This monograph covers ten lectures given by the author at the North Carolina
State University at Raleigh, NC, during the week May 16–20, 2011. The choice of
topics was a result of a compromise, given by the fact that the audience consisted
of both graduate students and specialists in the field. The author resisted the
temptation to devote the talks to his own results and attempted to present the
material scattered in the literature in a compact fashion.
Also, the level of generality was determined by the purpose of the book. We de-
cided to focus on deformations over local complete Noetherian rings (which of course
includes the Artin case), though more general bases, as formal dg-commutative al-
gebras or formal dg-schemes, can be considered. The complete Noetherian case
covers most types of deformations of algebraic structures a working mathematician
meets in his/her professional life.
The place for the conference was sensibly chosen, because the birth of de-
formation theory as we understand it today is related to this part of the globe.
Mike Schlessinger and Jim Stasheff worked at Chapel Hill, only 28 miles from
Raleigh—Mike contributed the Artin ring approach to deformation theory and, to-
gether with Jim, introduced the intrinsic bracket. They also wrote a seminal paper
that predated modern deformation theory with its emphasis on the moduli space
of the Maurer-Cartan equation. Tom Lada, who has spent most of his professional
career at Raleigh, together with Jim, introduced L∞-algebras. And, of course, the
life of Murray Gerstenhaber, the founding father of algebraic deformation theory,
is connected to Philadelphia, a 7 hour drive from Raleigh. Moreover, the author of
this monograph worked as a Fulbright fellow for two fruitful semesters in Chapel
Hill with Jim and Tom.
The book consists of 10 chapters which more or less correspond to the material
of the respective talks. Chapters 1–3 review classical Gerstenhaber’s deformation
theory of associative algebras over a local complete Noetherian ring. In chapters 4
and 5, which are devoted to Maurer-Cartan elements in differential graded Lie
algebras, the moduli space point of view begins to prevail. In chapter 6 we recall
L∞-algebras and, in chapter 7, the related simplicial version of the Maurer-Cartan
moduli space, and prove its homotopy invariance. As an application, we review
the main features of Kontsevich’s approach to deformation quantization of Poisson
manifolds. In chapters 8–9 we describe a construction of an L∞-algebra governing
deformations of a given class of (diagrams of) algebras. The last chapter contains
a couple of explicit examples and indicates possible generalizations.
Our intention was to make the presentation self-contained, assuming only ba-
sic knowledge of commutative algebra, homological algebra and category theory.
Suitable references are [AM69, HS71, ML63a, ML71]. We sometimes omitted