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This document is a thesis presented for the ``Habilitation \`a diriger des recherches''. The first chapter provides some background and sketch the story of the classical Schur-Weyl duality and its quantum analogue involving the Hecke algebra. Quantum groups and their centralisers are discussed along with their connections with the braid group, the Yang-Baxter equation and the construction of link invariants. A quick review of several related notions of Hecke algebras concludes the first chapter. The second chapter presents the fused Hecke algebras and an application to the construction of solutions for the Yang-Baxter equation. The third chapter summarises recent works on diagonal centralisers of Lie algebras, and gives an application to the study of the missing label for sl(3). The last chapter discusses algebraic results on the Yokonuma-Hecke algebras and their applications to the study of link invariants., Comment: 81 pages

These are the extended notes of a mini-course given at the school WinterBraids X. We discuss algebras simultaneously related to: the braid group, the Yang-Baxter equation and the representation theory of quantum groups. The main goal is to explain the idea of the fusion procedure for the Yang-Baxter equation and to show how it leads to new examples of such algebras: the fused Hecke algebras.

We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur--Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of $U_q(gl_N)$. Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitely the irreducible representations and the branching rules. Finally, we aim to an algebraic description of the centralisers of the tensor products of $U_q(gl_N)$-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of $U_q(sl_2)$., Comment: 52 pages

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., Comment: 8 pages

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 invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma--Hecke algebras. In general, the obtained invariants form an infinite family of 3-variables polynomials. As a consequence of the construction via the isomorphism, we reduce the number of invariants to study, given the number of connected components of a link. In particular, if the link is a classical link with N components, we show that N invariants generate the whole family., Comment: 15 pages

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 GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism., Comment: 34 pages