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research
Intermediate Subfactors with No Extra Structure
Authors
Pinhas Grossman
Vaughan F. R. Jones
Publication date
1 January 2004
Publisher
View
on
arXiv
Abstract
If
N
β
P
,
Q
β
M
N \subset P,Q \subset M
N
β
P
,
Q
β
M
are type II_1 factors with
N
β²
β©
M
=
C
i
d
N' \cap M = C id
N
β²
β©
M
=
C
i
d
and
[
M
:
N
]
[M:N]
[
M
:
N
]
finite we show that restrictions on the standard invariants of the elementary inclusions
N
β
P
N \subset P
N
β
P
,
N
β
Q
N \subset Q
N
β
Q
,
P
β
M
P \subset M
P
β
M
and
Q
β
M
Q \subset M
Q
β
M
imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto
P
P
P
and
Q
Q
Q
do not commute, then
[
M
:
N
]
[M:N]
[
M
:
N
]
is 6 or
6
+
4
2
6 + 4\sqrt 2
6
+
4
2
β
. In the former case
N
N
N
is the fixed point algebra for an outer action of
S
3
S_3
S
3
β
on
M
M
M
and the angle is
Ο
/
3
\pi/3
Ο
/3
, and in the latter case the angle is
c
o
s
β
1
(
2
β
1
)
cos^{-1}(\sqrt 2-1)
co
s
β
1
(
2
β
β
1
)
and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.Comment: 51 pages, 65 figure
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Last time updated on 22/10/2014