CHAPTER 1

Temperley-Lieb-Jones Theories

This chapter introduces Temperley-Lieb, Temperley-Lieb-Jones, and Jones al-

gebroids through planar diagrams. Temperley-Lieb-Jones (TLJ) algebroids gener-

alize the Jones polynomial of links to colored tangles. Jones algebroids, semisimple

quotients of TLJ algebroids at roots of unity, are the prototypical examples of rib-

bon fusion categories (RFCs) for application to topological quantum computation

(TQC). Some of them are conjectured to algebraically model anyonic systems in cer-

tain fractional quantum Hall (FQH) liquids, with Jones-Wenzl projectors (JWPs)

representing anyons. Our diagrammatic treatment exemplifies the graphical calcu-

lus for RFCs. Special cases of Jones algebroids include the Yang-Lee, Ising, and

Fibonacci theories.

Diagrammatic techniques were used by R. Penrose to represent angular momen-

tum tensors and popularized by L. Kauffman’s reformulation of the Temperley-Lieb

algebras. Recently they have witnessed great success through V. Jones’s planar al-

gebras and K. Walker’s blob homology.

1.1. Generic Temperley-Lieb-Jones algebroids

The goal of this section is to define the generic Jones representations of the braid

groups by showing that the generic Temperley-Lieb (TL) algebras are direct sums

of matrix algebras. Essential for understanding the structure of the TL algebras

are the Markov trace and the Jones-Wenzl idempotents or JWPs. We use the

magical properties of the JWPs to decompose TL algebras into matrix algebras.

Consequently we obtain explicit formulas for the Jones representations of the braid

groups.

1.1.1. Generic Temperley-Lieb algebroids.

Definition 1.1. Let F be a field. An F-algebroid Λ is a small F-linear category.

Recall that a category Λ is small if its objects, denoted as Λ0, form a set, rather

than a class. A category is F-linear if for any x, y ∈ Λ0 the morphism set Hom(x, y)

is an F-vector space, and for any x, y, z ∈ Λ0 the composition map

Hom(y, z) × Hom(x, y) → Hom(x, z)

is bilinear. We will denote Hom(x, y) sometimes as xΛy.

The term “F-algebroid” [BHMV] emphasizes the similarity between an F-

linear category and an F-algebra. Indeed, we have:

Proposition 1.2. Let Λ be an F-algebroid. Then for any x, y ∈

Λ0,

xΛx is an

F-algebra and xΛy is a yΛy − xΛx bimodule.

The proof is left to the reader, as we will do most of the time in the book. It

follows that an F-algebroid is a collection of algebras related by bimodules. In the

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http://dx.doi.org/10.1090/cbms/112/01