Cohomology theories for homotopy algebras and noncommutative geometry

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 generalises and puts in a conceptual framework previous work by Loday and Gerstenhaber–Schack.