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 groups, quantum groups and subfactors of von Neumann algebras.
Elements of this type form a basis of the Temperley-Lieb algebra.
, all semisimple Temperley-Lieb algebras are isomorphic.
may be represented diagrammatically as the vector space over noncrossing pairings of
The identity element is the diagram in which each point is connected to the one directly across the rectangle from it.
Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor
of simple modules is parametrized by integers
The dimension of a simple module is written in terms of binomial coefficients as[4] A basis of the simple module
[3] (Concatenation can produce non-monic pairings, which have to be modded out.)
is reducible, then its quotient by its maximal proper submodule is irreducible.
[1] Simple modules of the Brauer algebra
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: The coefficients
is the number of standard Young tableaux of shape
, and the Temperley-Lieb relations are supposed to hold for all
, sometimes called the unoriented Jones-Temperley-Lieb algebra,[6] is obtained by assuming
, and replacing non-contractible lines with the same factor
is even, there can even exist closed winding lines, which are non-contractible.
is generated by the set of monic pairings from
, and it is the addition of a non-contractible loop on the right which is identified with
The irreducible cell modules and quotients thereof form a complete set of irreducible modules of
[5] Cell modules of the unoriented Jones-Temperley-Lieb algebra must obey
with the lowest eigenvalue is known as the ground state.
An interesting observation is that the largest components of the ground state of
have a combinatorial enumeration as we vary the number of sites,[9] as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis.
[8] Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites
Surprisingly, these sequences corresponded to well known combinatorial objects.
even, this (sequence A051255 in the OEIS) corresponds to cyclically symmetric transpose complement plane partitions and for
odd, (sequence A005156 in the OEIS), these correspond to alternating sign matrices symmetric about the vertical axis.