Temperley–Lieb Algebra Overview

Updated 28 August 2025

Temperley–Lieb algebra is a finite-dimensional associative algebra defined via generators, relations, and diagrammatic realizations that yield combinatorial structures like the Catalan numbers.
It connects diverse fields such as statistical mechanics, knot theory, and representation theory by providing foundational links for transfer matrices, link invariants, and quantum topology.
Its cellular structure and categorification facilitate rigorous analysis of module theories and support extensions including cyclotomic, partition, and framisation generalizations.

The Temperley–Lieb algebra is a class of finite-dimensional associative algebras with deep connections to statistical mechanics, knot theory, representation theory, and categorification. Originally introduced in the context of exactly solvable lattice models, these algebras admit a compelling diagrammatic realization and play a central role in the structure and categorification of Hecke algebras, as well as in the construction of link invariants and quantum invariants in low-dimensional topology.

1. Algebraic Definition and Diagrammatic Realization

The Temperley–Lieb algebra $TL_n(\delta)$ can be defined in two fundamentally equivalent ways: via generators and relations or as a diagrammatic (planar) algebra.

Generators and Relations: $TL_n(\delta)$ is generated over a field $\mathbb{K}$ by $e_i$ for $1 \leq i \leq n-1$ , subject to:

$\begin{aligned} &\quad e_i^2 = \delta\, e_i, \ &\quad e_i e_j = e_j e_i \quad \text{if } |i-j| > 1, \ &\quad e_i e_{i\pm1} e_i = e_i. \end{aligned}$

The parameter $\delta$ is typically a function of a deformation parameter $q$ , often $\delta = q + q^{-1}$ .

Diagrammatic Realization: Elements are formal $\mathbb{K}$ -linear combinations of planar diagrams ("n-diagrams") with $n$ marked points on a top and bottom edge, each pairwise connected by non-crossing arcs. Multiplication corresponds to vertical stacking (top of one diagram glued to bottom of another), with any closed loops in the interior replaced by multiplication by $\delta$ . There exists a canonical isomorphism between the presented algebra and the diagram algebra via the mapping $e_i \mapsto E_i$ (diagram consisting of a cup at positions $i,i+1$ on top and a cap at those positions on the bottom).
Monomial and Normal Bases: The algebra has a canonical basis indexed by either certain reduced words in the $e_i$ generators (using the so-called "Jones normal form") or by crossingless pairings (or diagrams) of $2n$ points. The dimension of $TL_n(\delta)$ equals the $n$ th Catalan number.

2. Cellular Structure and Module Theory

$TL_n(\delta)$ is a prototypical example of a cellular algebra (in the sense of Graham–Lehrer), and possesses an explicit cellular basis.

Cell Datum: The algebra is endowed with a poset $\Lambda$ , collections $M(\lambda)$ for each $\lambda \in \Lambda$ , and basis elements $C_{S,T}^{\lambda}$ for $S,T \in M(\lambda)$ . The anti-involution $*$ is diagrammatic top–bottom reflection. Multiplication by algebra elements reduces modulo "lower cells," supporting an explicit construction of cell modules and facilitating the full classification of irreducibles.
Standard (Cell) Modules: For each propagation number $m$ (the number of through strands), one defines a standard module $S(n,m)$ with a basis of monic diagrams. An invariant symmetric bilinear form (the Gram form) determines the radical; the unique irreducible quotient $D(n,m) = S(n,m)/\text{rad} S(n,m)$ is a simple module. The decomposition numbers (composition multiplicities of simples in cell modules) are precisely characterized by diagrammatic linkage principles and combinatorial support sets.
Induction and Restriction: The inductive structure is encoded by exact sequences relating $TL_{n-1}$ -modules to $TL_n$ -modules, with standard modules $S(n,m)$ restricting to direct sums or extensions of $S(n-1,m)$ and $S(n-1,m-1)$ according to combinatorial rules. Splitting and non-splitting of these sequences depend on the semisimplicity of the algebra, controlled by the parameter $q$ .

3. Diagrammatics, Fully Commutative Elements, and Factorization

The diagram basis and the algebraic monomial basis are indexed by fully commutative (FC) elements of the symmetric group (type $A$ Coxeter group) $W(A_n)$ .

Bijective Correspondence: Each loop-free diagram corresponds uniquely to a FC Coxeter element, and each reduced expression (product of simple generators $s_i$ with only commutation relations) produces a diagram under $e_i \mapsto E_i$ . Factorization of diagrams into standard generators is algorithmic, relying on combinatorial extraction from diagrammatic structure via heap theory and explicit slicing by columns (aligned with vertical positions of simple jumps).
Alternate Bases and Bruhat Order: Closed-form formulas are available for change-of-basis matrices between several important bases (diagram/Kazhdan–Lusztig, Zinno's, and new interpolating bases) through noncrossing partitions and FC elements, with coefficients determined by the Bruhat order restricted to the noncrossing partitions (Gobet, 2014).

4. Categorification and Diagrammatic Approaches

Categorification of the Temperley–Lieb algebra is realized through diagrammatic monoidal categories and Soergel bimodules.

