Temperley–Lieb algebra

Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the unital associative R-algebra generated by the elements e_1, e_2, \ldots, e_{n-1}, subject to the Jones relations: • e_i^2 = \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| \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_{i_1}e_{i_1-1}\cdots e_{j_1}\big)\big(e_{i_2}e_{i_2-1}\cdots e_{j_2}\big)\cdots\big(e_{i_r}e_{i_r-1}\cdots e_{j_r}\big) where (i_1,i_2,\dots,i_r) and (j_1,j_2,\dots,j_r) are two strictly increasing sequences in \{1,2,\dots,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)) = \frac{(2n)!}{n!(n+1)!} The Temperley–Lieb algebra TL_n(\delta) is a subalgebra of the Brauer algebra \mathfrak{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\mathbb{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 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 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\times e_2 e_4 e_3=\delta\,e_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}{\frac{n-\ell}{2}} - \binom{n}{\frac{n-\ell}{2}-1} 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}\times 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 \text{ reducible } \iff \exists j\in\{1,2,\dots,\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 \mathfrak{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_\lambda\left(\mathfrak{B}_n(\delta)\right) = \bigoplus_{\begin{array}{c} |\lambda|\leq \ell\leq n \\ \ell\equiv |\lambda|\bmod 2\end{array}} c_\ell^\lambda W_\ell\left(TL_n(\delta)\right) The coefficients c_\ell^\lambda do not depend on n,\delta, and are given by : c_\ell^\lambda = f^\lambda\sum_{r=0}^{\frac{\ell-|\lambda|}{2}} (-1)^r \binom{\ell-r}{r}\binom{\ell-2r}{\ell-|\lambda|-2r}(\ell-|\lambda|-2r)!! where f^\lambda is the number of standard Young tableaux of shape \lambda, 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_n,\tau,\tau^{-1} such that • \tau e_i = e_{i+1}\tau for all 1 \leq i \leq n, • e_1\tau^2 = e_1e_2 \cdots e_{n-1}, • \tau \tau^{-1}=\tau^{-1}\tau = \text{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 by assuming \tau^n=\text{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\cdot\text{id} for some z\in\mathbb{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}{\frac{n-\ell}{2}} The cell module W_{\ell,z} is irreducible for all z\in\mathbb{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\neq 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 we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as \mathcal{H} = \sum_{j=1}^{n-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 \mathcal{H} = 2 - e_1 - e_2 , namely \mathcal{H} = 2 - - . We have two possible states, and . In acting by \mathcal{H} on these states, we find \mathcal{H} = 2 - - = - , and \mathcal{H} = 2 - - = - + . Writing \mathcal{H} as a matrix in the basis of possible states we have, \mathcal{H} = \left(\begin{array}{rr} 1 & -1\\ -1 & 1 \end{array}\right) The eigenvector of \mathcal{H} with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue \lambda_0 for \mathcal{H} is \lambda_0 = 0. The corresponding eigenvector is \psi_0 = (1, 1). As we vary the number of sites n we find the following table 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 \mathcal{H} have a combinatorial enumeration as we vary the number of sites, 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 sites 1, 2, 11, 170, \ldots = \prod_{j=0}^{\frac{n-2}{2}} \left( 3j + 1\right)\frac{ (2j)!(6j)!}{(4j)!(4j + 1)!} \qquad (n = 2, 4, 6,\dots) and for an odd numbers of sites 1, 3, 26, 646, \ldots = \prod_{j=0}^{\frac{n-3}{2}} (3j+2)\frac{ (2j + 1)!(6j + 3)!}{(4j + 2)!(4j + 3)!} \qquad (n=3, 5, 7, \dots) 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 ==References==