Temperley–Lieb algebra

Let ${\displaystyle R}$ be a commutative ring and fix ${\displaystyle \delta \in R}$. The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is the ${\displaystyle R}$-algebra generated by the elements ${\displaystyle U_{1},U_{2},\ldots ,U_{n-1}}$, subject to the Jones relations:

• ${\displaystyle U_{i}^{2}=\delta U_{i}}$ for all ${\displaystyle 1\leq i\leq n-1}$
• ${\displaystyle U_{i}U_{i+1}U_{i}=U_{i}}$ for all ${\displaystyle 1\leq i\leq n-2}$
• ${\displaystyle U_{i}U_{i-1}U_{i}=U_{i}}$ for all ${\displaystyle 2\leq i\leq n-1}$
• ${\displaystyle U_{i}U_{j}=U_{j}U_{i}}$ for all ${\displaystyle 1\leq i,j\leq n-1}$ such that ${\displaystyle |i-j|\neq 1}$

${\displaystyle TL_{n}(\delta )}$ may be represented diagrammatically as the vector space over noncrossing pairings of ${\displaystyle 2n}$ points on two opposite sides of a rectangle with n points on each of the two sides.

The identity element is the diagram in which each point is connected to the one directly across the rectangle from it. The generator ${\displaystyle U_{i}}$ is the diagram in which the ${\displaystyle i}$-th and ${\displaystyle (i+1)}$-th point on the left side are connected to each other, similarly the two points opposite to these on the right side, and all other points are connected to the point directly across the rectangle.

The generators of ${\displaystyle TL_{5}(\delta )}$ are:

From left to right, the unit 1 and the generators ${\displaystyle U_{1}}$, ${\displaystyle U_{2}}$, ${\displaystyle U_{3}}$, ${\displaystyle U_{4}}$.

Multiplication on basis elements can be performed by placing two rectangles side by side, and replacing any closed loops by a factor ${\displaystyle \delta }$, for example ${\displaystyle U_{1}U_{4}U_{3}U_{2}\times U_{2}U_{4}U_{3}=\delta \,U_{1}U_{4}U_{3}U_{2}U_{4}U_{3}}$:

× = = ${\displaystyle \delta }$ .

The Jones relations can be seen graphically:

= ${\displaystyle \delta }$

=

=

The five basis elements of ${\displaystyle TL_{3}(\delta )}$ are the following:

.

From left to right, the unit 1, the generators ${\displaystyle U_{2}}$, ${\displaystyle U_{1}}$, and ${\displaystyle U_{1}U_{2}}$, ${\displaystyle U_{2}U_{1}}$.

Consider an interaction-round-a-face model e.g. a square lattice model and let ${\displaystyle L}$ be the number of sites on the lattice. Following Temperley and Lieb[1] we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as

${\displaystyle {\mathcal {H}}=\sum _{j=1}^{L-1}(\delta -U_{j})}$

Applications

In what follows we consider the special case ${\displaystyle \delta =1}$.

We will firstly consider the case ${\displaystyle L=3}$. The TL Hamiltonian is ${\displaystyle {\mathcal {H}}=2-e_{1}-e_{2}}$, namely

${\displaystyle {\mathcal {H}}}$ = 2 - - .

We have two possible states,

and .

In acting by ${\displaystyle {\mathcal {H}}}$ on these states, we find

${\displaystyle {\mathcal {H}}}$ = 2 - - = - ,

and

${\displaystyle {\mathcal {H}}}$ = 2 - - = - + .

Writing ${\displaystyle {\mathcal {H}}}$ as a matrix in the basis of possible states we have,

${\displaystyle {\mathcal {H}}=\left({\begin{array}{rr}1&-1\\-1&1\end{array}}\right)}$

The eigenvector of ${\displaystyle {\mathcal {H}}}$ with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue ${\displaystyle \lambda _{0}}$ for ${\displaystyle {\mathcal {H}}}$ is ${\displaystyle \lambda _{0}=0}$. The corresponding eigenvector is ${\displaystyle \psi _{0}=(1,1)}$. As we vary the number of sites ${\displaystyle L}$ we find the following table[2]

${\displaystyle L}$	${\displaystyle \psi _{0}}$	${\displaystyle L}$	${\displaystyle \psi _{0}}$
2	(1)	3	(1, 1)
4	(2, 1)	5	${\displaystyle (3_{3},1_{2})}$
6	${\displaystyle (11,5_{2},4,1)}$	7	${\displaystyle (26_{4},10_{2},9_{2},8_{2},5_{2},1_{2})}$
8	${\displaystyle (170,75_{2},71,56_{2},50,30,14_{4},6,1)}$	9	${\displaystyle (646,\ldots )}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$

where we have used the notation ${\displaystyle m_{n}=(m,\ldots ,m)}$ ${\displaystyle n}$-times e.g., ${\displaystyle 5_{2}=(5,5)}$.

• Kauffman, Louis H. (1987). "State Models and the Jones Polynomial". Topology. 26 (3): 395–407. doi:10.1016/0040-9383(87)90009-7. MR 0899057.
• Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. London: Academic Press Inc. ISBN 0-12-083180-5. MR 0690578.