Cohomology theories for homotopy algebras and noncommutative geometry

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.

💡 Research Summary

The paper develops a unified framework for studying cohomology theories of strongly homotopy algebras—specifically $A_\infty$, $C_\infty$, and $L_\infty$ algebras—by embedding them into the language of non‑commutative geometry as formulated by Connes and Kontsevich. After a concise review of existing separate treatments (Hochschild cohomology for $A_\infty$, Chevalley‑Eilenberg cohomology for $L_\infty$, and cyclic cohomology for $C_\infty$), the authors introduce the non‑commutative differential forms $\Omega^\bullet A$ associated to a homotopy algebra $A$, together with Connes’ operators $b$ (the Hochschild differential) and $B$ (the Connes boundary). They show that when $A$ carries an $A_\infty$‑coalgebra structure, $\Omega^\bullet A$ inherits a compatible $A_\infty$‑coalgebra structure, allowing the definition of Hochschild cohomology as the homology of the $b$‑complex and cyclic cohomology as the homology of the total complex $(b+B)$.

For $L_\infty$‑algebras the same construction yields a non‑commutative Chevalley‑Eilenberg complex, and the total differential again gives a Lie‑cyclic cohomology theory. The most substantial contribution concerns $C_\infty$‑algebras, which simultaneously possess $A_\infty$ and $L_\infty$ structures together with a commutative symmetry (a co‑metric). The authors prove that this symmetry forces $b$ and $B$ to commute, which in turn enables a Hodge‑type decomposition of both Hochschild and cyclic cohomology. Concretely, they establish that \