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Books > Science & Mathematics > Mathematics > Algebra
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Lie Methods in Deformation Theory (Hardcover, 1st ed. 2022)
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Lie Methods in Deformation Theory (Hardcover, 1st ed. 2022)
Series: Springer Monographs in Mathematics
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This book furnishes a comprehensive treatment of differential
graded Lie algebras, L-infinity algebras, and their use in
deformation theory. We believe it is the first textbook devoted to
this subject, although the first chapters are also covered in other
sources with a different perspective. Deformation theory is an
important subject in algebra and algebraic geometry, with an origin
that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and
Grothendieck. In the last 30 years, a new approach, based on ideas
from rational homotopy theory, has made it possible not only to
solve long-standing open problems, but also to clarify the general
theory and to relate apparently different features. This approach
works over a field of characteristic 0, and the central role is
played by the notions of differential graded Lie algebra,
L-infinity algebra, and Maurer-Cartan equations. The book is
written keeping in mind graduate students with a basic knowledge of
homological algebra and complex algebraic geometry as utilized, for
instance, in the book by K. Kodaira, Complex Manifolds and
Deformation of Complex Structures. Although the main applications
in this book concern deformation theory of complex manifolds,
vector bundles, and holomorphic maps, the underlying algebraic
theory also applies to a wider class of deformation problems, and
it is a prerequisite for anyone interested in derived deformation
theory. Researchers in algebra, algebraic geometry, algebraic
topology, deformation theory, and noncommutative geometry are the
major targets for the book.
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