, π
where Sij is as in [3, Proposition 2.4℄.
Proof. Sin
e M =
j∈J NHj, we have
P = α+λ (M) =
λ(N )α+λ (Hj),
Q = α−λ (M) =
λ(N )α−λ (Hj).
Thanks to Remark 2.12, to 
ompute the eigenvalues of EPEQEP it su�
es to 
ompute
EPEQ on α
λ (Hj). We 
laim that for every t ∈ Hj the equalities
λ (t)) = φλ(ε(λ, λj)
∗ε(λj, λ)
∗)α−λ (t),
EP (α
λ (t)) = φλ(ε(λj, λ)ε(λ, λj))α
λ (t),
hold. Note that we have EQ = α
λ φα−
and φα−
= r∗λα
(·)rλ. Thus
(α+λ (t)) = r
(α+λ (t))rλ = r
(ε(λ, λj)
∗t)rλ = r
λλ̄(ε(λ, λj)
∗)ε(λj, λ)trλ
= r∗λλ̄(ε(λ, λj)
∗)ε(λj, λ)λj(rλ)t.
Sin
e
r∗λλ̄(ε(λ, λj)
∗)ε(λj, λ)λj(rλ) =
d(λj)
d(λj)
= φλ(ε(λ, λj)
∗ε(λj, λ)
we get the �rst equation. The se
ond one follows from a similar 
omputation. Using
the 
laim, we get
EPEQ(α
λ (t)) = |φλ(ε(λj, λ)ε(λ, λj))|2α+λ (t),
whi
h proves the statement. �
7.2. GHJ pairs. We �rst re
all the 
onstru
tion of the GHJ subfa
tors, whi
h �rst
appeared in the book of Goodman, de la Harpe and Jones [11℄. Our presentation is
based on [9℄.
Let G be one of the Dynkin diagrams of type A, D, or E, with a distinguished
vertex ∗ and let Ãn be the string algebra String(n)G (see [9, p.554℄ for the de�nition).
For i ≥ 1, we denote by ei the Jones proje
tion in Ãi+1∩Ã′i−1 de�ned by [9, De�nition
11.5℄. Then the Jones proje
tions {ek}∞k=1 satisfy the Temperley-Lieb relations:
(1) eiei±1ei = τei
(2) eiej = ejei if |i− j| ≥ 2
(3) tr(eiw) = τtr(w) if w is a word on e1, ..., ei−1.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 53
Here τ = 1/(4 cos2 π/m), where m is an integer asso
iated to G 
alled the Coxeter
number. For ea
h n, the Coxeter number of An is n+1, the Coxeter number of Dn is
2n− 2, and the Coxeter numbers of E6, E7, and E8 are 12, 18, and 30 respe
tively.
Let B̃i be the subalgebra of Ãi generated by 1 and e1, ..., ei−1, for ea
h i = 2, 3, ....
Then the towers B̃i ⊂ Ãi form 
ommuting squares, and the von Neumann algebra B̃
generated by ∪∞i=1B̃i is a subfa
tor of the fa
tor Ã generated by ∪∞i=1Ãi in the GNS
representation of the unique tra
e on ∪∞i=1Ãi. This is an irredu
ible subfa
tor with
�nite index, 
alled the GHJ subfa
tor for (G, ∗). The prin
ipal graphs of the GHJ
subfa
tors were 
omputed by Okamoto in [50℄.
Be
ause of the Temperley-Lieb relations, there is a unitary representation of the
braid group inside the algebras B̃i, sending the usual braid group generators σi to
gi = (t + 1)ei − 1, where t = e2πi/m, m again being the Coxeter number of G. Let
vi = g1g2 · · · gi−1 and wi = g−11 g−12 · · · g−1i−1. We set C̃i = viÃi−1v∗i and D̃i = wiÃi−1w∗i .
Then the towers C̃i ⊂ Ãi and D̃i ⊂ Ãi form 
ommuting squares, and the resulting
subfa
tor P̃ ⊂ M̃ and Q̃ ⊂ M̃ are irredu
ible and [M̃ : P̃ ] = [M̃ : Q̃] = 4 cos2 π/k.
For G = An the prin
ipal graphs of P̃ ⊂ M̃ and Q̃ ⊂ M̃ are An, for G = D2n+1 they
are A4n−1, for G = D2n they are D2n, for G = E7 they are A17 , and for G = En
with n = 6, 8 they are En. The pair P̃ and Q̃ is 
alled the GHJ pair [13℄. (Though
∗ is assumed to be an endpoint in [13, De�nition 6.2.5℄, we don't pose this 
ondition
here.) We set Ñ = P̃ ∩ Q̃ and set R̃ to be the subfa
tor generated by {ei}∞i=2. By

onstru
tion, we have R̃ ⊂ Ñ though these two fa
tors do not ne
essarily 
oin
ide
in general. For a spe
ial 
ase, the angles between P̃ and Q̃ are 
omputed in [13,
Theorem 6.3.2℄.
Now we show that
P̃ ⊂ M̃
Ñ ⊂ Q̃
is essentially the same obje
t as
P ⊂ M
N ⊂ Q

ussed in the previous subse
tion for an appropriate {λi}i∈I and N ⊂ M. Conse-
quently, the angles between P̃ and Q̃ 
an be easily 
omputed by Theorem 7.3. Our
argument below is inspired by [2, Appendix℄.
Let N be the AFD type III1 fa
tor and let {λi}ki=0 be a system of irredu
ible
endomorphisms isomorphi
 to the irredu
ible se
tors for the SU(2)k WZWmodel (see
[2℄ for details). Su
h a system 
an be obtained either by the loop group 
onstru
tion
[55℄ or by 
ombination of [57℄ and [14℄ using the quantum SU(2) at a root of unity.
Then the system {λi}ki=0 satis�es the assumptions (1)-(5) of the previous subse
tion.
The prin
ipal graph of λ1(N ) ⊂ N is Ak+1 and
[λiλj ] =
0≤2l≤min{|i+j|,k}−|i−j|
[λ|i−j|+2l],
d(λi) =
(i+1)π
sin π
Sij =
k + 2
((i+ 1)(j + 1)π
k + 2
54 PINHAS GROSSMAN AND MASAKI IZUMI
Assume that an in
lusion of fa
tors N ⊂ M satis�es the assumption (6). We set
i0 = 1 and we use the same notation as in the previous subse
tion. It is known that
su
h an in
lusion is 
ompletely 
lassi�ed by a pair (G, ∗) (up to graph automorphism)
where G is one of the Coxeter graphs of type A, D, or E [36, Theorem 2.1℄, so that we

