Quantum Maths Seminar, external speaker.
Speaker: Eli Hawkins, University of York
Title: Operations on the Hochschild bicomplex of a diagram of algebras
Abstract: A diagram of algebras is a functor from some small category to a category of associative algebras – for example, an algebraic quantum field theory. The defining properties include that the maps must be homomorphisms. This is a cubic condition, therefore deformations of a diagram of algebras cannot be described by the quadratic Maurer-Cartan equation of a differential graded Lie algebra. This suggests the presence of a natural L-infinity algebra.
Hochschild cohomology generalizes to diagrams of algebras, and this is constructed most naturally from a double complex.
In this talk, I will describe an operad, mQuilt, that acts on the Hochschild bicomplex of a diagram of algebras. I will describe an operad homomorphism from L-infinity to mQuilt, thus giving L-infinity algebra structures on both complexes. The Maurer-Cartan equation (on a certain subcomplex) describes deformations of diagrams of algebras.
I also construct an operad homomorphism from the (degree shifted) Gerstenhaber operad to the homology, H(mQuilt). This directly proves that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra.