Generalized Temperley-Lieb algebras and decorated tangles

Let (W,S) be a Coxeter system of affine type D, and let TL(W) the corresponding generalized Temperley-Lieb algebra. In this extended abstract we define an infinite dimensional associative algebra made of decorated diagrams which is isomorphic to TL(W). Moreover, we describe an explicit basis for such an algebra of diagrams which is in bijective correspondence with the classical monomial basis of TL(W), indexed by the fully commutative elements of W.

Let $(W,S)$ be an affine Coxeter system of type $\widetilde{B}$ or $\widetilde{D}$ and ${\rm TL}(W)$ the corresponding generalized Temperley-Lieb algebra. In this paper we define an infinite dimensional associative algebra made of decorated diagrams that is isomorphic to ${\rm TL}(W)$. Moreover, we describe an explicit basis for such an algebra consisting of special decorated diagrams that we call admissible. Such basis is in bijective correspondence with the classical monomi...

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine $C$. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the so-called...

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole or order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which in our set-up, occurs when the Coxeter type is of type A with index n-1. The proof involves a diagrammatic version of the Brauer algebra of type Dn in which the Temperley-Lieb alge...

Algebraic basics on Temperley-Lieb algebras are proved in an elementary and straightforward way with the help of tensor categories behind them.

In this paper, we present an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative elements (in the sense of Stembridge) of the Coxeter group of type affine $C$. Moreover, we provide an explicit description of a basis for the diagram algebra. In the sequel to this paper, we show that this diagrammatic representation is faithful. The results of this paper and its ...

We study the combinatorics of fully commutative elements in Coxeter groups of type $H_n$ for any $n > 2$. Using the results, we construct certain canonical bases (i.e., IC bases) for non-simply-laced generalized Temperley--Lieb algebras and show how to relate them to morphisms in the category of decorated tangles.

We define a new class of algebras, cyclotomic Temperley-Lieb algebras of type D, in a diagrammatic way, which is a generalization of Temperley-Lieb algebras of type D. We prove that the cyclotomic Temperley-Lieb algebras of type D are cellular. In fact, an explicit cellular basis is given by means of combinatorial methods. After determining all the irreducible representations of these algebras, we give a necessary and sufficient condition for a cyclotomic Temperley-Lieb algeb...

We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various constructions counted by the Catalan numbers. We use a method similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to construct an orthogonal basis for the vector space over these diagrams.

In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley--Lieb algebra of type affine $C$, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine $C$ contains an infinite number of fully commutative elements, and so the corresponding Temperley--Lieb algebra is of infinite rank. With the exception of type affine $A$, all other generalized Tem...