the Masters' Thesis of S. Sundar

Chapter 1
The Temperley-Lieb Algebra
1.1 The Temperley-Lieb algebra Tn(τ)
We consider only�algebras. Let τ be a nonzero complex number.
Definition 1. For n ≥ 2, let Tn(τ) be the�algebra generated by 1,e1,e2 · · · en−1
subject to the following relations :
e 2 i = ei for i ∈ {1,2, · · · ,n − 1}
eiej = ejei if |i − j| ≥ 2
eiejei = τei if |i − j| = 1
Tn(τ) has the following universal property. Let A be a unital�algebra.
Let f1,f2, · · · ,fn−1 ∈ A be such that
f 2 i = fi for i ∈ {1,2, · · · ,n − 1}
fifj = fjfi if |i − j| ≥ 2
fifjfi = τfi if |i − j| = 1
Then there exists a unique algebra homomorphism φ : Tn(τ) → A such that
φ(ei) = fi and φ(1) = 1A where 1A denotes the multiplicative identity of A.
We now proceed to prove that Tn(τ) is finite dimensional. By a word on
1,e1,e2, · · · ,en−1 we mean a product ei1 ei2 · · · eip. By convention empty
product denotes 1. Note that words on 1,e1,e2, · · · ,en−1 span Tn(τ).
Lemma 1. Let w be a word on 1,e1,e2 · · · ,en−1. Then
w = τ k (ei1 ei1−1 · · · ej1 )(ei2 ei2−1 · · · ej2 ) · · · (eipeip−1 · · · ejp)
2