Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Origins of the Temperley-Lieb algebra: early history

Stephen Doty, Anthony Giaquinto

TL;DR

This historical survey traces the origins and early algebraic and combinatorial foundations of the Temperley–Lieb algebras $TL_n(\delta)$, emphasizing diagrammatic realizations, the basic construction, and connections to Hecke algebras. It explains how $TL_n(\delta)$ arises as a diagram algebra and as a quotient of $\mathbf{H}_n$, analyzes semisimplicity via quantum-integer criteria, and develops the representation theory through cellularity and Schur–Weyl duality with $U_q(\mathfrak{gl}_2)$ or $U_q(\mathfrak{sl}_2)$. The work also integrates combinatorics with Dyck paths, skew shapes, and $321$-avoiding permutations, highlighting algorithmic tools like Kauffman’s diagrams and Bowman’s skew-shape method. Collectively, the paper preserves a coherent historical view of how early methods laid the groundwork for later quantum topology and representation-theoretic developments.

Abstract

We give an historical survey of some of the original basic algebraic and combinatorial results on Temperley-Lieb algebras, with a focus on certain results that have become folklore.

Origins of the Temperley-Lieb algebra: early history

TL;DR

This historical survey traces the origins and early algebraic and combinatorial foundations of the Temperley–Lieb algebras , emphasizing diagrammatic realizations, the basic construction, and connections to Hecke algebras. It explains how arises as a diagram algebra and as a quotient of , analyzes semisimplicity via quantum-integer criteria, and develops the representation theory through cellularity and Schur–Weyl duality with or . The work also integrates combinatorics with Dyck paths, skew shapes, and -avoiding permutations, highlighting algorithmic tools like Kauffman’s diagrams and Bowman’s skew-shape method. Collectively, the paper preserves a coherent historical view of how early methods laid the groundwork for later quantum topology and representation-theoretic developments.

Abstract

We give an historical survey of some of the original basic algebraic and combinatorial results on Temperley-Lieb algebras, with a focus on certain results that have become folklore.
Paper Structure (9 sections, 24 theorems, 103 equations, 2 figures)

This paper contains 9 sections, 24 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Temperley--Lieb algebra
  3. Diagrammatics
  4. The basic construction
  5. Relation between $\mathbb{A}_n$ and $\operatorname{TL}_n$
  6. $\operatorname{TL}_n$ as a quotient of the Iwahori--Hecke algebra
  7. Skew shapes and $321$-avoiding permutations
  8. Representations of $\operatorname{TL}_n$
  9. Schur--Weyl duality

Key Result

Lemma 2.2

If $w = e_{i_1} \cdots e_{i_l}$ is a reduced word in $\operatorname{TL}_n$ then $m := \max\{i_1, \dots, i_l\}$ occurs only once in the sequence $(i_1, \dots, i_l)$.

Figures (2)

  • Figure 1: Construction of the diagram $d$ from $w(d)$
  • Figure 2: Bratteli diagram up to level $7$

Theorems & Definitions (55)

  • Remark 2.1
  • Lemma 2.2: Jones' Lemma
  • proof
  • Theorem 2.3: Jones' normal form
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2: Kauffman K:90*Thm. 4.3
  • proof
  • Corollary 3.3
  • ...and 45 more