Planar Prop of Differential Operators

2022/04/24 by Slava Pimenov, Pimenov, Slava
Mathematics · Physics and Astronomy · #14A22 (Primary) #16S80 #18M85 (Secondary) #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2204.11297

openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Hom(A^{⊗ q}, A^{⊗ 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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