Grups de trenes, representació de Burau i categorificació de Khovanov-Seidel

Abstract

[en] The braid group, together with its representations, is a fascinating mathematical structure, studied from different fields, such as group theory, topology... Moreover, it is a theory that extends beyond itself, with relations that go from the theory of knots and their invariants to concepts of theoretical physics. The main objective of the paper is the introduction of the notion of the braid group, the Burau representation and a categorification of it. We will begin by presenting braids as a mathematical structure and the different ways of interpreting the group they form. Then, we introduce the non-reduced and reduced Burau representations. This family of representations is faithful for $n<4$ and not faithful for $n>4$, it is unknown if it is faithful for $n=4$. In this work, the case $n=5$ is not studied. Finally, the Seidel-Khovanov categorification of the Burau representation is presented, which, curiously, is faithful for all $n$.