Temperley–Lieb algebra explained

In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.

Structure

Generators and relations

Let

R be a commutative ring and fix \delta \in R . The Temperley–Lieb algebra TL _{n(\delta)} is the R -algebra generated by the elements e _{1,} e _{2,} \ldots, e _{n-1} , subject to the Jones relations: e ^{2} _{i} = \delta e _{i} for all 1 \leq i \leq n -1 e _{i} e _{i+1} e _{i} = e _{i} for all 1 \leq i \leq n -2 e _{i} e _{i-1} e _{i} = e _{i} for all 2 \leq i \leq n -1 e _{i} e _{j} = e _{j} e _{i} for all 1 \leq i,j \leq n -1 such that |i - j| ≠ 1 Using these relations, any product of generators e _{i} can be brought to Jones' normal form:E = ( e _{i _{1}} e _{i _{1-1}} … e _{j _{1}} ) ( e _{i _{2}} e _{i _{2-1}} … e _{j _{2}} ) … ( e _{i _{r}} e _{i _{r-1}} … e _{j _{r}} ) where (i _{1,i} _{2,...,i} _{r)} and (j _{1,j} _{2,...,j} _{r)} are two strictly increasing sequences in \{ 1 , 2 , ... ,n -1 \} . Elements of this type form a basis of the Temperley-Lieb algebra.

The dimensions of Temperley-Lieb algebras are Catalan numbers:

\dim(TL _{n(\delta))} = (( 2 n)!)/(n!(n +1 )!)

The Temperley–Lieb algebra

TL _{n(\delta)} is a subalgebra of the Brauer algebra ak{B} _{n(\delta)} , and therefore also of the partition algebra P _{n(\delta)} . The Temperley–Lieb algebra TL _{n(\delta)} is semisimple for \delta\in C - F _{n} where F _{n} is a known, finite set. For a given n , all semisimple Temperley-Lieb algebras are isomorphic.

Diagram algebra

TL _{n(\delta)} may be represented diagrammatically as the vector space over noncrossing pairings of 2 n 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

e _{i} is the diagram in which the i -th and (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

TL _{5(\delta)} are:

From left to right, the unit 1 and the generators

e _{1} , e _{2} , e _{3} , e _{4} .

Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor

\delta , for example e _{1} e _{4} e _{3} e _{2 x} e _{2} e _{4} e _{3=\deltae} _{1} e _{4} e _{3} e _{2} e _{4} e _{3} :

× = =

\delta .

The Jones relations can be seen graphically:

=

\delta

=

=

The five basis elements of

TL _{3(\delta)} are the following:

.

From left to right, the unit 1, the generators

e _{2} , e _{1} , and e _{1} e _{2} , e _{2} e _{1} .

Representations

Structure

For

\delta such that TL _{n(\delta)} is semisimple, a complete set \{W _{\ell\}} of simple modules is parametrized by integers 0 \leq \ell\leq n with \ell\equiv n\bmod 2 . The dimension of a simple module is written in terms of binomial coefficients as\dim(W _{\ell)} = \binom{n}{ (n - \ell)/(2) } - \binom

A basis of the simple module

W _{\ell} is the set M _{n,\ell} of monic noncrossing pairings from n points on the left to \ell points on the right. (Monic means that each point on the right is connected to a point on the left.) There is a natural bijection between \cup _{\begin{array} {c} 0 \leq \ell\leq n \ \ell\equiv n\bmod 2 \end{array}}M _{n,\ell} x M _{n,\ell} , and the set of diagrams that generate TL _{n(\delta)} : any such diagram can be cut into two elements of M _{n,\ell} for some \ell .

Then

TL _{n(\delta)} acts on W _{\ell} by diagram concatenation from the left. (Concatenation can produce non-monic pairings, which have to be modded out.) The module W _{\ell} may be called a standard module or link module.

If

\delta = q + q ^{-1} with q a root of unity, TL _{n(\delta)} may not be semisimple, and W _{\ell} may not be irreducible:W _{\ell} reducible \iff \exists j\in\{ 1 , 2 , ... ,\ell\}, q ^{2n-4\ell+2+2j} =1 If W _{\ell} is reducible, then its quotient by its maximal proper submodule is irreducible.

Branching rules from the Brauer algebra

Simple modules of the Brauer algebra

ak{B} _{n(\delta)} can be decomposed into simple modules of the Temperley-Lieb algebra. The decomposition is called a branching rule, and it is a direct sum with positive integer coefficients:W _{λ\left(ak{B}} _{n(\delta)\right)} = o plus _{\begin{array} {c} | λ |\leq \ell\leq n \ \ell\equiv | λ |\bmod 2 \end{array}} c ^{λ} _{\ell} W _{\ell\left(TL} _{n(\delta)\right)} The coefficients c ^{λ} _{\ell} do not depend on n,\delta , and are given byc ^{λ} _{\ell} = f ^{(\ell - | λ |)/(2)} _{r =0} ( -1 ) ^{r} \binom{\ell - r}{r}\binom{\ell -2 r}{\ell - | λ | -2 r}(\ell - | λ | -2 r)!! where f ^{λ} is the number of standard Young tableaux of shape λ , given by the hook length formula.

