Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry

  title={Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry},
  author={Maxim Kontsevich and Yan S. Soibelman},
  journal={Lecture Notes in Physics},
We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from the geometric point of view. The chapter contains homological versions of the notions of properness and smoothness of projective varieties as… 
Infinity structures and higher products in rational homotopy theory
Rational homotopy theory classically studies the torsion free phenomena in the homotopy category of topological spaces and continuous maps. Its success is mainly due to the existence of relatively
On the triangulated category of DQ-modules
The main subject of this thesis is the study of deformation quantization modules or DQ-modules. This thesis investigates to which extent some theorems of algebraic geometry can be generalized to
We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞–multiplication—we think of such algebras as A∞–algebras “with
Cyclic A8-algebras and cyclic homology
We provide a new description of the complex computing the Hochschild homology of an H-unitary A8-algebra A as a derived tensor product A bAe A such that: (1) there is a canonical morphism from it to
Chern Character, Loop Spaces and Derived Algebraic Geometry
In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified
A primer on A-infinity-algebras and their Hochschild homology
We present an elementary and self-contained construction of $A_\infty$-algebras, $A_\infty$-bimodules and their Hochschild homology and cohomology groups. In addition, we discuss the cup product in
Pre-Calabi-Yau algebras and ξ∂-calculus on higher cyclic Hochschild cohomology
We formulate the notion of pre-Calabi-Yau structure via the higher cyclic Hochschild complex and study its cohomology. A small quasi-isomorphic subcomplex in higher cyclic Hochschild complex gives
On algebraic structures of the Hochschild complex
We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincare duality hypothesis, such as Calabi-Yau algebras, derived
A-infinity-bimodules and Serre A-infinity-functors
This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which


Algebra extensions and nonsingularity
This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the
Deformations of algebras over operads and Deligne's conjecture
In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces
Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions
Cyclic homology and nonsingularity
From the pioneering work of Connes [Col] one knows that periodic cyclic homology can be regarded as a natural extension of de Rham cohomology to the realm of noncommutative geometry. Our aim in this
Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads
This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a
Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: correlators
This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a
Double derivations and Cyclic homology
We give a new construction of cyclic homology of an associative algebra A that does not involve Connes' differential. Our approach is based on an extended version of the complex \Omega A, of
Deformation Quantization of Poisson Manifolds
I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the
Noncommutative homotopy algebras associated with open strings
We discuss general properties of $A_\infty$-algebras and their applications to the theory of open strings. The properties of cyclicity for $A_\infty$-algebras are examined in detail. We prove the