Bruno Vallette (Université de Nice-Sophia Antipolis), Associative algebras


  • Feb. 28, 2012, 17:30 - 18:30: Homotopy associative algebra
  • March 1, 2012, 15:00 - 16:00: Koszul duality and Diamond Lemma


Homotopy associative algebra

When one wants to transfer associative algebra structures though homotopy equivalences, one automatically discovers the notion of homotopy associative algebra. In order to conceptually understand such a result, and later to generalize it, I will introduce a simple operad, based on planar rooted trees. Finally, I will explain how to build a resolution of this operad, due to Jim Stasheff, which is based on polytopes, called the associaedra.

Koszul duality and Diamond Lemma

Starting from the symmetric algebra as the toy model for quadratic algebras, I will introduce the notion of Poincaré-Birkhoff-Witt bases. I will then describe the Koszul duality theory for associative algebras. Finally, I will show that an algebra which admits a PBW basis is Koszul. This result is based the Diamond lemma and the rewriting method thereby drawing a concrete link with the other talks of the workshop.