## INTRODUCTION TO A-INFINITY ALGEBRAS AND MODULES

### Bernhard Keller

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

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:

- the reconstruction of a complex from its homology,
- 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.

http://www.math.jussieu.fr/~keller/publ/ioanabs.html

Bernhard Keller, le 10 janvier, 2001.

keller@math.jussieu.fr

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