arXiv research

Maps Heegaard Floer homology to Hecke algebras for surfaces.

Defines braids with double lines for links in a surface times circle and connects it to the affine Hecke algebra.

A mathematical isomorphism connects Floer homology to DAHA representations.

Generalizes Hecke algebra for double torus, linking to skein algebra.

New basis and Schur-Weyl duality for loop Hecke algebra defined.

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 of these algebras and construct several bases for them. We then show how we can…

The thesis explores centralisers and Hecke algebras in representation theory with applications to knots and physics.

We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce …

Floer homology connects to quiver Hecke algebras in Coulomb branches.

This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…

New algebra for twice-punctured torus curves.

Heegaard Floer homology connects to polynomial representations of Hecke algebras.

In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the $p$--adic framed braids and the $p$--adic Yokonuma--Hecke algebras in…

We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of…

Study Morse theory on loop spaces and Hecke algebras.

In this paper we consider all possible generalizations of the B-type Hecke algebras, namely the cyclotomic and what we call 'generalized', and we construct Markov traces on each of them, so as to obtain all possible different levels of homfly-pt analogues in the solid torus related to the (Hecke) algebras of B-type.

In this work we introduce the concept of Modular Framization or simply Framization. We construct a framization $F_{d,n}$ of the Birman--Wenzl--Murakami algebra, also known as BMW algebra, and start a systematic study of this framization. We show that $F_{d,n}$ is finite dimensional and the \lq braid generators\rq\ of t…

This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadrat…

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

A new approach to Morse theory using folded ribbon trees.

Maps from buildings to spaces study K-theory of Hecke algebras.

We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.

This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Marko…

In this paper we announce the existence of a family of new $2$-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type $A$. Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the fa…

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebra…

We study a relation between the Hecke groups and the index of subfactors in a von Neumann algebra. Such a problem was raised by V. F. R. Jones. We solve the problem using the notion of a cluster C*-algebra.

We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combini…

We define the singular Hecke algebra ${\mathcal H} (SB_n)$ as the quotient of the singular braid monoid algebra ${\mathbb C} (q) [SB_n]$ by the Hecke relations $σ_k^2 = (q-1) σ_k +q$, $1 \le k\le n-1$, and define the Markov traces on the sequence $\{{\mathcal H}(SB_n)\}_{n=1}^{+\infty}$ in the same way as for the Marko…

Representations of the Iwahori-Hecke algebra of type A_{n-1} are equivalent to representations of the braid group B_n for which the generators satisfy a certain quadratic relation. We show how to construct such representations from the natural action of B_n on the homology of configuration spaces of the punctured disk.…

In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yok…

The trace of the affine Hecke category is compared with the elliptic Hall algebra.

In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the so…

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogu…

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

In this paper we classify all Markov traces on Iwahori-Hecke algebras associated with the finite Coxeter groups of type B. We then use these traces for constructing Jones-type invariants for oriented knots inside a solid torus, and finally we give skein-theoretical interpretations.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

We compare the invariant for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. We show that the two invariants, as maps on the set ${\mathcal L}$ of oriented link types in $S^3$, do not coincide except in a few trivial cases.

The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the $p$-adic framed braids and the $p$-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulo…

Introduces a new algebra from loop braid groups with properties similar to Hecke algebras.

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case …

Study framizations of algebras using Schur--Weyl duality and tied braids.

Paper constructs a HOMFLYPT-type invariant for pseudo links.

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the…

In this paper we study the kernel of the homomorphism $B_{g,n} \to B_n$ of the braid group $B_{g,n}$ in the handlebody $\mathcal{H}_g$ to the braid group $B_n$. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra $H_{g,n}(q)$, which is some analog of the Hecke algebra $H_n(q…

The mixed braid groups $B_{2,n}, \ n \in \mathbb{N}$, with two fixed strands and $n$ moving ones, are known to be related to the knot theory of certain families of $3$-manifolds. In this paper we define the mixed Hecke algebra $\mathrm{H}_{2,n}(q)$ as the quotient of the group algebra ${\mathbb Z}\, [q^{\pm 1}] \, B_{2…

In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type $A$ as a quotient algebra of the Yokonuma-Hecke algebra of type $A$. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…

Explains fusion for Yang-Baxter equation and braid group.