<s>
In	O
mathematics	O
,	O
planar	B-Application
algebras	I-Application
first	O
appeared	O
in	O
the	O
work	O
of	O
Vaughan	O
Jones	O
on	O
the	O
standard	O
invariant	O
of	O
a	O
II1	O
subfactor	O
.	O
</s>
<s>
Any	O
subfactor	O
planar	B-Application
algebra	I-Application
provides	O
a	O
family	O
of	O
unitary	O
representations	O
of	O
Thompson	O
groups	O
.	O
</s>
<s>
Any	O
finite	O
group	O
(	O
and	O
quantum	O
generalization	O
)	O
can	O
be	O
encoded	O
as	O
a	O
planar	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
idea	O
of	O
the	O
planar	B-Application
algebra	I-Application
is	O
to	O
be	O
a	O
diagrammatic	O
axiomatization	O
of	O
the	O
standard	O
invariant	O
.	O
</s>
<s>
A	O
planar	B-Application
algebra	I-Application
is	O
a	O
representation	O
of	O
the	O
planar	O
operad	O
;	O
more	O
precisely	O
,	O
it	O
is	O
a	O
family	O
of	O
vector	O
spaces	O
,	O
called	O
-box	O
spaces	O
,	O
on	O
which	O
acts	O
the	O
planar	O
operad	O
,	O
i.e.	O
</s>
<s>
The	O
family	O
of	O
vector	O
spaces	O
generated	O
by	O
the	O
planar	O
tangles	O
having	O
intervals	O
on	O
their	O
output	O
disk	O
and	O
a	O
white	O
(	O
or	O
black	O
)	O
-marked	O
interval	O
,	O
admits	O
a	O
planar	B-Application
algebra	I-Application
structure	O
.	O
</s>
<s>
This	O
planar	B-Application
algebra	I-Application
encodes	O
the	O
notion	O
of	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
.	O
</s>
<s>
A	O
semisimple	O
and	O
cosemisimple	O
Hopf	O
algebra	O
over	O
an	O
algebraically	O
closed	O
field	O
is	O
encoded	O
in	O
a	O
planar	B-Application
algebra	I-Application
defined	O
by	O
generators	O
and	O
relations	O
,	O
and	O
"	O
corresponds	O
"	O
(	O
up	O
to	O
isomorphism	O
)	O
to	O
a	O
connected	O
,	O
irreducible	O
,	O
spherical	O
,	O
non	O
degenerate	O
planar	B-Application
algebra	I-Application
with	O
non	O
zero	O
modulus	O
and	O
of	O
depth	O
two	O
.	O
</s>
<s>
A	O
subfactor	O
planar	B-Application
algebra	I-Application
is	O
a	O
planar	O
-algebra	O
which	O
is	O
:	O
</s>
<s>
No-ghost	O
theorem	O
:	O
The	O
planar	B-Application
algebra	I-Application
has	O
no	O
ghost	O
(	O
i.e.	O
</s>
<s>
Then	O
the	O
quotient	O
is	O
a	O
subfactor	O
planar	B-Application
algebra	I-Application
,	O
called	O
the	O
Temperley	O
–	O
Lieb-Jones	O
subfactor	O
planar	B-Application
algebra	I-Application
.	O
</s>
<s>
Any	O
subfactor	O
planar	B-Application
algebra	I-Application
with	O
constant	O
admits	O
as	O
planar	O
subalgebra	O
.	O
</s>
<s>
A	O
planar	B-Application
algebra	I-Application
is	O
a	O
subfactor	O
planar	B-Application
algebra	I-Application
if	O
and	O
only	O
if	O
it	O
is	O
the	O
standard	O
invariant	O
of	O
an	O
extremal	O
subfactor	O
of	O
index	O
,	O
with	O
and	O
.	O
</s>
<s>
There	O
is	O
a	O
subfactor	O
planar	B-Application
algebra	I-Application
encoding	O
any	O
finite	O
group	O
(	O
and	O
more	O
generally	O
,	O
any	O
finite	O
dimensional	O
Hopf	O
-algebra	O
,	O
called	O
Kac	O
algebra	O
)	O
,	O
defined	O
by	O
generators	O
and	O
relations	O
.	O
</s>
<s>
A	O
(	O
finite	O
dimensional	O
)	O
Kac	O
algebra	O
"	O
corresponds	O
"	O
(	O
up	O
to	O
isomorphism	O
)	O
to	O
an	O
irreducible	O
subfactor	O
planar	B-Application
algebra	I-Application
of	O
depth	O
two	O
.	O
</s>
<s>
The	O
subfactor	O
planar	B-Application
algebra	I-Application
associated	O
to	O
an	O
inclusion	O
of	O
finite	O
groups	O
,	O
</s>
<s>
A	O
Bisch-Jones	O
subfactor	O
planar	B-Application
algebra	I-Application
(	O
sometimes	O
called	O
Fuss-Catalan	O
)	O
is	O
defined	O
as	O
for	O
but	O
by	O
allowing	O
two	O
colors	O
of	O
string	O
with	O
their	O
own	O
constant	O
and	O
,	O
with	O
as	O
above	O
.	O
</s>
<s>
It	O
is	O
a	O
planar	O
subalgebra	O
of	O
any	O
subfactor	O
planar	B-Application
algebra	I-Application
with	O
an	O
intermediate	O
such	O
that	O
and	O
.	O
</s>
<s>
The	O
first	O
finite	O
depth	O
subfactor	O
planar	B-Application
algebra	I-Application
of	O
index	O
is	O
called	O
the	O
Haagerup	O
subfactor	O
planar	B-Application
algebra	I-Application
.	O
</s>
<s>
A	O
subfactor	O
planar	B-Application
algebra	I-Application
remembers	O
the	O
subfactor	O
(	O
i.e.	O
</s>
<s>
Let	O
be	O
a	O
finite	O
index	O
subfactor	O
,	O
and	O
the	O
corresponding	O
subfactor	O
planar	B-Application
algebra	I-Application
.	O
</s>
<s>
For	O
any	O
irreducible	O
subfactor	O
planar	B-Application
algebra	I-Application
,	O
the	O
set	O
of	O
biprojections	O
is	O
a	O
finite	O
lattice	O
,	O
of	O
the	O
form	O
,	O
as	O
for	O
an	O
interval	O
of	O
finite	O
groups	O
.	O
</s>
<s>
Using	O
the	O
biprojections	O
,	O
we	O
can	O
make	O
the	O
intermediate	O
subfactor	O
planar	B-Application
algebras	I-Application
.	O
</s>
<s>
The	O
uncertainty	O
principle	O
extends	O
to	O
any	O
irreducible	O
subfactor	O
planar	B-Application
algebra	I-Application
:	O
</s>