Categorification via Soergel Bimodules: The monoidal category of Soergel bimodules categorifies the Hecke algebra; its quotient (eliminating six-valent vertices) yields a planar diagrammatic category $\mathrm{TLC}_1$ , with relations reducing morphisms to "simple forests" decorated by polynomials. The quotient by the TL ideal, generated by explicit symmetric polynomials $y_{i,j}$ and $z_{i,j}$ , recovers the TL structure at the categorical level (Elias, 2010).
Module Categorification: The cell (sign) modules of TL can also be categorified by constructing quotient categories or subquotients corresponding to induced modules and filtration by standard modules. The Grothendieck group of the diagrammatic category is isomorphic to TL with compatible trace.
Foam/Topological Functors: Functors from the diagrammatic TL category to foam categories realize the ideals geometrically in terms of the vanishing of certain topological invariants (2D cobordisms, Weyl lines in the reflection representation).
Exceptional Collections and Khovanov Homology: Minimal idempotents (Jones–Wenzl projectors) in TL categorify to chain complexes with semi-orthogonal decompositions, giving rise to Postnikov towers and exceptional collections in the categorified setting, fundamental for the construction of Khovanov homology and tangle invariants (Cooper et al., 2012).

5. Extensions and Generalizations

Significant algebraic generalizations expand the classical TL framework using decorated diagrams, ties, partition structures, framisation, and cyclotomic enhancements.

Cyclotomic and Type D TL Algebras: Cyclotomic Temperley–Lieb algebras of type D, constructed via $m$ -decorated tangles (with dot and blob decorations), generalize the blob algebra and allow a cellular structure reflecting the representation theory of higher type Hecke algebras and associated blob algebras (Sun, 2010). Cellular quotients in type D are characterized by combinatorially indexed bases and diagrammatic presentations (Davis, 2015).
Partition and Tie Extensions: Partition TL algebras include extra tie generators, relating TL to partition algebras through generalized Steinberg relations, extending its relevance for combinatorics and partition symmetries (Juyumaya, 2013).
Framisation and Matrix Algebra Isomorphism: Framisation refers to the passage from the Hecke algebra to the Yokonuma–Hecke algebra, and further to its Temperley–Lieb quotients, with classification of representations in terms of multipartitions with bounded columns, and decomposition into matrix algebras over tensor products of classical TL algebras (Chlouveraki et al., 2015).
Valenced and Modular Generalizations: The valenced TL algebra arises by allowing boundary nodes to carry arbitrary "valences", relevant in conformal field theory and encoding fusion rules for tilting modules of $U_q(\mathfrak{sl}_2)$ . The modular representation theory, over positive characteristic, is developed by entirely diagrammatic (cellular) methods, answering the decomposition number problem for all simples in terms of support sets determined by $(\ell,p)$ -expansions of the cell parameters (Flores et al., 2018, Spencer, 2020, Spencer, 2021).

6. Physical and Representation-Theoretic Applications

The Temperley–Lieb algebra unifies algebraic, combinatorial, and physical perspectives, with multiple key applications:

Statistical Mechanics and Integrable Models: TL algebras underlie the transfer matrix and Hamiltonian constructions in models such as the 6-vertex, XXZ, and Potts models. Bond algebra approaches directly relate the TL algebra to the Ising and XXZ Hamiltonians via representations of the generating relations on spin chains (Imamura, 2022).
Knot Theory and Quantum Invariants: The Jones polynomial arises through Markov traces (Ocneanu traces) on TL algebras; precise passage conditions (e.g., $\zeta^2 = q^4$ ) determine when the trace factors to the TL quotient and when the invariant matches the Jones polynomial (Goundaroulis et al., 2013).
Fusion Categories and TQFT: At roots of unity, the semisimple quotients of TL algebras (Jones quotients) correspond to endomorphism algebras of tilting modules in truncated fusion categories, yielding connections with affine Lie algebras and the Virasoro minimal models in conformal field theory (Iohara et al., 2017).
Categorical Schur–Weyl Dualities: Schur–Weyl duality extends to infinite Temperley–Lieb algebras, with a conjectured equivalence between certain Serre subcategories of TL-infinite modules and quantum group modules, indicating broader links between diagram algebras and quantum symmetries (Moore, 2019).

7. Computational Methods: Rewriting Systems and Bases

Basis construction in TL algebras is algorithmically tractable using rewriting theory.

Reduction Systems: With generators $\{e_i\}$ and relations, a convergent rewriting system produces unique reduced normal forms (Jones normal forms) for words, with confluence and termination established via lexicographic orderings and supporting algorithms for explicit basis extraction (Thiebaut, 26 Aug 2025).
Diagrammatic Algorithms: Canonical decompositions of diagrams (by rightmost simple jumps or region counting) allow reconstruction of reduced factorizations in terms of simple diagrams, fully aligning the combinatorics of diagrams, Coxeter theory, and monomial bases (Ernst et al., 2015).
Categorical and Oriented Generalizations: Category theory provides the abstract framework for both the classic and oriented TL categories as monoidal categories with prescribed generating morphisms (cups, caps) and local relations (zig-zag, bubble, and exchange), supporting extended rewriting algorithms for basis computation in categories with additional structure (Thiebaut, 26 Aug 2025).

In summary, the Temperley–Lieb algebra constitutes a profoundly influential object at the intersection of diagrammatics, algebra, and physics. Its structure is governed by rich combinatorics of noncrossing partitions, fully commutative Coxeter elements, and recursive local relations. The cellular paradigm enables a transparent, diagrammatic approach to module theory, decomposition numbers, and representation classification. Extensions and categorifications further entrench the TL algebra at the foundation of modern quantum algebra, topological quantum field theory, and categorical representation theory.