Algebraic Operads
An Algorithmic Companion
1st Edition
Published on 14. October 2024
Book
Paperback/Softback
384 pages
978-1-032-92108-2 (ISBN)
Description
Algebraic Operads: An Algorithmic Companion presents a systematic treatment of Groebner bases in several contexts. The book builds up to the theory of Groebner 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 Groebner bases and their applications to the construction of universal enveloping algebras; Groebner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Groebner bases, important for applications to algebraic topology, and homological and homotopical algebra.
The last chapters of the book combine classical commutative Groebner 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.
The authors present a variety of topics including: noncommutative Groebner bases and their applications to the construction of universal enveloping algebras; Groebner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Groebner bases, important for applications to algebraic topology, and homological and homotopical algebra.
The last chapters of the book combine classical commutative Groebner 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.
Reviews / Votes
"This book presents a systematic treatment of Groebner 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."- Lo?c 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 "This book presents a systematic treatment of Groebner 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."- Lo?c 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
More details
Language
English
Place of publication
Boca Raton
United Kingdom
Publishing group
Taylor & Francis Ltd
Target group
Professional and scholarly
Academic
Illustrations
18 s/w Abbildungen
18 Illustrations, black and white
Dimensions
Height: 234 mm
Width: 156 mm
Weight
548 gr
ISBN-13
978-1-032-92108-2 (9781032921082)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
E-Book
04/2016
Chapman & Hall/CRC
€103.99
Available for download
E-Book
04/2016
1st Edition
Chapman and Hall
€103.99
Available for download
Book
04/2016
1st Edition
Chapman & Hall/CRC
€187.10
Article not available for order
Persons
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.
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.
Content
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 Groebner Bases. Commutative Groebner 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.