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Algebraic Operads
An Algorithmic Companion




ISBN 9781482248562
Published April 5, 2016 by Chapman and Hall/CRC
365 Pages - 18 B/W Illustrations

 
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Book Description

Algebraic Operads: An Algorithmic Companion presents a systematic treatment of Gröbner bases in several contexts. The book builds up to the theory of Gröbner 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: noncommutative Gröbner bases and their applications to the construction of universal enveloping algebras; Gröbner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Gröbner bases, important for applications to algebraic topology, and homological and homotopical algebra.

The last chapters of the book combine classical commutative Gröbner bases with operadic ones to approach some classification problems for operads. 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.

Table of Contents

Normal Forms for Vectors and Univariate Polynomials
Standard Forms
Normal Forms

Noncommutative Associative Algebras
Introduction
Free Associative Algebras
Normal Forms
Computing Gröbner Bases
Examples of Gröbner Bases and Their Applications
Rewriting Systems and Gröbner Bases
Exercises

Nonsymmetric Operads

Introduction
Nonsymmetric Operads
Free Nonsymmetric Operads
Normal Forms
Computing Gröbner Bases
Examples of Gröbner Bases for Nonsymmetric Operads
Normal Forms for Algebras over Nonsymmetric Operads
Exercises

Twisted Associative Algebras and Shuffle Algebras
Introduction
Twisted Associative Algebras and Shuffle Algebras
Free Shuffle Algebras
Normal Forms
Computing Gröbner Bases
Examples of Shuffle Algebras and their Applications
Exercises

Symmetric Operads and Shuffle Operads
Introduction
Symmetric Operads and Shuffle Operads
Free Shuffle Operads
Normal Forms
Computing Gröbner Bases
Examples of Gröbner Bases for Shuffle Operads
Exercises

Operadic Homological Algebra and Gröbner Bases
Introduction
First Instances of Koszul Signs for Graded Operads
Koszul Duality for Operads
Models for Operads from Gröbner Bases
Exercises

Commutative Gröbner Bases
Introduction
Commutative Associative Polynomials
Equivalent Definitions of Commutative Gröbner Bases
Classification of Commutative Monomial Orders
Zero-Dimensional Ideals
Complexity of Gröbner Bases: A Historical Survey
Exercises

Linear Algebra over Polynomial Rings
Introduction
Rank of a Polynomial Matrix; Determinantal Ideals
Some Elementary Examples
Algorithms for Linear Algebra over Polynomial Rings
Bibliographical Comments
Exercises

Case Study of Nonsymmetric Binary Cubic Operads
Introduction
Toy Model: The Quadratic Case
The Cubic Case
Exercises

Case Study of Nonsymmetric Ternary Quadratic Operads
Introduction
Generalities on Nonsymmetric Operads with One Generator
Nonsymmetric Ternary Operads
Further Directions
Exercises

Appendices: Maple Code for Buchberger’s Algorithm
First Block: Initialization
Second Block: Monomial Orders
Third Block: Sorting Polynomials
Fourth Block: Standard Forms of Polynomials
Fifth Block: Reduce and Self-Reduce
Sixth Block: Main Loop — Buchberger’s Algorithm

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Author(s)

Biography

Murray R. Bremner, PhD, is a professor at the University of Saskatchewan in Canada. He attended that university as an undergraduate, and received an M. Comp. Sc. degree at Concordia University in Montréal. He obtained a doctorate in mathematics at Yale University with a thesis entitled On Tensor Products of Modules over the Virasoro Algebra. Prior to returning to Saskatchewan, he held shorter positions at MSRI in Berkeley and at the University of Toronto. Dr. Bremner authored the book Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications and is a co-translator with M. V. Kotchetov of Selected Works of A. I. Shirshov in English Translation. His primary research interests are algebraic operads, nonassociative algebra, representation theory, and computer algebra.

Vladimir Dotsenko, PhD, is an assistant professor in pure mathematics at Trinity College Dublin in Ireland. He studied at the Mathematical High School 57 in Moscow, Independent University of Moscow, and Moscow State University. His PhD thesis is titled Analogues of Orlik–Solomon Algebras and Related Operads. Dr. Dotsenko also held shorter positions at Dublin Institute for Advanced Studies and the University of Luxembourg. His collaboration with Murray started in February 2013 in CIMAT (Guanajuato, Mexico), where they both lectured in the research school "Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms." His primary research interests are algebraic operads, homotopical algebra, combinatorics, and representation theory.

Reviews

"This book presents a systematic treatment of Gröbner bases, and more generally of the problem of normal forms, departing from linear algebra, going through commutative and noncommutative algebra, to operads. The algorithmic aspects are especially developed, with numerous examples and exercises."- Loϊc Foissy

"By balancing computational methods and abstract reasoning, the authors of the book under review have written an excellent up-to-date introduction to Grobner basis methods applicable to associative structures, especially including operads. The book will be of interest to a wide range of readers, from undergraduates to experts in the field."

~ Ralf Holtkamp, Mathematical Reviews, March 2018