<s>
In	O
statistical	O
mechanics	O
,	O
the	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
an	O
algebra	O
from	O
which	O
are	O
built	O
certain	O
transfer	B-Algorithm
matrices	I-Algorithm
,	O
invented	O
by	O
Neville	O
Temperley	O
and	O
Elliott	O
Lieb	O
.	O
</s>
<s>
It	O
is	O
also	O
related	O
to	O
integrable	O
models	O
,	O
knot	O
theory	O
and	O
the	O
braid	B-Application
group	I-Application
,	O
quantum	O
groups	O
and	O
subfactors	O
of	O
von	O
Neumann	O
algebras	O
.	O
</s>
<s>
The	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
the	O
-algebra	O
generated	O
by	O
the	O
elements	O
,	O
subject	O
to	O
the	O
Jones	O
relations	O
:	O
</s>
<s>
Elements	O
of	O
this	O
type	O
form	O
a	O
basis	O
of	O
the	O
Temperley-Lieb	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
dimensions	O
of	O
Temperley-Lieb	B-Application
algebras	I-Application
are	O
Catalan	O
numbers	O
:	O
</s>
<s>
The	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
a	O
subalgebra	O
of	O
the	O
Brauer	B-Application
algebra	I-Application
,	O
and	O
therefore	O
also	O
of	O
the	O
partition	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
semisimple	O
for	O
where	O
is	O
a	O
known	O
,	O
finite	O
set	O
.	O
</s>
<s>
For	O
a	O
given	O
,	O
all	O
semisimple	O
Temperley-Lieb	B-Application
algebras	I-Application
are	O
isomorphic	O
.	O
</s>
<s>
Simple	O
modules	O
of	O
the	O
Brauer	B-Application
algebra	I-Application
can	O
be	O
decomposed	O
into	O
simple	O
modules	O
of	O
the	O
Temperley-Lieb	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
affine	O
Temperley-Lieb	B-Application
algebra	I-Application
is	O
an	O
infinite-dimensional	O
algebra	O
such	O
that	O
.	O
</s>
<s>
The	O
Temperley-Lieb	B-Application
algebra	I-Application
is	O
a	O
quotient	O
of	O
the	O
corresponding	O
affine	O
Temperley-Lieb	B-Application
algebra	I-Application
.	O
</s>
<s>
a	O
square	O
lattice	B-Algorithm
model	I-Algorithm
and	O
let	O
be	O
the	O
number	O
of	O
sites	O
on	O
the	O
lattice	O
.	O
</s>
<s>
For	O
even	O
,	O
this	O
corresponds	O
to	O
cyclically	O
symmetric	O
transpose	O
complement	O
plane	O
partitions	O
and	O
for	O
odd	O
,	O
,	O
these	O
correspond	O
to	O
alternating	B-Algorithm
sign	I-Algorithm
matrices	I-Algorithm
symmetric	O
about	O
the	O
vertical	O
axis	O
.	O
</s>