an identify Ãn with (νλ
n, νλn) and B̃n with ν((λ
n, λn)). Sin
e νλ = α±λ ν, we have
the in
lusion relation α±λ (Ãn) ⊂ (α±λ νλn, α±λ νλn) = (νλn+1, νλn+1) = Ãn+1. Re
all
that the tra
e tr on Ãn is given by φνλn .
Lemma 7.4. Let the notation be as above. Then
(1) For x ∈ Ãn, we have α±λ (x) = ε±(λ, λn)∗xε±(λ, λn). In 
onsequen
e, the
restri
tion of α±λ to Ãn, as a map from Ãn to Ãn+1, is tra
e preserving.
(2) For x ∈ Ãn we have EP (x) ⊂ α+λ (Ãn) and EQ(x) ⊂ α−λ (Ãn). The restri
tions
of EP and EQ to Ãn is tra
e preserving.
Proof. (1) Let x ∈ Ãn. Then ν̄(x) ∈ (γ̂λn, γ̂λn) and
ε(λ, γ̂)λ(ν̄(x))ε(λ, γ̂)∗ = ν̄(x)
γ̂ λ λn
= ν̄(x)
γ̂ λ λ
ν λ λ
= ν̄(ε(λ, λn)∗xε(λ, λn)),
whi
h shows α+λ (x) = ε(λ, λ
n)∗xε(λ, λn). The se
ond equality 
an be shown in the
same way.
(2) Sin
e EP (x) = α
λ φα+
(x) = α+λ (r
λ (x)rλ), it su�
es to show r
λ (x)rλ ∈ Ãn
for x ∈ Ãn, whi
h 
an be easily shown by using (1). This also shows
tr(EP (x)) = tr(α
λ φα+
(x)) = tr(φα+
(x)) = φνλnφα+
(x) = φα+
νλn(x)
= φνλn+1(x) = tr(x).
Sin
e ε(λ, λn)∗ = ε(λ, λ)∗λ(ε(λ, λ)∗) · · ·λn−1(ε(λ, λ)∗), we 
an identify α+λ (Ãn−1)
with C̃n and α
λ (Ãn−1) with D̃n (for an appropriate 
hoi
e of the braiding).
We now show that
P ⊂ M
N ⊂ Q
P̃ ⊂ M̃
Ñ ⊂ Q̃
are 
onjugate after tensor produ
t
with the AFD type III1 fa
tor.
Lemma 7.5. Let L be a fa
tor and σ ∈ Mor0(N ,L). We set An = (σλn, σλn)
and Bn = σ((λn, λn)). Let A and B be the II1 fa
tors generated by ∪nAn and ∪nBn
respe
tively on the GNS Hilbert spa
e of the unique tra
e on ∪nAn. Then A ∩ B′ =
(σ, σ).
Proof. Sin
e (λn, λn) is generated by Jones proje
tions, the statement follows from
the �atness of the Jones proje
tions [9, Chapter 12℄. �
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 55
Theorem 7.6. Let the notation be as above. Then
P̃ ⊗ R∞ ⊂ M̃ ⊗ R∞
Ñ ⊗ R∞ ⊂ Q̃⊗ R∞
P ⊂ M
N ⊂ Q
are 
onjugate where R∞ is the AFD type III1 fa
tor.
Proof. First we show that R ⊂ M and R̃ ⊂ M̃ have the same standard invariant. We
�x a positive integer n larger than half of the depth of R ⊂ M , whi
h is �nite now.
Let ρ = νλ. Then R = ρ(N ) and
(ρ̄ρ)n−1ρ̄((ρ, ρ)) ⊂ (ρ̄ρ)n−1((ρ̄ρ, ρ̄ρ)) ⊂ · · · ⊂ ((ρ̄ρ)n, (ρ̄ρ)n)
∪ ∪ ∪
C ⊂ (ρ̄ρ)n−1((ρ̄, ρ̄)) ⊂ · · · ⊂ ((ρ̄ρ)n−1ρ̄, (ρ̄ρ)n−1ρ̄)
is the standard invariant of R ⊂ M (or rather (ρ̄ρ)n(N ) ⊂ (ρ̄ρ)n−1ρ̄(M)). We set
A2j,m = (ρ̄ρ)n−j(((ρ̄ρ)jλm, (ρ̄ρ)jλm)),
A2j+1,m = (ρ̄ρ)n−j−1ρ̄((ρ(ρ̄ρ)jλm, ρ(ρ̄ρ)jλm)).
Note that ρ̄ρ is a dire
t sum of irredu
ibles in {λi}ki=0. When m is larger than the
depth of λ(N ) ⊂ N , the in
lusion graph for Aj,m ⊂ Aj+1,m is the re�e
tion of that
for Aj+1,m ⊂ Aj+2,m thanks to Frobenius re
ipro
ity. Therefore
Aj,m ⊂ Aj+1,m ⊂ Aj+2,m
is the basi
 
onstru
tion.
Let Aj,∞ be the fa
tor generated by ∪mAj,m in the GNS representation of the
unique tra
e of ∪mA2n,m. Then we may identify R̃ ⊂ M̃ with A0,∞ ⊂ A1,∞ and
A0,∞ ⊂ A1,∞ ⊂ A2,∞ ⊂ · · · ⊂ A2n,∞
is the Jones tower. Lemma 7.5 implies
A2j,∞ ∩ A′0,∞ = (ρ̄ρ)n−j(((ρ̄ρ)j , (ρ̄ρ)j)),
A2j+1,∞ ∩ A′0,∞ = (ρ̄ρ)n−j−1ρ̄((ρ(ρ̄ρ)j, ρ(ρ̄ρ)j)),
whi
h shows that R ⊂ M and R̃ ⊂ M̃ have the same prin
ipal graph. On the other
hand, we have
A2j,∞ ∩ A′1,∞ ⊃ (ρ̄ρ)n−j(((ρ̄ρ)j−1ρ̄, (ρ̄ρ)j−1ρ̄)),
A2j+1,∞ ∩ A′1,∞ ⊃ (ρ̄ρ)n−j−1ρ̄((ρρ̄)j, (ρρ̄)j)).
Sin
e [M : R] = [M̃ : R̃], the equality holds in the above in
lusions and the two
subfa
tors R ⊂ M and R̃ ⊂ M̃ have the same standard invariant. Therefore we

on
lude that the two in
lusions R ⊂ M and R̃ ⊗ R∞ ⊂ M̃ ⊗ R∞ are 
onjugate
thanks to Popa's result [53℄ with [19℄ (see [47℄ too).
To �nish the proof, it su�
es to show that eP = eP̃ and eQ = eQ̃ hold in the
above identi�
ation of the standard invariants, whi
h 
an be done by using Lemma
7.4,(2). �
56 PINHAS GROSSMAN AND MASAKI IZUMI
Remark 7.7. Sin
e
|φλ(ε(λj, λ)ε(λ, λj))| =
|S1j |
d(λ1)d(λj)|S00|
| sin π
2(j+1)π
| sin 2π
(j+1)π
| cos (j+1)π
| cos π
the operator ε(λj, λ)ε(λ, λj) is a s
alar if and only if j = 0, k. Thus we have Ji0 =
J ∩ {0, k}. It is known that k ∈ J o

urs only if k is even. In this 
ase, we may and
do assume λ2k = idN by 
hoosing a representative λk satisfying λkλk/2 = λk/2, whi
h
implies that there exists a unitary uk ∈ Hk satisfying u2k = 1. Therefore we may
regard N +NHk as the 
rossed produ
t N ⋊λk Z/2Z. We make this assumption and
identi�
ation in what follows.
The following 
orollary answers the question raised in [13, 
omment after Propo-
sition 6.2.4℄ in the negative. (The subfa
tor R̃ is denoted by TL2 in [13℄.)
Corollary 7.8. Let the notation be as above. Then R̃ = P̃ ∩ Q̃ if and only if k /∈ J .
If k ∈ J , the index of R̃ ⊂ P̃ ∩ Q̃ is 2.
Proof. Thanks to Theorem 7.6, we may work on R and N instead of R̃ and Ñ , whi
h
together with Lemma 7.2 show the �rst statement. When k ∈ J , Lemma 7.2 shows
N = α±λ (N +NHk) and the se
ond statement follows from Remark 7.7. �
The following 
orollary is a generalization of [13, Theorem 6.3.2℄.
Corollary 7.9. Let the notation be as above. Then
Ang(P̃ , Q̃) = {cos−1
| cos (j+1)π
| cos π
| ; j ∈ J} \ {0,
Example 7.10. Let N ⊂ M be the in
lusion of the AFD type III1 fa
tors asso
iated
with the 
onformal embedding SU(2)10 ⊂ SO(5)1 (see [58, p.381℄). Then the 
orre-
sponding GHJ subfa
tor is given by E6 with ∗ equal to the vertex of the minimum
value of the Perron-Frobenius eigenve
tor, and so its prin
ipal graph is as in Theorem
5.4,(2) (see [9, p.726℄,[50℄). Sin
e [γ̂] = [λ0]⊕ [λ6], we have R̃ = Ñ , and so Ñ ⊂ P̃ is

onjugate to the GHJ subfa
tor, while P̃ ⊂ M̃ is the E6 subfa
tor. Sin
e
dim(νλ, νλ) = dim(γ̂, λ2) = dim(λ0 ⊕ λ6, λ0 ⊕ λ2) = 1,
the in
lusion Ñ ⊂ M̃ is irredu
ible. The angle between P̃ and Q̃ is 
omputed as
Θ(P̃ , Q̃) = cos−1(2 −
3) by using either of [13, Theorem 6.3.2℄, Theorem 4.3, or
Corollary 7.9.
It is observed in [13, Proposition 6.2.6℄ that R̃ ⊂ M̃ is irredu
ible if and only if
∗ is an endpoint. However, even when ∗ is not an endpoint, the in
lusion Ñ ⊂ M̃
may be irredu
ible (though it is very subtle in general to de
ide whether Ñ ⊂ M̃
is irredu
ible or not as we will see below). We use this observation to 
ompute the
opposite angles of quadrilaterals 
onstru
ted in [12℄. The following theorem is shown
in [12℄.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 57
Theorem 7.11. Let
B ⊂ A
D ⊂ C
be an irredu
ible non
ommuting quadrilateral of the
hyper�nite II1 fa
tors su
h that D ⊂ B and D ⊂ C are supertransitive and [B : D]
and [C : D] are less than 4. Then the prin
ipal graphs of D ⊂ B and D ⊂ C are
A2n+1 with an integer n ≥ 2. For ea
h integer n ≥ 2, su
h a quadrilateral exists and
is unique up to 
onjuga
y.
We now 
ompute the �opposite angles� of the above quadrilateral.
Theorem 7.12. Let
Ĉ ⊂ D̂
A ⊂ B̂
be the dual quadrilateral of the quadrilateral in the
previous theorem. Then
Ang(B̂, Ĉ) = {cos−1
(2j+1)π
cos π
; j = 1, 2, · · · , [n− 1
where [(n− 1)/2] is the integer part of (n− 1)/2.
Proof. We show that the above quadrilateral is given by the GHJ pair of A2n+1 with
∗ equal to the midpoint. Thanks to Theorem 7.6, we 
an use α-indu
tion instead of
the GHJ 
onstru
tion for this purpose. Let N and {λi}ki=0 be as before with k = 2n.
Let (M, {λ̃i}ki=0) be a 
opy of (N , {λi}ki=0). We embed N into M by
ν : N ∋ x 7→ λn(x) ∈ M.
Then we have ν̄ = λ̃n regarded as a map from M into N , and so γ̂ = λ2n and γ = λ̃2n.
Sin
e
[λ2n] =
[λ2i],
we have J = {0, 2, 4, · · · , k} and Ji0 = {0, k}. Therefore thanks to Corollary 7.8,
N = α±λ (N ⋊λk Z/2Z) = α±λ (ν(N ) + ν(N )uk),
where uk ∈ (ν, νλk) = (λ̃n, λ̃nλ̃k). As before, we set M = M, P = α+λ (M), Q =
α−λ (M), N = P ∩ Q, and Q =
P ⊂ M
N ⊂ Q
. Then the prin
ipal graphs of P ⊂ M
and Q ⊂ M are A2n+1. To prove the theorem, it su�
es to show that N ⊂ M is
irredu
ible and Q is not 
o
ommuting thanks to Corollary 7.9 and Theorem 7.11.
We 
laim α±λ = Adε
±(λ, λn)λ̃. Indeed, sin
e ε
±(λ, γ̂) = λn(ε
±(λ, λn))ε
±(λ, λn) and
ε±(λ, λn) ∈ (λλn, λnλ) = (λν̄, ν̄λ̃), we have
Ad(ε±(λ, γ̂))λν̄(x) = Adλn(ε
±(λ, λn))Adε
±(λ, λn)λν̄(x) = Adν̄(ε
±(λ, λn))ν̄(λ̃(x))
for all x ∈ M, whi
h shows the 
laim. In parti
ular, the two subfa
tors P and Q are
inner 
onjugate in M . Therefore if N ⊂ M is irredu
ible, Theorem 3.10 applied to
the dual quadrilateral Q̂ shows that Q is not 
o
ommuting.
58 PINHAS GROSSMAN AND MASAKI IZUMI
Now our only task is to show M ∩N ′ = C. Note that N is generated by ν(λ(N ))
and ν(ε(λ, λk)
∗)uk with uk ∈ (ν, νλk), and so
M ∩N ′ = (νλ, νλ) ∩ {ν(ε(λ, λk)∗)uk}′.
Therefore it su�
es to show
(λnλ, λnλ) ∩ {λn(ε(λ, λk)∗)u}′ = C.
where u is a unitary in (λn, λnλk). By Frobenius re
ipro
ity, we have
dim(λnλ, λnλ) = dim(λ
2) = dim(λ2n, λ0 ⊕ λ2) = 2,
and we 
an 
hoose a basis {1, s} of (λnλ, λnλ) with
where v ∈ (λn, λnλ2) and w ∈ (λ2λ, λ). On one hand, we have
sλn(ε(λ, λk)
∗)u = v
where ṽ = ε(λ2, λn)
∗v. On the other hand,
λn(ε(λ, λk)
∗)us =
Sin
e dim(λn, λ2λnλk) = 1, there exists a s
alar c su
h that
λ2 λn
λn = c
λ2 λn
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 59
whi
h implies sλn(ε(λ, λk)
∗)u = cλn(ε(λ, λk)
∗)us. The number c is given by Kirillov-
Reshetikhin's quantum 6j-symbol with q = eπi/(n+1) (see [37, (5.4)℄). Using the
notation in [37℄ (irredu
ible representations are parameterized by half integers there),
we get
Note that although an expli
it formula of the 6j-symbols depends on the 
hoi
e of
intertwiners in general, the above one does not be
ause the same intertwiners u and
ṽ appear in the both sides. This 
an be 
omputed by [37, Theorem 5.1℄ and it turns
out that we have c = −1. Therefore N ⊂ M is irredu
ible. �
7.3. Asymptoti
 in
lusions and angles. Another ri
h sour
e of non
ommuting
quadrilaterals is asymptoti
 in
lusions. We show that a te
hnique similar to the one
we used in the previous subse
tion works in this 
ase too.
Let N ⊂ M be an irredu
ible in
lusion of the hyper�nite II1 fa
tors with �nite
index and of �nite depth and let
N ⊂ M ⊂ M1 ⊂ · · · ⊂ Mn ⊂ · · ·
be the Jones tower. The algebra ∪nMn has a unique tra
e and we denote by M∞
the fa
tor generated by ∪nMn in the GNS representation of the unique tra
e of
∪nMn. Then the in
lusion M ∨ (M∞ ∩ M′) ⊂ M∞ is 
alled the asymptoti
in
lusion of N ⊂ M (see [9, Chapter 12.6℄ for details). It is easy to show that
N ∨ (M∞ ∩ N ′) ⊂ M∞
N ∨ (M∞ ∩M′) ⊂ M∨ (M∞ ∩M′)
is an irredu
ible 
ommuting quadrilateral
of fa
tors. We show how to 
ompute the �opposite angles" of this quadrilateral.
Example 7.13. When the prin
ipal graph of N ⊂ M is An, there exists a sequen
e
of Jones proje
tion {ei}∞i=1 su
h that N = {ei}∞i=2
′′ ⊂ M = {ei}∞i=1
. We set e0 = eN ,
e−1 = eM and e−(i+1) = eMi for i ≥ 1. Then M∞ = {ei}∞i=−∞
, M∨ (M∞ ∩M′) =
{ei}′′i 6=0, N ∨ (M∞ ∩ N ′) = {ei}′′i 6=1, and N ∨ (M∞ ∩M′) = {ei}′′i 6=0,1. The index of
the asymptoti
 in
lusion in this 
ase is 
omputed in [8℄ and the prin
ipal graphs are
obtained in [10℄.
It is well-known that M∞ ∩M′ is naturally identi�ed with the opposite algebra
Mopp of M and M ∨ (M∞ ∩ M′) is naturally identi�ed with M ⊗ Mopp, and
so we identify N ∨ (M∞ ∩ M′) with N ⊗ Mopp too. However, we never identify
N ∨ (M∞∩N ′) with N ⊗N opp as we 
annot take 
ommon fa
torizations M⊗Mopp
and N ⊗ N opp for M∨ (M∞ ∩M′) and N ∨ (M∞ ∩ N ′). By 
utting the Hilbert
spa
e L2(M∞) by an appropriate proje
tion in M′∞, we may assume that M∞ a
ts
on L2(M⊗Mopp) = L2(M)⊗ L2(Mopp). Sin
e eN may be 
onfused with eN ⊗ 1 in
this representation, we use the symbol e for eN in M∞. Let J = JM ⊗ JMopp be the

