Volume 9, issue 3 (2009)

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Cohomology theories for homotopy algebras and noncommutative geometry

Alastair Hamilton and Andrey Lazarev

Algebraic & Geometric Topology 9 (2009) 1503–1583
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.

Keywords
infinity-algebra, cyclic cohomology, Harrison cohomology, symplectic structure, Hodge decomposition
Mathematical Subject Classification 2000
Primary: 13D03, 13D10
Secondary: 46L87