Related papers: Temperley-Lieb and Birman-Murakami-Wenzl like rela…
We construct a representation of the Temperley-Lieb algebra from a multiplicity-free semisimple monoidal Abelian category ${\cal C}$, with two simple objects $\lambda$ and $\nu$ such that $\lambda\otimes\nu$ is simple and Hom$_{\cal…
The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum sl(2) on the n-th tensor power of the 2-dimensional irreducible module. We…
The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study admissibility conditions on the ground ring for these algebras, and show that the…
Algebraic basics on Temperley-Lieb algebras are proved in an elementary and straightforward way with the help of tensor categories behind them.
The Birman-Murakami-Wenzl algebras (BMW algebras) of type En for n=6,7,8 are shown to be semisimple and free over a quotient of a polynomial algebra of ranks 1,440,585; 139,613,625; and 53,328,069,225. We also show they are cellular over…
Generalised Temperley-Lieb categories with regions labelled by elements of a commutative algebra were introduced by M. Khovanov and the second author in [Pure Appl. Math. Q. 19 (2023), no. 5]. We consider the case where the regions are…
A relation for the Jones-Wenzl projector is proven. It has the following consequence for representations of the Temperley-Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian $n \times n$ matrix $T$ of…
The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a…
The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators…
Guided by consideration of problems in 2 and 3 dimensional lattice model computation, we are led to define a number of new categories, and functors between these categories and the partition category, culminating in the introduction of two…
The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra $\TL_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector…
We provide a necessary and sufficient condition for a type D Temperley-Lieb algebra ${\rm TLD}_n(\delta)$ being semi-simple by studying branching rule for cell modules. As a byproduct, our result is used to study the so-called forked…
The two-colored Temperley-Lieb algebra $2\mathrm{TL}_R({}_s {n})$ is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector $\mathrm{JW}_R({}_s {n}) \in 2\mathrm{TL}_R({}_s {n})$ plays an important…
The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We show that the cyclotomic BMW algebras are free modules over any (admissible, integral)…
We show that the Temperley--Lieb category $\mathbf{TL}(q;\mathbb{C})$ embeds in an ultraproduct of modular tensor categories when $q$ is not a root of unity. As a result, we show that its Drinfeld center is semisimple and describe its…
We investigate the representation theory of the Temperley-Lieb algebra, $TL_n(\delta)$, defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for…
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic…
We give a new and conceptually straightforward proof of the well-known presentation for the Temperley-Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new…
We explicitly describe the category of modules of the Temperley-Lieb algebra $\mathrm{TL}_n(\beta)$ under specialization $\beta=0$ for even $n$ in terms of a quiver algebra, analogous to a result of Berest-Etingof-Ginzburg. In particular,…
We construct combinatorial (involutory) Gelfand models for the following diagram algebras in the case when they are semi-simple: Brauer algebra, its partial analogue, walled Brauer algebra, its partial analogue, Temperley-Lieb algebra, its…