An Introduction to Nonassociative Algebras

by Richard D. Schafer

Overview

"An important addition to the mathematical literature … contains very interesting results not available in other books; written in a plain and clear style, it reads very smoothly." — Bulletin of the American Mathematical Society
This concise study was the first book to bring together material on the theory of nonassociative algebras, which had previously been scattered throughout the literature. It emphasizes algebras that are, for the most part, finite-dimensional over a field. Written as an introduction for graduate students and other mathematicians meeting the subject for the first time, the treatment's prerequisites include an acquaintance with the fundamentals of abstract and linear algebra.
After an introductory chapter, the book explores arbitrary nonassociative algebras and alternative algebras. Subsequent chapters concentrate on Jordan algebras and power-associative algebras. Throughout, an effort has been made to present the basic ideas, techniques, and flavor of what happens when the associative law is not assumed. Many of the proofs are given in complete detail.

Product Details

ISBN-13:	9780486164175
Publisher:	Dover Publications
Publication date:	11/15/2017
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	176
File size:	32 MB
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About the Author

Richard D. Schafer (1918–2014) was a Professor of Mathematics at the Massachusetts Institute of Technology from 1959 until his retirement in 1988.

Read an Excerpt

PREFACE

This little book is an expanded version of the lectures on nonassociative algebras which I gave at an Advanced Subject Matter Institute in Algebra, which was held at Oklahoma State University in the summer of 1961 under the sponsorship of the National Science Foundation.

I have had no desire to write a treatise on this subject. Instead I have tried to present here in an elementary way some topics which have been of interest to me, and which will be helpful to graduate students who are encountering nonassociative algebras for the first time. Proofs are not given of all of the results cited, but a number of the proofs which are included illustrate techniques which are important for the study of nonassociative algebras.

Alternative algebras are presented in some detail. I have treated Jordan algebras in a somewhat more cursory way, except for describing their relationships to the exceptional simple Lie algebras. A considerably deeper account of Jordan algebras will be found in the forthcoming book by Jacobson.

I expect that any reader will be acquainted with the content of a beginning course in abstract algebra and linear algebra. Portions of six somewhat more advanced books are recommended for background reading, and at appropriate places reference is made to these books for results concerning quadratic forms, fields, associative algebras, and Lie algebras. The books are:

Albert, A. A., "Structure of Algebras," Vol. 24. American Mathematical Society Colloquium Publications, New York, 1939;

Artin, Emil, "Galois Theory," No. 2, 2nd ed. Notre Dame Mathematical Lectures, Notre Dame, 1948;

Artin, Emil, "Geometric Algebra," No. 3 (Interscience Tracts in Pure and Applied Mathematics). Wiley (Interscience), London and New York, 1957;

Jacobson, Nathan, "Lectures in Abstract Algebra," Vol. II (Linear Algebra). