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 x Λy. 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, x Λx is an F-algebra and x Λy is a y Λy x Λx 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 1
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