In many areas of mathematics some "higher operations" are
arising. These havebecome so important that several research
projects refer to such expressions. Higher operationsform new types
of algebras. The key to understanding and comparing them, to
creating invariants of their action is operad theory. This is a
point of view that is 40 years old in algebraic topology, but the
new trend is its appearance in several other areas, such as
algebraic geometry, mathematical physics, differential geometry,
and combinatorics. The present volume is the first comprehensive
and systematic approach to algebraic operads. An operad is an
algebraic device that serves to study all kinds of algebras
(associative, commutative, Lie, Poisson, A-infinity, etc.) from a
conceptual point of view. The book presents this topic with an
emphasis on Koszul duality theory. After a modern treatment of
Koszul duality for associative algebras, the theory is extended to
operads. Applications to homotopy algebra are given, for instance
the Homotopy Transfer Theorem. Although the necessary notions of
algebra are recalled, readers are expected to be familiar with
elementary homological algebra. Each chapter ends with a helpful
summary and exercises. A full chapter is devoted to examples, and
numerous figures are included.
After a low-level chapter on Algebra, accessible to (advanced)
undergraduate students, the level increases gradually through the
book. However, the authors have done their best to make it suitable
for graduate students: three appendicesreview the basic results
needed in order to understand the various chapters. Since higher
algebra is becoming essential in several research areas like
deformation theory, algebraic geometry, representation theory,
differential geometry, algebraic combinatorics, and mathematical
physics, the book can also be used as a reference work by
researchers.
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