Homotopy algebras and noncommutative geometry

Abstract

We study cohomology theories of strongly homotopy algebras, namely $A_{\infty },C_{\infty }$ and $L_{\infty }$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_{\infty }$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_{\infty }$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_{\infty }$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincare duality space.

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• Deformation quantization of Poisson manifolds, I
• Introduction to string field theory
• Closed String Field Theory: Quantum Action and the BV Master Equation
• The Geometry of the Master Equation and Topological Quantum Field Theory
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Lie theory for nilpotent L-infinity algebras
• On a theorem of Kontsevich
• Duality in Gerstenhaber algebras
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Cited by in corpus (14)

• Matrix factorisations and open topological string theory
• Topological conformal field theories and Calabi-Yau categories
• Involutive A-infinity algebras and dihedral cohomology
• Graph cohomology classes in the Batalin-Vilkovisky formalism
• Unimodular homotopy algebras and Chern-Simons theory
• Generating the superpotential on a D-brane category: I
• Sheaves on local Calabi-Yau varieties
• Operads and moduli spaces
• Calabi-Yau Frobenius algebras
• Lie bialgebras and the cyclic homology of $A_{\infty }$ structures in topology
• Characteristic classes of A-infinity algebras
• Characteristic classes of $\ai$-algebras
• Equalizers in the category of cocomplete cocategories
• Non-commutative moduli spaces of topological D-branes