Deformations of Diagrams of Algebras and How I Learned to Love Operads

by Eli Hawkins

Thursday 2 May 2024, 16:15 → 17:15 Europe/Zurich

Sitzungszimmer (MI)

A diagram of algebras is a collection of algebras with some specified homomorphisms between them. These appear in Algebraic quantum field theory. I am interested in deformations of these structures.

Deformations of a single algebra can be understood using the Maurer-Cartan equation in the Hochschild complex, which is a differential graded Lie algebra. The Hochschild cohomology of an algebra also has the structure of a Gerstenhaber algebra (a compatible product and graded Lie bracket).

In the condition that a linear map be a homomorphism, on one side of the equation, the homomorphism appears twice and the multiplication map once, so this is a cubic equation. However, the Maurer-Cartan equation in a differential graded Lie algebra is always a quadratic equation, so it cannot describe deformations of diagrams of algebras. This suggests the presence of something more general: an L-infinity algebra!

In this talk, I will sketch the theory of operads, and how I used them to construct an L-infinity structure on the Hochschild complex of a diagram of algebras and directly prove that the Hochschild cohomology is a Gerstenhaber algebra.