Planar Prop of Differential Operators

Slava Pimenov

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arXiv · 2204.11297

Planar Prop of Differential Operators

Slava Pimenov

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Abstract

We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces $\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure of a planar prop, i.e. each differential operator has multiple inputs and outputs and they can be composed along planar graphs. Furthermore, for a formally smooth algebra we have the surjective symbol map from $D(A)$ to the space of poly-derivations. We also consider another planar prop $E(A)$ generated by automorphisms of the trivial associative deformation of $A$ over the completion of a free associative algebra. We construct a natural map from $E(A)$ to $D(A)$ and identify its image.

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Slava Pimenov. 2022-04-24. Planar Prop of Differential Operators. https://arxiv.org/abs/2204.11297

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