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Deformation Theory Seminar

Monday, December 5, 2022 - 1:45pm

Eli Hawkins

York U, UK


University of Pennsylvania

This will be the ultimate Deformation seminar for this semester. Time and day for next semester to be determined.

A diagram of algebras is a functor from some small category to a category of associative algebras. 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. The “asimplicial” subcomplex is given by deleting the bottom row, and first order deformations are described in asimplicial cohomology.
In this talk, I will describe an operad, mQuilt, that acts on the Hochschild bicomplex (and asimplicial subcomplex) 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 the asimplicial complex 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 and asimplicial cohomologies of a diagram of algebras are Gerstenhaber algebras.