Grundlehren der mathematischen Wissenschaften

Algebraic Operads

Authors: Loday, Jean-Louis, Vallette, Bruno

  • Being the first book on algebraic operads, will be used as a reference work in this field
  • Each chapter contains a list of exercises and a résumé
  • A new and conceptual presentation of the Koszul duality theory is presented​
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About this book

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 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.

About the authors

Jean-Louis Loday has worked in algebraic K-theory, algebraic topology, noncommutative geometry and higher algebra. He is the author of "Cyclic Homology" a worldwide reference, published as volume 301 in the Grundleheren der mathematischen Wissenschadften. He is an editor of several mathematical journals and several Proceedings. He has organized six of the annual conferences "Operads 20xx" (starting in 2004).

Bruno Vallette has worked on operads since his first years in mathematical research. He is the author of 10 research articles and a specialist of properads and of Batalin-Vilkovisky algebras.

Loday and Vallette organized several summer schools and international conferences on the subject (France, China, Norway) and published the Proceedings of these conferences.

Reviews

From the reviews:

“It is a welcome addition to the existing literature and will, no doubt, become a standard reference for many authors working in this quickly developing field. … it is an impressive piece of work, which gives a comprehensive account of the foundations of the theory of algebraic operads, starting from the most basic notions, such as associative algebras and modules. It will be of interest to a broad swath of mathematicians: from undergraduate students to experts in the field.” (Andrey Yu. Lazarev, Mathematical Reviews, March, 2013)

Table of contents (13 chapters)

  • Algebras, Coalgebras, Homology

    Loday, Jean-Louis (et al.)

    Pages 1-36

  • Twisting Morphisms

    Loday, Jean-Louis (et al.)

    Pages 37-59

  • Koszul Duality for Associative Algebras

    Loday, Jean-Louis (et al.)

    Pages 61-87

  • Methods to Prove Koszulity of an Algebra

    Loday, Jean-Louis (et al.)

    Pages 89-118

  • Algebraic Operad

    Loday, Jean-Louis (et al.)

    Pages 119-192

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-642-30362-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-642-30361-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • ISBN 978-3-642-44835-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Algebraic Operads
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
346
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-30362-3
DOI
10.1007/978-3-642-30362-3
Hardcover ISBN
978-3-642-30361-6
Softcover ISBN
978-3-642-44835-5
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XXIV, 636
Topics