Affine Temperley-Lieb algebra

The affine Temperley-Lieb algebra

aTL _{n(\delta)} is an infinite-dimensional algebra such that TL _{n(\delta)\subset} aTL _{n(\delta)} . It is obtained by adding generators e ^{-1} _{n,\tau,\tau} such that\tau e _{i} = e _{i+1} \tau for all 1 \leq i \leq n ,e ^{2} _{1 \tau} = e _{1e} _{2} … e _{n-1} ,\tau \tau ^{-1} = \tau ^{-1} \tau = id .The indices are supposed to be periodic i.e. e _{n+1} = e _{1,e} _{n=e} _{0} , and the Temperley-Lieb relations are supposed to hold for all 1 \leq i \leq n . Then \tau ^{n} is central. A finite-dimensional quotient of the algebra aTL _{n(\delta)} , sometimes called the unoriented Jones-Temperley-Lieb algebra, is obtained byassuming \tau ^{n=id} , and replacing non-contractible lines with the same factor \delta as contractible lines (for example, in the case n =4 , this implies e _{1e} _{3e} _{2e} _{4e} _{1e} _{3} = \delta ^{2} e _{1e} _{3} ).

The diagram algebra for

aTL _{n(\delta)} is deduced from the diagram algebra for TL _{n(\delta)} by turning rectangles into cylinders. The algebra aTL _{n(\delta)} is infinite-dimensional because lines can wind around the cylinder. If n is even, there can even exist closed winding lines, which are non-contractible.

The Temperley-Lieb algebra is a quotient of the corresponding affine Temperley-Lieb algebra.

The cell module

W _{\ell,z} of aTL _{n(\delta)} is generated by the set of monic pairings from n points to \ell points, just like the module W _{\ell} of TL _{n(\delta)} . However, the pairings are now on a cylinder, and the right-multiplication with \tau is identified with z ⋅ id for some z\in C ^{*} . If \ell =0 , there is no right-multiplication by \tau , and it is the addition of a non-contractible loop on the right which is identified with z + z ^{-1} . Cell modules are finite-dimensional, with \dim(W _{\ell,z} ) = \binom{n}{ (n - \ell)/(2) }The cell module W _{\ell,z} is irreducible for all z\in C ^{*-R(\delta)} , where the set R(\delta) is countable. For z\in R(\delta) , W _{\ell,z} has an irreducible quotient. The irreducible cell modules and quotients thereof form a complete set of irreducible modules of aTL _{n(\delta)} . Cell modules of the unoriented Jones-Temperley-Lieb algebra must obey z ^{\ell=1} if \ell ≠ 0 , and z + z ^{-1} = \delta if \ell =0 .

Applications

Temperley–Lieb Hamiltonian

Consider an interaction-round-a-face model e.g. a square lattice model and let

n be the number of sites on the lattice. Following Temperley and Lieb^[1] we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) asl{H} = \sum ^{n -1} _{j =1} (\delta - e _{j)}

In what follows we consider the special case

\delta =1 .

We will firstly consider the case

n = 3 . The TL Hamiltonian is l{H} = 2 - e _{1} - e _{2} , namelyl{H} = 2 - - .

We have two possible states,

and .

In acting by

l{H} on these states, we findl{H} = 2 - - = -,

and

l{H} = 2 - - = - + .

Writing

l{H} as a matrix in the basis of possible states we have,l{H} = \left(\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right)

The eigenvector of

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

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

where we have used the notation

m _{j} = (m, \ldots, m) j -times e.g., 5 _{2} = ( 5 , 5 ) .

An interesting observation is that the largest components of the ground state of

l{H} have a combinatorial enumeration as we vary the number of sites,^[3] as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis. Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites1 , 2 , 11 , 170 , \ldots = \prod ^{(n -2)/(2)} _{j =0} \left( 3 j +1 \right) (( 2 j)!( 6 j)!)/(( 4 j)!( 4 j +1 )!) (n = 2 , 4 , 6 , ... )

and for an odd numbers of sites

1 , 3 , 26 , 646 , \ldots = \prod ^{(n -3)/(2)} _{j =0} ( 3 j +2 ) (( 2 j +1 )!( 6 j +3 )!)/(( 4 j +2 )!( 4 j +3 )!) (n =3 , 5 , 7 , ... )

Surprisingly, these sequences corresponded to well known combinatorial objects. For

n even, this corresponds to cyclically symmetric transpose complement plane partitions and for n odd,, these correspond to alternating sign matrices symmetric about the vertical axis.

XXZ spin chain

Further reading

• Book: Kauffman, Louis H. . Knots and Physics . 1991 . World Scientific . 978-981-02-0343-6 . en.
• Louis H.. Kauffman. Louis H. Kauffman. State Models and the Jones Polynomial. Topology. 1987. 26. 3. 395–407. 10.1016/0040-9383(87)90009-7. 0899057. free.
• Book: Baxter, Rodney J.. Rodney J. Baxter. Exactly solved models in statistical mechanics. Academic Press Inc.. London . 1982. 0690578. 0-12-083180-5 .

Notes and References

1. Neville. Temperley. Harold Neville Vazeille Temperley. Elliott H. Lieb. Elliott. Lieb. Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 322. 1549 . 1971. 251–280. 0498284 . 10.1098/rspa.1971.0067. 77727. 1971RSPSA.322..251T. 122770421.
2. Batchelor. Murray. Murray Batchelor. de Gier. Jan . Nienhuis. Bernard. 2001. The quantum symmetric XXZ chain at \Delta = -1 / 2 , alternating-sign matrices and plane partitions. . 34. 19. L265–L270. 10.1088/0305-4470/34/19/101. 1836155. cond-mat/0101385. 118048447.
3. de Gier. Jan. 2005. Loops, matchings and alternating-sign matrices. . 298. 1–3. 365–388. 10.1016/j.disc.2003.11.060. 2163456. math/0211285. 2129159.