Algebraic Operads

Author : Jean-Louis Loday,Bruno Vallette
Publisher : Springer Science & Business Media
Page : 649 pages
File Size : 44,8 Mb
Release : 2012-08-08
Category : Mathematics
ISBN : 9783642303623

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Algebraic Operads by Jean-Louis Loday,Bruno Vallette Pdf

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.

Author : Murray R. Bremner,Vladimir Dotsenko
Publisher : CRC Press
Page : 382 pages
File Size : 55,8 Mb
Release : 2016-04-06
Category : Mathematics
ISBN : 9781482248579

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Algebraic Operads by Murray R. Bremner,Vladimir Dotsenko Pdf

Algebraic Operads: An Algorithmic Companion 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. The authors present a variety of topics including: non

Author : Martin Markl,Steven Shnider,James D. Stasheff
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 43,7 Mb
Release : 2002
Category : Operads
ISBN : 9780821843628

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Operads in Algebra, Topology and Physics by Martin Markl,Steven Shnider,James D. Stasheff Pdf

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.

Author : Chengming Bai
Publisher : World Scientific
Page : 318 pages
File Size : 43,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814365123

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Operads and Universal Algebra by Chengming Bai Pdf

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.

Author : J.-L. Loday,A. Frabetti,F. Chapoton,F. Goichot
Publisher : Springer
Page : 140 pages
File Size : 47,6 Mb
Release : 2003-07-01
Category : Mathematics
ISBN : 9783540453284

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Dialgebras and Related Operads by J.-L. Loday,A. Frabetti,F. Chapoton,F. Goichot Pdf

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.

Author : Jean-Louis Loday
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 40,6 Mb
Release : 1997
Category : Electronic
ISBN : 9780821805138

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Operads: Proceedings of Renaissance Conferences by Jean-Louis Loday Pdf

Author : Benoit Fresse
Publisher : American Mathematical Soc.
Page : 704 pages
File Size : 51,9 Mb
Release : 2017-05-22
Category : Grothendieck groups
ISBN : 9781470434823

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Homotopy of Operads and Grothendieck-Teichmuller Groups by Benoit Fresse Pdf

The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.

Author : Bernard Russo,Asuman Güven Aksoy,Ravshan Ashurov,Shavkat Ayupov
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 51,7 Mb
Release : 2016-08-25
Category : Banach algebras
ISBN : 9781470419288

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Topics in Functional Analysis and Algebra by Bernard Russo,Asuman Güven Aksoy,Ravshan Ashurov,Shavkat Ayupov Pdf

The USA-Uzbekistan Conference on Analysis and Mathematical Physics, focusing on contemporary issues in dynamical systems, mathematical physics, operator algebras, and several complex variables, was hosted by California State University, Fullerton, from May 20–23, 2014. The main objective of the conference was to facilitate scientific communication and collaboration between mathematicians from the USA and Uzbekistan. This volume contains the proceedings of the Special Session on Algebra and Functional Analysis. The theory of operator algebras is the unified theme for many papers in this volume. Out of four extensive survey papers, two cover problems related to derivation of various algebras of functions. The other two surveys are on classification of Leibniz algebras and on evolution algebras. The sixteen research articles are devoted to certain analytic topics, such as minimal projections with respect to numerical radius, functional equations and discontinuous polynomials, Fourier inversion for distributions, Schrödinger operators, convexity and dynamical systems.

Author : Bai Chengming,Guo Li,Loday Jean-louis
Publisher : World Scientific
Page : 320 pages
File Size : 42,6 Mb
Release : 2012-02-23
Category : Mathematics
ISBN : 9789814458337

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Operads And Universal Algebra - Proceedings Of The International Conference by Bai Chengming,Guo Li,Loday Jean-louis Pdf

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.

