1st Edition

# Algebraic Operads An Algorithmic Companion

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**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

**Introduction**

Noncommutative Associative Algebras

Noncommutative Associative Algebras

Free Associative Algebras

Normal Forms

Computing Gröbner Bases

Examples of Gröbner Bases and Their Applications

Rewriting Systems and Gröbner Bases

Exercises

Nonsymmetric Operads

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

**Introduction**

Twisted Associative Algebras and Shuffle Algebras

Twisted Associative Algebras and Shuffle Algebras

Twisted Associative Algebras and Shuffle Algebras

Free Shuffle Algebras

Normal Forms

Computing Gröbner Bases

Examples of Shuffle Algebras and their Applications

Exercises

**Introduction**

Symmetric Operads and Shuffle Operads

Symmetric Operads and Shuffle Operads

Symmetric Operads and Shuffle Operads

Free Shuffle Operads

Normal Forms

Computing Gröbner Bases

Examples of Gröbner Bases for Shuffle Operads

Exercises

**Introduction**

Operadic Homological Algebra and Gröbner Bases

Operadic Homological Algebra and Gröbner Bases

First Instances of Koszul Signs for Graded Operads

Koszul Duality for Operads

Models for Operads from Gröbner Bases

Exercises

**Introduction**

Commutative Gröbner Bases

Commutative Gröbner Bases

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

**Introduction**

Linear Algebra over Polynomial Rings

Linear Algebra over Polynomial Rings

Rank of a Polynomial Matrix; Determinantal Ideals

Some Elementary Examples

Algorithms for Linear Algebra over Polynomial Rings

Bibliographical Comments

Exercises

**Introduction**

Case Study of Nonsymmetric Binary Cubic Operads

Case Study of Nonsymmetric Binary Cubic Operads

Toy Model: The Quadratic Case

The Cubic Case

Exercises

**Introduction**

Case Study of Nonsymmetric Ternary Quadratic Operads

Case Study of Nonsymmetric Ternary Quadratic Operads

Generalities on Nonsymmetric Operads with One Generator

Nonsymmetric Ternary Operads

Further Directions

Exercises

**First Block: Initialization**

Appendices: Maple Code for Buchberger’s Algorithm

Appendices: Maple Code for Buchberger’s Algorithm

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