Algebraic Operads: An Algorithmic Companion

by Murray R. Bremner, Vladimir Dotsenko

Overview

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.

Product Details

ISBN-13:	9781040055120
Publisher:	CRC Press
Publication date:	04/06/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	383
File size:	3 MB

About the Author

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 Montreal. 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.

Table of Contents

Normal Forms for Vectors and Univariate Polynomials. Noncommutative Associative Algebras. Nonsymmetric Operads. Twisted Associative Algebras and Shuffle Algebras. Symmetric Operads and Shuffle Operads. Operadic Homological Algebra and Grobner Bases. Commutative Grobner Bases. Linear Algebra over Polynomial Rings. Case Study of Nonsymmetric Binary Cubic Operads. Case Study of Nonsymmetric Ternary Quadratic Operads. Appendices: Maple Code for Buchberger's Algorithm.

Editorial 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 Gröbner 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

"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 Gröbner 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

From the Publisher