Combinatorial Hopf algebras: definition and examples

30.04.2021, 11:00  –  Online Seminar
Arbeitsgruppenseminar Analysis

Cécile Mammez (University of Lille)

The theory of Hopf algebras, initially motivated by investigations on compact Lie groups, has been intensively studied. Many connections with areas of mathematics (algebraic combinatorics, representation theory, category theory, operad theory... ),physics (renormalisation, quantum field theory...), theoretical computer science (words,semantics. . . ).. were discovered and many questions still remain open. In this talk, we will first recall the definition of a combinatorial Hopf algebra and a dual Hopf algebra. We will illustrate these notions with the tensor Hopf algebra and the co-tensor Hopf algebra. Then, we will list some questions  that usually arise when studying a  Hopf algebra. After that, we will present the Hopf algebra on rooted trees. It is also called the Connes-Kreimer Hopf algebra, an algebraic structure which the authors introduced  in quantum field theory in order to renormalise of integrals associated to Feynmann diagrams. Finally, motivated by the reconstruction of walks from self-avoiding walks and simple cycles, we will present the construction of Hopf algebras on walks of graphs.

Work in collaboration with L. Foissy, P.-L. Giscard and M. Ronco.

Forthcoming speakers are  Viet Dang on May 7th, Yannic Vargas on May 14th, John Barrett on May 21st,  Malte Leimbach on May 28th, Alfonso Garmendia on June 4th, Konrad Waldorf on June 11th and Bernadette Lessel on July 2nd. You are welcome to suggest speakers or topics for the forthcoming sessions.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks,  please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.