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 generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

This 54 pages paper is a substantial revision of the part of math.QA/0410621 dealing with algebraic Hodge decompositions of Hochschild and cyclic cohomology theories. The main addition is the treatment of cohomology theories corresponding to unital infinity-structures

References in corpus (9)

• 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
• Transferring homotopy commutative algebraic structures
• A super-analogue of Kontsevich's theorem on graph homology

Cited by in corpus (19)

• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Classes on compactifications of the moduli space of curves through solutions to the quantum master equation
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Maurer-Cartan moduli and theorems of Riemann-Hilbert type
• Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
• Noncommutative geometry and compactifications of the moduli space of curves
• Curved infinity-algebras and their characteristic classes
• Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras
• L-infinity maps and twistings
• Hodge decomposition of string topology
• A Poincaré-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras
• Gauge equivalence for complete $L_{\infty }$-algebras
• Unbased rational homotopy theory: a Lie algebra approach
• Models for classifying spaces and derived deformation theory
• Symplectic $C_{\infty }$-algebras
• Maurer-Cartan moduli and models for function spaces
• On the obstructed Lagrangian Floer theory