anoni
al 
onjugation of M⊗Mopp. We set
M := J
N ∨ (M∞ ∩M′)
J = J(N ⊗Mopp)′J = M1 ⊗Mopp,
P := J
M∨ (M∞ ∩M′)
J = J(M⊗Mopp)′J = M⊗Mopp,
60 PINHAS GROSSMAN AND MASAKI IZUMI
Q := J
N ∨ (M∞ ∩ N ′)
J, N := JM′∞J.
Note that Q is not equal to M1 ⊗ N opp as we formally (just formally) broke the
symmetry between M∨ (M∞ ∩M′) and N ∨ (M∞ ∩ N ′) by representing M∞ on
L2(M⊗Mopp). However, we still have the following:
Lemma 7.14. Let the notation be as above. Then Q and M1 ⊗ N opp are inner

onjugate in M . In parti
ular, we have M⊗QM ∼= M1⊗Mopp1 as M−M bimodules.
Proof. Note that
N ⊗N opp ⊂ N ⊗Mopp ⊂ N ∨ (M∞ ∩ N ′) = (N ⊗Mopp) ∨ {e}
is the basi
 
onstru
tion with the Jones proje
tion e. Thus
Q ⊂ M1 ⊗Mopp ⊂ J(N ⊗N opp)′J = M1 ⊗Mopp1
is the basi
 
onstru
tion too, whi
h shows that Q and M1⊗N opp are inner 
onjugate
in M . �
Our purpose of this subse
tion is to show the following theorem:
Theorem 7.15. Let the notation be as above. If the prin
ipal graph of N ⊂ M is
An, then
Ang(P,Q) = {cos−1
(j+1)π
cos π
; j = 1, 2, · · · [n− 2
We go ba
k to the general (not ne
essarily An) 
ase. Let Q =
P ⊂ M
N ⊂ Q
be as
above. To 
ompute Ang(P,Q) we may repla
e M with
M ⊗ B(ℓ2)⊗ B(ℓ2) ∼= M⊗ B(ℓ2)⊗ (M⊗ B(ℓ2))opp,
and so we may and do assume that N ⊂ M is an in
lusion of properly in�nite
fa
tors from now on. In this 
ase, the in
lusion N ⊂ P is des
ribed as the Longo-
Rehren in
lusion [46℄ whose stru
ture is well-studied [22℄. Therefore the stru
ture of
N ⊂ P ⊂ M is 
ompletely understood. It is a little tri
ky to de
ide the position of
Q, but we 
an handle it by using Lemma 7.14. We use the symbols ι, ιQ, κ, and κQ
as in Se
tion 4.
We re
all the notation in [22℄ to des
ribe the Longo-Rehren in
lusion N ⊂ P . Let
ν : M →֒ M1 be the in
lusion map and let ∆ = {ρξ}ξ∈∆0 be the set of irredu
ibles

ontained in ∪n(ν̄ν)n. We may assume that ρe = idM with e ∈ ∆0 and [ρξ] = [ρξ̄].
We identifyMopp withM′ = JMMJM and de�ne j : M → Mopp by j(x) = JMxJM
for x ∈ M. For x ∈ M, the 
orresponding element xopp ∈ Mopp is identi�ed with
j(x∗). We denote by ρ
ξ the endomorphism of Mopp 
orresponding to ρξ. In our
situation, we have ρ
ξ = jρξj
. It is known that we have [ιῑ] =
[ρ̂ξ], where
ρ̂ξ = ρξ⊗ρoppξ , and so d(ι)2 =
d(ξ)2. We 
hoose an isometry Vξ ∈ (ρξ⊗ρoppξ , ιῑ)
su
h that rῑ = Ve, whi
h will be denoted by V for simpli
ity. We denote rι ∈ N by W ,
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 61
whi
h is des
ribed as follows. Let {T (ζξ,η)i}
i=1 be an orthonormal basis of (ρζ , ρξρη),
and set
ξ,η =
ξ,η)i ⊗ j(T (
ξ,η)i),
whi
h does not depend on the 
hoi
e of the basis. Then we have
ξ,η,ζ
d(ξ)d(η)
Vξρ̂ξ(Vη)T
To des
ribe the 
onditional expe
tation EQ, we use the following lemma:
Lemma 7.16. Let B ⊂ A be an irredu
ible in
lusion of property in�nite fa
tors with
�nite index and let µ : B →֒ A be the in
lusion map. Let [µµ̄] =
i=0 ni[σi] be the
irredu
ible de
omposition and let {s(i)l}nil=1 be an orthonormal basis of (µ, σiµ). Then
EB(x) =
[A : B]
d(σi)s(i)
l σi(x)s(i)l, ∀x ∈ A.
Proof. Let s̃(i)l =
d(σi)s(i)
l σi(rµ̄). Then {s̃(i)l}
l=1 is an orthonormal basis of
(σi, µµ̄) thanks to Frobenius re
ipro
ity. Thus
EB(x) = µ(r
µµ̄(x)rµ) = µ(r
µ)µµ̄(x)µ(rµ) =
µ(r∗µ)s̃(i)lσi(x)s̃(i)
l µ(rµ).
On the other hand,
µ(r∗µ)s̃(i)l =
d(σi)µ(r
µ)s(i)
l σi(rµ̄) =
d(σi)s(i)
l σi(µ(r
µ)rµ̄) =
d(σi)
s(i)∗l ,
whi
h shows the statement. �
Re
all that to 
ompute Ang(P,Q), it su�
es to determine the eigenvalues of EPEQ
restri
ted to P .
Lemma 7.17. Let [ν̄ν] =
nξ[ρξ] be the irredu
ible de
omposition. Then for
x ∈ P , we have
EPEQ(x) =
d(ι)2
[M : N ]
W ∗Vξρ̂ξ(x)V
Proof. Thanks to Lemma 7.14, we have [κQκ̄Q] =
nξ[idM1 ⊗ ρ
ξ ]. Thus to

