Bernhard Keller

May 18, 1999, last modified on January 10, 2001

Final version, to appear in Homology, Homotopy and Applications

These are expanded notes of a minicourse of three lectures given at the Euroconference "Homological Invariants in Representation Theory" in Ioannina, Greece, March 16 to 21, 1999, and of a talk at the Instituto de Matemáticas, UNAM, México, on April 28, 1999. They present basic results on A-infinity-algebras and their modules. They are inspired and motivated by a chapter of a course that M. Kontsevich gave in spring 1998 at the Ecole normale supérieure.

Section 1 gives minimal historical background. In section 2, we motivate the introduction of A-infinity-algebras and modules by two basic problems from homological algebra:

  1. the reconstruction of a complex from its homology,
  2. the reconstruction of the category of iterated selfextensions of a module from its extension algebra.

Then we briefly present the topological origin of A-infinity-structures. Section 3 is devoted to A-infinity-algebras and their morphisms. The central result is the theorem on the existence of minimal models. In sections 4 and 5, we introduce the derived category of an A-infinity-algebra and we present the natural framework for the solution of problem 1. In section 6, we sketch the formalism of standard functors and arrive at the solution of problem 2. Section 7 presents the category of twisted objects, which is of importance because it is "computable".

Omissions: The links with Morita theory for derived categories or dg categories have not been made explicit. The relevance of A-infinity-algebras to boxes and matrix problems was discovered by S. Ovsienko. We only give a hint of this important development in our last example in section 7. The notions of A-infinity-equivalence and of A-infinity-enhanced triangulated categories have not yet been included.
Bernhard Keller, le 10 janvier, 2001.

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