1st Edition

Algebraic Operads An Algorithmic Companion

    383 Pages 18 B/W Illustrations
    by Chapman & Hall

    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.

    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

    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.

    "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