ompute EQ using Lemma 7.16, it su�
es to obtain (κQ, (idM1⊗ρoppξ )κQ) for ξ ∈ ∆1.
Although it is not so easy to 
apture this spa
e dire
tly, we have
(κQ, (idM1 ⊗ ρ
ξ )κQ) ⊂ (κQιQ, (idM1 ⊗ ρ
ξ )κQιQ)
= ((ν ⊗ idMopp)ι, (ν ⊗ ρoppξ )ι)
∼= (ιῑ, ν̄ν ⊗ ρoppξ ).
62 PINHAS GROSSMAN AND MASAKI IZUMI
Comparing the dimensions of the both sides above, we get equality for the above
in
lusion.
Let {t(ξ)i}
i=1 be an orthonormal basis of (ν, νρξ). We 
laim that
{ d(ι)√
(t(ξ)∗i ⊗ 1)V ∗ξ W}
is an orthonormal basis of ((ν⊗ idMopp)ι, (ν⊗ ρoppξ )ι). Indeed, it is easy to show that
(t(ξ)∗i ⊗ 1)V ∗ξ W belongs to the intertwiner spa
e ((ν ⊗ idMopp)ι, (ν ⊗ ρ
ξ )ι), and so
the operator W ∗Vξ(t(ξ)lt(ξ)
i ⊗ 1)V ∗ξ W is already a s
alar, whi
h is equal to
ENEP (W
∗Vξ(t(ξ)lt(ξ)
i ⊗ 1)V ∗ξ W ) = EN(W ∗VξEP (t(ξ)lt(ξ)∗i ⊗ 1)V ∗ξ W )
〈t(ξ)l, t(ξ)i〉
EN (W
ξ W ) =
W ∗EN (VξV
ξ )W =
d(ξ)δi,l
d(ι)2
d(ξ)δi,l
d(ι)2
This shows the 
laim. Thus Lemma 7.16 shows that for x ∈ P we have
EQ(x) =
[M : N ]
d(ι)2W ∗Vξ(t(ξ)i ⊗ 1)(idM ⊗ ρoppξ )(x)(t(ξ)∗i ⊗ 1)V ∗ξ W
d(ι)2
[M : N ]
W ∗Vξρ̂ξ(x)(t(ξ)it(ξ)
i ⊗ 1)V ∗ξ W.
Therefore we get
EPEQ(x) =
d(ι)2
[M : N ]
W ∗Vξρ̂ξ(x)EP (t(ξ)it(ξ)
i ⊗ 1)V ∗ξ W
d(ι)2
[M : N ]
W ∗Vξρ̂ξ(x)V
To 
ompute the eigenvalues of EPEQ|P , we need the 
rossed produ
t-like de
om-
position as in Theorem 2.9 based on the irredu
ible de
omposition of ῑι. It is known
that every se
tor 
ontained in a power of ῑι is of the form σ̃α, where σ is a �nite dire
t
sum of endomorphisms in ∆ and α is a parameter of the 
orresponding half-braiding
εασ(ξ) ∈ (σρξ, ρξσ) (see [22, Se
tion 4℄ for details). Let
Uα(σ) =
σ(ξ)⊗ 1)(σ ⊗ idMopp)(V ∗ξ ).
Then σ̃α is the restri
tion of AdUα(σ)(σ ⊗ idMopp) to N . Irredu
ible σ̃α is 
ontained
in ῑι if and only if σ 
ontains idM. In this 
ase, we have
(ι, ισ̃α) = Uα(σ)
(idM, σ)⊗ C
Therefore to 
ompute Ang(P,Q), it su�
es to 
ompute EPEQ on this spa
e thanks
to Remark 2.11.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 63
Lemma 7.18. Let σ̃α be an irredu
ible endomorphism of N 
ontained in ῑι. Then
for v ∈ (idM, σ),
EPEQ(U
α(σ)(v ⊗ 1)) = Uα(σ)(φν̄ν(εασ(ν̄ν)v)⊗ 1).
The map (idM, σ) ∋ v 7→ φν̄ν(εασ(ν̄ν)v) depends only on the se
tor of ν̄ν and
φν̄ν(ε
σ(ν̄ν)v) =
nξd(ξ)
[M : N ]φρξ(ε
σ(ξ)v).
Proof. It is routine work to show the se
ond statement. Sin
e we have
Uα(σ)(v ⊗ 1) =
σ(η)v ⊗ 1)V ∗η ,
Lemma 7.17 implies
EPEQ(U
α(σ)(v ⊗ 1))
d(ι)2
[M : N ]
W ∗Vξρ̂ξ(Vη(ε
σ(η)v ⊗ 1)V ∗η )V ∗ξ W
[M : N ]
η,ζ∈∆0
nξd(η)
iρξ(ε
σ(η)v)T (
ξ,η)i ⊗ 1
V ∗ζ .
Thanks to the half-braiding relation, we have
i ρξ(ε
σ(η)v) = ε
σ(ζ)σ(T (
ξ,η))
∗ρξ(v).
Thus Frobenius re
ipro
ity implies
d(η)T (
iρξ(ε
σ(η)v)T (
ξ,η)i
= εασ(ζ)
d(η)σ(T (
ξ,η))
∗ρξ(v)T (
ξ,η)i
= d(ξ)d(ζ)εασ(ζ)
σ(r∗ρξ̄ρξ(T (
))i))ε
∗ρξ(vT (
)∗i )rρξ̄
= d(ξ)d(ζ)εασ(ζ)
σ(r∗ρξ̄ρξ(T (
))i))ε
∗ρξ(σ(T (
)∗i )v)rρξ̄
= d(ξ)d(ζ)εασ(ζ)
σ(r∗ρξ̄ρξ(T (
)iT (
)∗i ))ε
∗ρξ(v)rρξ̄
= d(ξ)d(ζ)εασ(ζ)σ(r
)εασ(ξ)
∗ρξ(v)rρξ̄ .
64 PINHAS GROSSMAN AND MASAKI IZUMI
Using the half-braiding relation again, we see that this is equal to
d(ξ)d(ζ)εασ(ζ)r
σ(ξ̄)v)rρξ̄ = d(ξ)d(ζ)ε
σ(ζ)φρξ̄(ε
σ(ξ̄)v),
whi
h �nishes the proof. �
Proof of Theorem 7.15. Assume that the prin
ipal graph of N ⊂ M is An. The
stru
ture of N −N se
tors asso
iated with the in
lusion N ⊂ P is des
ribed in [22,
Se
tion 7℄. We may assume that there exists a system of endomorphism {λi}n−1i=0
of M isomorphi
 to the irredu
ible se
tors of the SU(2)n−1 WZW model su
h that
N = λ1(M). Then we have [ν̄ν] = [λ21] and ∆ = {λ2j}
[(n−1)/2]
j=0 . When i − j is
even, we set σi,j = λiλj and εσi,j(λl) = ε
+(λi, λl)λi(ε
−(λj, λl)). Then {εσi,j (λl)}l is a
half-braiding for σi,j and we denote by σ̃i,j the 
orresponding endomorphism of N .
Assume �rst that n is even. In this 
ase, the endomorphism σ̃2i,2i is irredu
ible
[ῑι] =
n/2−1⊕
[σ̃2i,2i].
Sin
e (idM, σ2i,2i) = Crλ2i , we get
Ang(P,Q) = {cos−1
r∗λ2iφλ21(εσ2i,2i(λ
1)rλ2i); i = 0, 1, · · · ,
} \ {0, π
It is easy to show r∗λ2iφλ21(εσ2i,2i(λ
1)rλ2i) = φλ1(ε(λ2i, λ1)ε(λ1, λ2i))
and we get the
statement.
Assume now that n = 2s+1 is odd. Then σ̃i,i is irredu
ible for i = 0, 1, 2, · · ·s− 1
while σ̃s,s is de
omposed into two se
tors, say µ0 and µ1 su
h that
[ῑι] =
[σ̃i,i]⊕ [µ0].
Then a similar 
omputation as above shows
Ang(P,Q) = {cos−1 |φλ1(ε(λi, λ1)ε(λ1, λi))|; i = 1, 2, · · · , s},
(sin
e EPEQ((ι, ισ̃s,s)) = 0 we have EPEQ((ι, ιµ0)) = 0 too) whi
h shows the state-
ment. �
8. Appendix
In the proof of Theorem 5.19 we use the fa
t that there exists a unique Q-system
for idP ⊕ κ̄θκ up to equivalen
e. We give a proof of this statement here.
The next lemma follows from Frobenius re
ipro
ity [20, Se
tion 2.3℄, whi
h says
that the right normalization of the norm of an element v ∈ (τ, ρσ) is
d(ρ)d(σ)
Lemma 8.1. Let L, M, and N be properly in�nite fa
tors and ρ ∈ Mor0(N ,M),
σ ∈ Mor0(L,N ), and τ ∈ (L,M) be irredu
ible morphisms. If v ∈ (τ, ρσ) satis�es
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 65
||v|| =
d(ρ)d(σ)
, then
(d(τ)d(σ)
(d(τ)d(ρ)
We re
all the 
onstru
tion of the Haagerup subfa
tor in [23, Se
tion 7℄. Let O4 be
the Cuntz algebra generated by isometries {S0, T0, T1, T2} and let d = (3 +
13)/2
(whi
h will be d(ρ)). We introdu
e an endomorphism ρ and a period 3 automorphism
α of O4 by setting
(8.1) α(S0) = S0, α(Ti) = Ti+2,
(8.2) ρ(S0) =
i∈Z/3Z
TiTi,
(8.3) ρ(Ti) =
−i + T−iS0S
j,k∈Z/3Z
A(i+ j, i+ k)TjTi+j+kT
where i is understood as an element of Z/3Z and
A(0, 0) = 1− 1
d− 1 ,
A(0, 1) = A(0, 2) = A(1, 0) = A(2, 0) = A(1, 1) = A(2, 2) =
d− 1 ,
A(1, 2) = A(2, 1) =
4d− 1
2(d− 1) .
Then ρ and α extend to an irredu
ible endomorphism and an outer automorphism
respe
tively of the weak 
losure M of O4 in the GNS representation of some KMS
state, whi
h will be denoted by the same symbols. Moreover we have S0 ∈ (idM , ρ2),
T0 ∈ (ρ, ρ2), T1 ∈ (αρ, ρ2), and T2 ∈ (α2ρ, ρ2) and αρ = ρα2. It is easy to show that
ρ satis�es the 
ondition of Lemma 3.5, namely
(8.4) ρ(T0)S0 = T0S0,
(8.5)
dS0 + (d− 1)T 20 =
dρ(S0) + (d− 1)ρ(T0)T0.
