Cohomology theories for homotopy algebras and noncommutative geometry

TL;DR

This paper develops a unified framework for cohomology theories of homotopy algebras using noncommutative geometry, leading to a generalized Hodge decomposition for Hochschild and cyclic cohomology.

Contribution

It introduces a novel general framework connecting homotopy algebras and noncommutative geometry, extending previous cohomology results.

Findings

Established a general form of Hodge decomposition for Hochschild and cyclic cohomology of $C_ Infty$-algebras.

Unified cohomology theories for $A_ Infty$, $C_ Infty$, and $L_ Infty$-algebras.

Extended previous work by Loday and Gerstenhaber-Schack within a new conceptual framework.

Abstract

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.