Algebraic Operads [electronic resource] /
Loday, Jean-Louis.author.
Vallette, Bruno.author.
In many areas of mathematics some “higher operations” are arising. These have become so important that several research projects refer to such expressions. Higher operations form 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 HomotopyTransfer Theorem. Although the necessary notions of algebra are recalled, readers areexpected 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 an elementary chapter on classical algebra, accessible to undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices  review 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.  .
Description based upon print version of record.
Includes bibliographical references (p. 609-623) and index.
Preface -- 1.Algebras, coalgebras, homology -- 2.Twisting morphisms -- 3.Koszul duality for associative algebras -- 4.Methods to prove Koszulity of an algebra -- 5.Algebraic operad -- 6 Operadic homological algebra -- 7.Koszul duality of operads -- 8.Methods to prove Koszulity of an operad -- 9.The operads As and A\infty -- 10.Homotopy operadic algebras -- 11.Bar and cobar construction of an algebra over an operad -- 12.(Co)homology of algebras over an operad -- 13.Examples of algebraic operads -- Apendices: A.The symmetric group -- B.Categories -- C.Trees -- References -- Index -- List of Notation.
In many areas of mathematics some “higher operations” are arising. These have become so important that several research projects refer to such expressions. Higher operations form 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 HomotopyTransfer Theorem. Although the necessary notions of algebra are recalled, readers areexpected 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 an elementary chapter on classical algebra, accessible to undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices  review 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.  .
Category theory (Mathematics).
Homological algebra.
Nonassociative rings.
Rings (Algebra).
Algebraic topology.
Manifolds (Mathematics).
Complex manifolds.
Category Theory, Homological Algebra.
Non-associative Rings and Algebras.
Algebraic Topology.
Manifolds and Cell Complexes (incl. Diff.Topology).