Therefore there exist a Q-system for idM ⊕ ρ and a subfa
tor P ⊂ M su
h that
[κκ̄] = [idM ] ⊕ [ρ], where κ : P →֒ M is the in
lusion map. Note that ρ = η̂
66 PINHAS GROSSMAN AND MASAKI IZUMI
and α = θ in the notation of Theorem 5.19. Lemma 3.5 shows that we 
an 
hoose
isometries v = rκ̄, w = κ(rκ), and v1 ∈ (ρ, κκ̄) su
h that (3.3) holds:
(8.6) w =
v1ρ(v)v
v1ρ(v1)S0v
v1ρ(v1)T0v
To �nish the proof of Theorem 5.19, it su�
es to show that there exists a unique
Q-system for idP ⊕ κ̄ακ up to equivalen
e. For this purpose we solve the equations
(3.1) and (3.2) with σ := κ̄ακ. For this, we use the following notation:
(d(κ)2
v∗1 =
(d+ 1
v∗1 ,
(d+ 1
κ̄ ρ ρκ̄
Every vertex expressing an element in (αρ, ρα2), (ρ, αρα) et
. su
h as
will always mean 1. We 
hoose S0 as rρ = r̄ρ and set
:= d1/4T0,
= d1/4T ∗0 .
= d1/2S∗0ρ(d
1/4T0) = d
1/4T ∗0 =
= ρ(d1/2S∗0)d
1/4T0 = d
1/4T ∗0 =
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 67
In the same way, we have
Equation (8.4) means
ρ ρ ρ ρ ρ ρ
We use a similar expression for d1/4T1 ∈ (αρ, ρ2) = (ρα2, ρ2) and d1/4T2 ∈ (α2ρ, ρ2) =
(ρα, ρ2).
Sin
e we have
κρκ̄ κρκ̄
κ̄ ρ κ
we simply express this intertwiner by
. In a similar way, we have
κ̄ κα
κ̄ κα
and we simply express it by
κ̄ κα
. The diagram
κ̄ κα
is also interpreted
in the same way.
We set
κ̄ α κ κ̄ α κ
ρ ∈ (idP , σ2).
Then thanks to Lemma 8.1, we have
||R|| =
d(κ)2
d(κ)d(ρ)
d(κ)2
= d(κ) =
d+ 1.
All we have to show is the following:
Theorem 8.2. Let the notation be as above. Then there exist exa
tly two elements
S ∈ (σ, σ2) satisfying ||S|| = d(σ)1/4 = (d+ 1)1/4 and
(8.7) σ(S)R = SR,
68 PINHAS GROSSMAN AND MASAKI IZUMI
(8.8)
(R− σ(R)) = σ(S)S − S2.
(Note that if S satis�es the above 
ondition, so does −S.)
To prove the theorem, we 
hoose a basis of (σ, σ2). Note that we have
dim(σ, σ2) = dim(κ̄ακ, κ̄ακκ̄ακ̄) = dim(κκ̄ακκ̄, ακκ̄α)
= dim(α⊕ αρ⊕ α2ρ⊕ α2ρ2, α2 ⊕ ρ) = 2.
We set
α α κκ̄
∈ (σ, σ2),
α α κκ̄
ρ ∈ (σ, σ2).
Then sin
e rκ̄ and v1 are orthogonal, so are S1 and S2.
Lemma 8.3. Let the notation be as above. Then ||S1|| = ||S2|| = d(σ)1/4 = (d+1)1/4.
Proof. For S1, we have
||S1||2 = d(κ) κκ̄
α κκ̄
Sin
e this is already a s
alar, it is equal to
κ = κκ̄
d(κ)√
= d(κ).
For S2,
||S2||2 =
d(κ)√
α κκ̄
= d(κ).
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 69
The above lemma shows that every element S ∈ (σ, σ2) satisfying ||S|| = d(σ)1/4 is
uniquely expressed as S = aS1 + bS2, where a and b are 
omplex numbers satisfying
|a|2 + |b|2 = 1. Therefore (8.7) and (8.8) are equivalent to the following equations
respe
tively:
aS1R + bS2R = aσ(S1)R + bσ(S2)R,
(R− σ(R)) = a2(σ(S1)S1 − S21) + ab(σ(S1)S2 − S1S2 + σ(S2)S1 − S2S1)
+ b2(σ(S2)S2 − S22).
The following lemma will be frequently used in what follows.
Lemma 8.4. Let the notation be as above. Then
Proof. The �rst equality is easy. For the se
ond, we have
Thanks to Equation (8.6), we have
(d(κ)2
d(κ)1/2v∗1ρ(v
1)wv1 =
d2 − 1
d1/4T0,
whi
h shows the statement. �
Lemma 8.5. Let the notation be as above. Then we have
S1R = σ(S1)R, S2R = σ(S2)R.
In parti
ular, (8.7) always holds.
Proof. We introdu
e a linear isomorphism F : (idP , σ
3) → (κκ̄, ακκ̄ακκ̄α) by
F (x) =
α κ κ̄
κ κ̄α α
Sin
e every intertwiner above belongs to the spa
e (idP , σ
3), it su�
es to show the
equalities after applying F to the intertwiners.
70 PINHAS GROSSMAN AND MASAKI IZUMI
For F (S1R), we have
F (S1R) =
α α κ
α α κ
Using Lemma 8.4 with 
onsideration of [α2(idM ⊕ ρ)α] = [idM ]⊕ [αρ], we get
F (S1R) =
α α κ
In a similar way, we have
F (σ(S1)R) =
Sin
e we have
= dα2(S∗0)S0 = d,
= dα(S∗0)S0 = d,
we get F (S1R) = F (σ(S1)R).
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 71
For F (S2R), we have
F (S2R) =
α α κ κ̄
α α κ
Using Lemma 8.4 with 
onsideration of [ρ idMα] = [αρ] and [ρρα] = [α]⊕ [ρ]⊕ [αρ]⊕
[α2ρ], we get
F (S2R) =
α α κ
In a similar way,
F (σ(S2)R) =
Sin
e we have
= d3/4ρ(T ∗0 )T2T0 = d
3/4A(2, 1)T1 = d
3/4A(1, 2)T1,
= d3/4α(T ∗0 )αρ(T1)T0 = d
3/4T ∗2 ρ(T2)T0 = d
3/4A(1, 2)T1,
we get F (S2R) = F (σ(S2)R). �
Lemma 8.6. Let S = aS1 + bS2 with a, b ∈ C. Then Equation (8.8) is equivalent to
(8.9)
= a2 +
d− 1ab,
72 PINHAS GROSSMAN AND MASAKI IZUMI
(8.10)
= ab− b
(8.11) 0 = a2 − ab√
− A(1, 2)b2,
(8.12) 0 = ab+
1 + (d− 1)A(1, 2)
(d− 1)3/2 b
Proof. Let G : (σ, σ3) → (κκ̄ακκ̄, ακκ̄ακκ̄α) be the linear isomorphism de�ned in a
similar way as in the proof of the previous lemma. We �rst 
ompute G(R), G(σ(R)),
G(S21) et
.
For G(R), we have
G(R) =
α κ κ̄ α κ
α κ κ̄ α κ
α κ κ̄ α κ
Applying Lemma 8.4 to the se
ond term, we get
G(R) =
α κ κ̄ α κ
α κ κ̄ α κ
α κ κ̄ α κ
The intertwiner G(σ(R)) is the mirror image of G(R). Sin
e
α α ρ αρ
dS0 =
α α ρ αρ
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 73
we get
G(R− σ(R))
α κ κ̄ α κ
α κ κ̄ ακ
α κ κ̄ α κ
α κ κ̄ α κ
For G(S21), Lemma 8.4 implies
G(S21) =
α α κ κ̄
α κ κ̄κ κ̄
α α κ κ̄
α κ κ̄κ κ̄
α α κ κ̄ α
α κ κ̄κ κ̄
α α κ κ̄ α
α κ κ̄κ κ̄
α α κ κ̄ α
α κ κ̄κ κ̄
α α κ κ̄ α
α κ κ̄κ κ̄
74 PINHAS GROSSMAN AND MASAKI IZUMI
where we use α(S0) = S0. The intertwiner G(σ(S1)S1) is the mirror image of G(S
A similar argument shows
G(S2S1) =
α α κ κ̄ α
α κ κ̄κ κ̄
α α κ κ̄ α
α κ κ̄κ κ̄
and G(σ(S2)S1) is its mirror image.
For G(S1S2), we have
G(S1S2) =
α α κ
κακ κ̄
α α κ
κακ κ̄
α α κ
κακ κ̄
α α κ
κακ κ̄
α α κ
κακ κ̄
α α κ
κακ κ̄
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 75
Sin
e (α2α, αρ) = (α2α, ραρ) = 0, we get
G(S1S2) =
α α κ
κακ κ̄
α α κ
κακ κ̄
The intertwiner G(σ(S1)S2) is the mirror image of G(S1S2). A similar argument
shows
G(S22) =
α α κ
κακ κ̄
α α κ
κακ κ̄
α α κ
κακ κ̄
and G(σ(S2)S2) is the mirror image of G(S
Now (8.8) is equivalent to
α αα ρ
α αα ρ
α αα ρ
αα α ρ
αα α ρ
76 PINHAS GROSSMAN AND MASAKI IZUMI
α α α
α α α
α α α
ρ α ρ
ρ α ρ
ρ α ρ
ρα α α
ρ α αα
ρ α ρ
ρα α α
ρ α ρ
− d− 1
ρ α αα
ρ α ρ
It is easy to show that the �rst and the se
ond equations are equivalent to (8.9),
the third and fourth are equivalent to (8.10), and the �fth and sixth are equivalent
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 77
