Related papers: Framization of the Temperley-Lieb Algebra
In an earlier work, we defined a ``generalised Temperley-Lieb algebra'' $TL_{r,1,n}$ corresponding to the imprimitive reflection group $G(r,1,n)$ as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised…
We define a tower of affine Temperley-Lieb algebras of type $\tilde{A_{n}}$. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov-Ocneanu-Jones trace on the tower of Temperley-Lieb algebras of…
In this paper we define the $p$-adic framed braid group ${\mathcal F}_{\infty,n}$, arising as the inverse limit of the modular framed braids and we give topological generators for ${\mathcal F}_{\infty, n}$. We also give geometric…
The Gromov-Witten invariants of \mathbb{C}^3 with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al in the fermionic…
Both the original Temperley-Lieb algebras $\mathsf{TL}_{n}$ and their dilute counterparts $\mathsf{dTL}_{n}$ form families of filtered algebras: $\mathsf{TL}_{n}\subset \mathsf{TL}_{n+1}$ and $\mathsf{dTL}_{n}\subset\mathsf{dTL}_{n+1}$, for…
In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e., looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily…
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We…
We present in detail a recent conjecture on self-adjoint representations of the Temperley-Lieb algebra for particular values on the unit circle. The formulation in terms of graphical calculus is emphasized and discussed for several…
We show that the Temperley-Lieb algebra is constructed from the generators of the transverse-field Ising-type bond algebra. It is also shown that, when we take the representation of the generators to the one-dimensional transverse-field…
In this sequel to my previous paper, "Is String Theory in Knots?" I explore ways of constructing symmetries through an algebraic stepping process using knotted graphs. The hope is that this may lead to an algebraic formulation of string…
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or…
In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove…
There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…
We introduce the ramified partition algebra, which is a physically motivated and natural generalization of the partition algebra. We investigate its representation theory and demonstrate quasi--heredity under certain conditions. Under these…
It was studied the growth of Temperley-Lieb type algebras with orthogonal and commutative relations associated with 2-colored edges graphs. Depending on the structure of the graphs they can be finite-dimensional or to have linear or…
In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory…
We complete our analysis of the Temperley-Lieb subproduct systems, which define quantum analogues of Arveson's $2$-shift, by extending the main results of the previous paper to the general parameter case. Specifically, we show that the…
Invariant affine reflection algebras are the last and the most general known extension of affine Kac-Moody Lie algebras, introduced in recent years. We develop a method known as "affinization" to the class of invariant affine reflection…
Construction of the diagrammatic version of the affine Temperley-Lieb algebra of type $\widetilde{A_N}$ as a subring of matrices over the Laurent polynomials is given. We move towards geometrical understanding of cellular structure of the…
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical…