E-Book, Englisch, 383 Seiten
Bremner / Dotsenko Algebraic Operads
Erscheinungsjahr 2016
ISBN: 978-1-4822-4857-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
An Algorithmic Companion
E-Book, Englisch, 383 Seiten
ISBN: 978-1-4822-4857-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
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.
Zielgruppe
This book is intended for mathematicians and first-year graduate students.
Autoren/Hrsg.
Weitere Infos & Material
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