to (8.11). For the last equation, we have
ρα α α
ρ α αα
ρ α ρ
2 = dT2ρα
2(S∗0)T0 − dα(T1)S∗0ρα(T0)
= dT2ρ(S
0)T0 − dT0S0ρ(T2) =
d(T2T
0 − T0T ∗1 ),
ρα α α
ρ α ρ
ρ α αα
ρ α ρ
= dρ(T ∗0 )T2ρ(T
2 )T0 − dα(T ∗0 )αρ(T1)T ∗2 ρα(T0)
= dρ(T ∗0 )T2ρ(T
2 )T0 − dT ∗2 ρ(T2)T ∗2 ρ(T2)
l∈Z/3Z
A(2, l)T2+lT
)∗( ∑
k∈Z/3Z
A(2, 2 + k)T2+kT
l∈Z/3Z
A(1, l + 2)T1+lT
k∈Z/3Z
A(1, k + 2)T1+kT
l∈Z/3Z
A(2, l)A(2, l + 1)TlT
l+1 − d
k∈Z/3Z
A(1, k)A(1, k + 2)Tk−1T
(d− 1)2 +
A(1, 2)
0 − T0T ∗1 ).
Therefore the last equation is equivalent to (8.12). �
Proof of Theorem 8.2. Equation (8.11) shows b 6= 0 and so (8.12) implies
(8.13) a = − B + 1
(d − 1)3/2 b.
78 PINHAS GROSSMAN AND MASAKI IZUMI
where B = (d− 1)A(1, 2). Using the fa
t that B satis�es B2−B+ d = 0, we 
an see
that this is 
ompatible with (8.11). Iterating (8.13) into (8.9), we get
(B + 1)2
(d− 1)3 b
2 − B + 1
d− 1 b
(d− 1)3
B2 + 2B + 1− (d− 1)2(B + 1)
(d− 1)3
3B − d+ 1− (d+ 2)(B + 1)
(d− 1)3
(1− d)B + d− d2
= − 1
(d− 1)2
B + d)b2,
whi
h shows
(8.14) b2 = − (d− 1)
(B + d)
We get the same equation from (8.10).
Now all we have to show is that (8.13) and (8.14) are 
ompatible with |a|2+|b|2 = 1.
The equation (8.13) and |a|2 + |b|2 = 1 imply
|b|2 = 1 +
(d− 1)3 |B + 1|
2 = 1 +
(d− 1)3 = 1 +
d− 1 =
d− 1 .
Sin
e |B + d|2 = d2 + 2d = d(d− 1)2, this is 
ompatible with (8.14). �
Referen
es
1. Asaeda, M. and Haagerup, U., Exoti
 subfa
tors of �nite depth with Jones indi
es (5+
13)/2
and (5 +
17)/2. Comm. Math. Phys. 202 (1999), 1�63.
2. Bö
kenhauer, J., Evans, D. E., and Kawahigashi, Y. Chiral stru
ture of modular invariants for
subfa
tors. Comm. Math. Phys. 210 (2000), 733�784.
3. Bö
kenhauer, J., Evans, D. E., and Kawahigashi, Y. On α-indu
tion, 
hiral generators and
modular invariants for subfa
tors. Comm. Math. Phys. 208 (1999), no. 429�487.
4. Bis
h, D. A note on intermediate subfa
tors. Pa
i�
 Journal of Mathemati
s, 163 (1994),
201�216.
5. Bis
h, D., Bimodules, higher relative 
ommutants and the fusion algebra asso
iated to a sub-
fa
tor. In Operator algebras and their appli
ations. Fields Institute Communi
ations, Vol. 13,
Ameri
an Math. So
. (1997), 13�63.
6. Bis
h, D. and Haagerup, U., Composition of subfa
tors: new examples of in�nite depth sub-
fa
tors. Ann. S
i. É
ole Norm. Sup. (4) 29 (1996), 329�383.
7. Bis
h, D. and Jones, V. F. R., Algebras asso
iated to intermediate subfa
tors. Inventiones
Mathemati
ae, 128 (1997), 89�157.
8. Choda, M. Index for fa
tors generated by Jones' two sided sequen
e of proje
tions. Pa
i�
 J.
Math. 139 (1989), 1�16.
9. Evans, D. E. and Kawahigashi, Y., Quantum symmetries on operator algebras. Oxford Uni-
versity Press (1998).
10. Evans, D. E. and Kawahigashi, Y., Orbifold subfa
tors from He
ke algebras. II. Quantum
doubles and braiding. Comm. Math. Phys. 196 (1998), 331�361.
11. Goodman, F., de la Harpe, P. and Jones, V. F. R., Coxeter graphs and towers of algebras.
MSRI Publi
ations (Springer) (1989), 14.
CLASSIFICATION OF NONCOMMUTING QUADRILATERALS OF FACTORS 79
12. Grossman, P. (in press), Forked Temperley-Lieb algebras and intermediate subfa
tors. J. Fun
t.
Anal. (2007), doi:10.1016/j.jfa.2007.03.014
13. Grossman, P. and Jones, V. F. R., Intermediate subfa
tors with no extra stru
ture. J. Amer.
Math. So
. 20 (2007), 219�265.
14. Hayashi, T. and Yamagami, S., Amenable tensor 
ategories and their realizations as AFD
bimodules. J. Fun
t. Anal. 172 (2000), 19�75.
15. Hiai, F., Minimizing indi
es of 
onditional expe
tations onto a subfa
tor. Publ. Res. Inst. Math.
S
i. 24 (1988), 673�678.
16. Hong, J. H., Subfa
tors with prin
ipal graph E
6 . A
ta Appl. Math. 40 (1995), 255�264.
17. Izumi, M., Appli
ation of fusion rules to 
lassi�
ation of subfa
tors. Publ. Res. Inst. Math.
S
i., 27 (1991), 953�994.
18. Izumi, M., Goldman's type theorem for index 3. Publ. Res. Inst. Math. S
i. 28 (1992), 833�843.
19. Izumi, M., On type II and type III prin
ipal graphs of subfa
tors. Math. S
and. 73 (1993),
307�319.
20. Izumi, M., Subalgebras of in�nite C
-algebras with �nite Watatani indi
es II. Cuntz-Krieger
algebras. Duke J. Math. 91 (1998), 409-461.
21. Izumi, M., Goldman's type theorems in index theory. Operator algebras and quantum �eld
theory (Rome, 1996), 249�269, Int. Press, Cambridge, MA, 1997.
22. Izumi, M., The stru
ture of se
tors asso
iated with Longo-Rehren in
lusions. I. General theory.
Comm. Math. Phys. 213 (2000), 127�179.
23. Izumi, M., The stru
ture of se
tors asso
iated with Longo-Rehren in
lusions. II. Examples.
Rev. Math. Phys. 13 (2001), 603�674.
24. Izumi, M., Chara
terization of isomorphi
 group-subgroup subfa
tors. Int. Math. Res. Not.
2002, no. 34, 1791�1803.
25. Izumi, M. and Kawahigashi, Y., Classi�
ation of subfa
tors with the prin
ipal graph D
n . J.
Fun
t. Anal. 112 (1993), 257�286.
26. Izumi, M. and Kosaki, H., Ka
 algebras arising from 
omposition of subfa
tors: general theory
and 
lassi�
ation. Mem. Amer. Math. So
. 158 (2002), no. 750.
27. Izumi, M. and Kosaki, H., On a subfa
tor analogue of the se
ond 
ohomology. Reviews in
Mathemati
al Physi
s, 14 (2002), 733�757.
28. Izumi, M., Longo, R., and Popa, S. A Galois 
orresponden
e for 
ompa
t groups of automor-
phisms of von Neumann algebras with a generalization to Ka
 algebras. J. Fun
t. Anal. 155
(1998), 25�63.
29. Jones, V. F. R. (1980). A
tions of �nite groups on the hyper�nite type II1 fa
tor. Memoirs of
the Ameri
an Mathemati
al So
iety, 237.
30. Jones, V. F. R., Index for subfa
tors. Inventiones Mathemati
ae, 72 (1983), 1�25.
31. Jones, V. F. R. (in press). Planar algebras I. New Zealand Journal of Mathemati
s. QA/9909027
32. Jones, V. F. R., Quadrati
 tangles in planar algebras. In preparation, (2003),
http://math.berkeley.edu/∼vfr
33. Jones, V. F. R. and Xu, F., Interse
tions of �nite families of �nite index subfa
tors. Interna-
tional Journal of Mathemati
s, 15 (2004), 717�733.
34. Kawahigashi, Y., On �atness of O
neanu's 
onne
tions on the Dynkin diagrams and 
lassi�-

ation of subfa
tors. J. Fun
t. Anal. 127 (1995), 63�107.
35. Kawahigashi, Y. and Longo, R., Classi�
ation of two-dimensional lo
al 
onformal nets with
c < 1 and 2-
ohomology vanishing for tensor 
ategories. Comm. Math. Phys. 244 (2004),
63�97.
36. Kawahigashi, Y. and Longo, R., Pennig, U., and Rehren, K.-H., The 
lassi�
ation of non-lo
al

hiral CFT with c < 1. math.OA/0505130, to appear in Comm. Math. Phys.
37. Kirillov, A. N. and Reshetikhin, N. Yu., Representations of the algebra Uq(sl(2)), q-orthogonal
polynomials and invariants of links. In�nite-dimensional Lie algebras and groups (Luminy-
Marseille, 1988), 285�339, Adv. Ser. Math. Phys., 7, World S
i. Publ., Teane
k, NJ, 1989.
http://math.berkeley.edu/~vfr
http://arxiv.org/abs/math/0505130
80 PINHAS GROSSMAN AND MASAKI IZUMI
38. Kosaki, H. Extension of Jones' theory on index to arbitrary fa
tors. J. Fun
t. Anal. 66 (1986),
123�140.
39. Kosaki, H. and Yamagami, S., Irredu
ible bimodules asso
iated with 
rossed produ
t algebras.
Internat. J. Math. 3 (1992), 661�676.
40. Landau, Z., Fuss-Catalan algebras and 
hains of intermediate subfa
tors. Pa
i�
 J. Math., 197
(2001), 325�367.
41. Landau, Z., Ex
hange relation planar algebras. J. of Fun
t. Anal., 195 (2002), 71�88.
42. Longo, R., Index of subfa
tors and statisti
s of quantum �elds. II. Corresponden
es, braid
group statisti
s and Jones polynomial. Comm. Math. Phys. 130 (1990), 285�309.
43. Longo, R., A duality for Hopf algebras and for subfa
tors. I. Comm. Math. Phys. 159 (1994),
133�150.
44. Longo, R. and Rehren, K.-H. (1995). Nets of subfa
tors. Reviews in Mathemati
al Physi
s, 7,
567�597.
45. Longo, R. and Roberts, J. E. (1997). A theory of dimension. K-theory, 11, 103�159.
46. Masuda, T., An analogue of Longo's 
anoni
al endomorphism for bimodule theory and its
appli
ation to asymptoti
 in
lusions. Internat. J. Math. 8 (1997), 249�265.
47. Masuda, T., An analogue of Connes-Haagerup approa
h for 
lassi�
ation of subfa
tors of type
III1. J. Math. So
. Japan, 57 (2005), 959�1001.
48. Nakamura, M. and Takeda, Z., A Galois theory for �nite fa
tors. Pro
. Japan A
ad. 36 (1960),
258�260.
49. O
neanu, A. Quantum symmetry, di�erential geometry of �nite graphs and 
lassi�
ation of
subfa
tors, University of Tokyo Seminary Notes 45, (1991), (Notes re
orded by Kawahigashi,
50. Okamoto, S., Invariants for subfa
tors arising from Coxeter graphs. Current Topi
s in Operator
Algebras, World S
ienti�
 Publishing, (1991), 84�103.
51. Pimsner, M. and Popa, S., Entropy and index for subfa
tors. Annales S
ienti�ques de l'É
ole
Normale Superieur, 19 (1986), 57�106.
52. Popa, S., Classi�
ation of amenable subfa
tors of type II. A
ta Math. 172 (1994), 163�255.
53. Popa, S., Classi�
ation of subfa
tors and their endomorphisms. CBMS Regional Conferen
e
Series in Mathemati
s, 86. Published for the Conferen
e Board of the Mathemati
al S
ien
es,
Washington, DC; by the Ameri
an Mathemati
al So
iety, Providen
e, RI, 1995.
54. Sano, T. and Watatani, Y., Angles between two subfa
tors. Journal of Operator Theory, 32
(1994), 209�241.
55. Wassermann, A., Operator algebras and 
onformal �eld theory. III. Fusion of positive energy
representations of LSU(N) using bounded operators. Invent. Math. 133 (1998), 467�538.
56. Watatani, Y., Latti
es of intermediate subfa
tors. J. Fun
t. Anal., 140 (1996), 312�334.
57. Wenzl, H., C∗ tensor 
ategories from quantum groups. J. Amer. Math. So
. 11 (1998), 261�282.
58. Xu, Feng, New braided endomorphisms from 
onformal in
lusions. Comm. Math. Phys. 192
(1998), 349�403.
Pinhas Grossman, Department of Mathemati
s, 1326 Stevenson Center, Vanderbilt
University, Nashville, TN, 37203
e-mail: pinhas.grossman�vanderbilt.edu
Masaki Izumi, Department of Mathemati
s, Graduate S
hool of S
ien
e, Kyoto University,
Sakyo-ku, Kyoto 606-8502, Japan
e-mail: izumi�math.kyoto-u.a
.jp
	1. Introduction
	2. Preliminaries
	2.1. Quadrilaterals
	2.2. Sectors
	3. Q-systems and angles
	4. (2,2)-supertransitive quadrilaterals
	5. Classification I
	5.1. Class I
	5.2. Class II
	5.3. Class III
	5.4. Class IV
	6. Classification II
	7. -induction and GHJ pairs
	7.1. -induction and angles
	7.2. GHJ pairs
	7.3. Asymptotic inclusions and angles
	8. Appendix
	References
