<s>
In	O
mathematics	O
,	O
a	O
Bratteli	B-Algorithm
diagram	I-Algorithm
is	O
a	O
combinatorial	O
structure	O
:	O
a	O
graph	O
composed	O
of	O
vertices	O
labelled	O
by	O
positive	O
integers	O
(	O
"	O
level	O
"	O
)	O
and	O
unoriented	O
edges	O
between	O
vertices	O
having	O
levels	O
differing	O
by	O
one	O
.	O
</s>
<s>
The	O
notion	O
was	O
introduced	O
by	O
Ola	O
Bratteli	O
in	O
1972	O
in	O
the	O
theory	B-Algorithm
of	I-Algorithm
operator	I-Algorithm
algebras	I-Algorithm
to	O
describe	O
directed	O
sequences	O
of	O
finite-dimensional	O
algebras	O
:	O
it	O
played	O
an	O
important	O
role	O
in	O
Elliott	O
's	O
classification	O
of	O
AF-algebras	O
and	O
the	O
theory	O
of	O
subfactors	O
.	O
</s>
<s>
A	O
Bratteli	B-Algorithm
diagram	I-Algorithm
is	O
given	O
by	O
the	O
following	O
objects	O
:	O
</s>
<s>
An	O
ordered	O
Bratteli	B-Algorithm
diagram	I-Algorithm
is	O
a	O
Bratteli	B-Algorithm
diagram	I-Algorithm
together	O
with	O
a	O
partial	O
order	O
on	O
En	O
such	O
that	O
for	O
any	O
v∈Vn	O
the	O
set	O
{	O
e∈En−	O
1:r(e )	O
=v	O
}	O
is	O
totally	O
ordered	O
.	O
</s>
<s>
A	O
Bratteli	B-Algorithm
diagram	I-Algorithm
with	O
a	O
unique	O
infinitely	O
long	O
path	O
in	O
Emax	O
and	O
Emin	O
is	O
called	O
essentially	O
simple	O
.	O
</s>
<s>
(	O
1	O
)	O
If	O
,	O
the	O
ith	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
corresponding	O
Bratteli	B-Algorithm
diagram	I-Algorithm
is	O
the	O
same	O
as	O
Young	O
's	O
lattice	O
.	O
</s>
<s>
(	O
2	O
)	O
If	O
is	O
the	O
Brauer	B-Application
algebra	I-Application
or	O
the	O
Birman	O
–	O
Wenzl	O
algebra	O
on	O
i	O
strands	O
,	O
then	O
the	O
resulting	O
Bratteli	B-Algorithm
diagram	I-Algorithm
has	O
partitions	O
of	O
i	O
–	O
2k	O
(	O
for	O
)	O
with	O
one	O
edge	O
between	O
partitions	O
on	O
adjacent	O
levels	O
if	O
one	O
can	O
be	O
obtained	O
from	O
the	O
other	O
by	O
adding	O
or	O
subtracting	O
1	O
from	O
a	O
single	O
part	O
.	O
</s>
<s>
(	O
3	O
)	O
If	O
is	O
the	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
on	O
i	O
strands	O
,	O
the	O
resulting	O
Bratteli	O
has	O
integers	O
i–2k''	O
(	O
for	O
)	O
with	O
one	O
edge	O
between	O
integers	O
on	O
adjacent	O
levels	O
if	O
one	O
can	O
be	O
obtained	O
from	O
the	O
other	O
by	O
adding	O
or	O
subtracting	O
1	O
.	O
</s>
