Dialgebras and Related Operads

Dialgebras and Related Operads
Title Dialgebras and Related Operads PDF eBook
Author J.-L. Loday
Publisher Springer
Pages 138
Release 2003-07-01
Genre Mathematics
ISBN 3540453288

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The main object of study of these four papers is the notion of associative dialgebras which are algebras equipped with two associative operations satisfying some more relations of the associative type. This notion is studied from a) the homological point of view: construction of the (co)homology theory with trivial coefficients and general coefficients, b) the operadic point of view: determination of the dual operad, that is the dendriform dialgebras which are strongly related with the planar binary trees, c) the algebraic point of view: Hopf structure and Milnor-Moore type theorem.

Algebraic Operads

Algebraic Operads
Title Algebraic Operads PDF eBook
Author Jean-Louis Loday
Publisher Springer Science & Business Media
Pages 649
Release 2012-08-08
Genre Mathematics
ISBN 3642303625

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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.

Nonsymmetric Operads in Combinatorics

Nonsymmetric Operads in Combinatorics
Title Nonsymmetric Operads in Combinatorics PDF eBook
Author Samuele Giraudo
Publisher Springer
Pages 176
Release 2019-01-04
Genre Mathematics
ISBN 3030020746

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Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. A lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book portrays the main elements of this theory under a combinatorial point of view and exposes the links it maintains with computer science and combinatorics. Examples of operads appearing in combinatorics are studied. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed.

Algebraic Operads

Algebraic Operads
Title Algebraic Operads PDF eBook
Author Murray R. Bremner
Publisher CRC Press
Pages 382
Release 2016-04-06
Genre Mathematics
ISBN 1482248573

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This book presents a systematic treatment of Grobner bases in several contexts. The book builds up to the theory of Grobner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory.

Operads and Universal Algebra

Operads and Universal Algebra
Title Operads and Universal Algebra PDF eBook
Author Chengming Bai
Publisher World Scientific
Pages 318
Release 2012
Genre Mathematics
ISBN 9814365114

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The book aims to exemplify the recent developments in operad theory, in universal algebra and related topics in algebraic topology and theoretical physics. The conference has established a better connection between mathematicians working on operads (mainly the French team) and mathematicians working in universal algebra (primarily the Chinese team), and to exchange problems, methods and techniques from these two subject areas.

Operads in Algebra, Topology and Physics

Operads in Algebra, Topology and Physics
Title Operads in Algebra, Topology and Physics PDF eBook
Author Martin Markl
Publisher American Mathematical Soc.
Pages 362
Release 2002
Genre Mathematics
ISBN 0821843621

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Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. From their beginnings in the 1960s, they have developed to encompass such areas as combinatorics, knot theory, moduli spaces, string field theory and deformation quantization.

Dialgebras and Related Operads

Dialgebras and Related Operads
Title Dialgebras and Related Operads PDF eBook
Author J.-L. Loday
Publisher Springer
Pages 140
Release 2001-07-17
Genre Mathematics
ISBN 9783540421948

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The main object of study of these four papers is the notion of associative dialgebras which are algebras equipped with two associative operations satisfying some more relations of the associative type. This notion is studied from a) the homological point of view: construction of the (co)homology theory with trivial coefficients and general coefficients, b) the operadic point of view: determination of the dual operad, that is the dendriform dialgebras which are strongly related with the planar binary trees, c) the algebraic point of view: Hopf structure and Milnor-Moore type theorem.