Author : Nicolás Andruskiewitsch,Gongxiang Liu,Susan Montgomery,Yinhuo Zhang
Publisher : American Mathematical Soc.
Page : 359 pages
File Size : 44,8 Mb
Release : 2021-07-06
Category : Education
ISBN : 9781470456245

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Hopf Algebras, Tensor Categories and Related Topics by Nicolás Andruskiewitsch,Gongxiang Liu,Susan Montgomery,Yinhuo Zhang Pdf

The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.

Author : Tom Leinster
Publisher : Cambridge University Press
Page : 451 pages
File Size : 43,6 Mb
Release : 2004-07-22
Category : Mathematics
ISBN : 9780521532150

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Higher Operads, Higher Categories by Tom Leinster Pdf

Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics.

Author : Theodore Voronov
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 41,7 Mb
Release : 2002
Category : Geometric quantization
ISBN : 9780821832011

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Quantization, Poisson Brackets and Beyond by Theodore Voronov Pdf

The papers in this volume are based on talks given at the 2001 Manchester Meeting of the London Mathematical Society, which was followed by an international workshop on Quantization, Deformations, and New Homological and Categorical Methods in Mathematical Physics. Focus is on the topics suggested by the title: quantization in its various aspects, Poisson brackets and generalizations, and structures beyond'' this, including symplectic supermanifolds, operads, Lie groupoids and Lie (bi)algebroids, and algebras with $n$-ary operations. The book offers accounts of up-to-date results as well as accessible expositions aimed at a broad reading audience of researchers in differential geometry, algebraic topology and mathematical physics.

Author : Shavkat Ayupov,Bakhrom Omirov,Isamiddin Rakhimov
Publisher : CRC Press
Page : 157 pages
File Size : 54,9 Mb
Release : 2019-11-11
Category : Mathematics
ISBN : 9781000740400

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Leibniz Algebras by Shavkat Ayupov,Bakhrom Omirov,Isamiddin Rakhimov Pdf

Leibniz Algebras: Structure and Classification is designed to introduce the reader to the theory of Leibniz algebras. Leibniz algebra is the generalization of Lie algebras. These algebras preserve a unique property of Lie algebras that the right multiplication operators are derivations. They first appeared in papers of A.M Blokh in the 1960s, under the name D-algebras, emphasizing their close relationship with derivations. The theory of D-algebras did not get as thorough an examination as it deserved immediately after its introduction. Later, the same algebras were introduced in 1993 by Jean-Louis Loday , who called them Leibniz algebras due to the identity they satisfy. The main motivation for the introduction of Leibniz algebras was to study the periodicity phenomena in algebraic K-theory. Nowadays, the theory of Leibniz algebras is one of the more actively developing areas of modern algebra. Along with (co)homological, structural and classification results on Leibniz algebras, some papers with various applications of the Leibniz algebras also appear now. However, the focus of this book is mainly on the classification problems of Leibniz algebras. Particularly, the authors propose a method of classification of a subclass of Leibniz algebras based on algebraic invariants. The method is applicable in the Lie algebras case as well. Features: Provides a systematic exposition of the theory of Leibniz algebras and recent results on Leibniz algebras Suitable for final year bachelor's students, master's students and PhD students going into research in the structural theory of finite-dimensional algebras, particularly, Lie and Leibniz algebras Covers important and more general parts of the structural theory of Leibniz algebras that are not addressed in other texts

Author : Ieke Moerdijk,Bertrand Toën
Publisher : Springer Science & Business Media
Page : 186 pages
File Size : 52,7 Mb
Release : 2010-12-01
Category : Mathematics
ISBN : 9783034800525

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Simplicial Methods for Operads and Algebraic Geometry by Ieke Moerdijk,Bertrand Toën Pdf

"This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures delivered at the Centre de Recerca Matemàtica in February 2008, as part of a special year on Homotopy Theory and Higher Categories"--Foreword

Author : Samuele Giraudo
Publisher : Springer
Page : 172 pages
File Size : 42,9 Mb
Release : 2019-01-04
Category : Mathematics
ISBN : 9783030020743

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Nonsymmetric Operads in Combinatorics by Samuele Giraudo Pdf

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.