Assoc. Prof. Dr. Kenny De Commer, Vrije Universiteit Brussel: The braid equation and the reflection equation

On November 3rd, 2022, at 10:30 a.m., in room “pi” in the Mathematica building, the interested are invited to lecture:

Title: The braid equation and the reflection equation

Guest: Dr. Kenny De Commer, associate professor at VUB, department of Mathematics and Data Science, Brussels, Belgium.

Abstract : There is currently a lot of interest in algebraic structures which can mimic topological operations. One particular such operation is  braiding : take the ends of two strands which are hanging down, and turn around 180 degrees. If we replace each strand by a set X, one mimics the braiding operation by an invertible map r from the Cartesian square X² to itself. The crucial requirement of this map is that it satisfies the  braid equation  (also known as  Yang-Baxter equation). It turns out that such a map r leads to a wealth of interesting algebraic structures. We will give an overview of the different incarnations of these structures, and their surprising relation with other well-known areas of algebra (Galois theory, radical rings, …) We then look at a closely related topological operation: take the ends of two strands and turn around 360 degrees! Transferring this operation to the set-theoretic world, we are now looking at maps satisfying a different equation, known as the  reflection equation . As our own small contribution to this area, we will comment on some algebraic structures associated with this equation.