Algebraic Operads: An Algorithmic Companion

Murray R. Bremner, Vladimir Dotsenko

April 5, 2016 by Chapman and Hall/CRC
Reference - 365 Pages - 18 B/W Illustrations
ISBN 9781482248562 - CAT# K23679


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  • Provides an introduction to normal forms and Gröbner bases for graduate students and research mathematicians
  • Presents a systematic treatment of the theory of Gröbner bases for associative algebras, nonsymmetric operads, shuffle algebras, and shuffle operads, emphasizing the common features of these theories, especially various "diamond lemmas", which guarantee the existence of normal forms
  • Includes original applications of commutative Gröbner bases to classification of nonsymmetric operads
  • Illustrates the theories with numerous detailed computational examples


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.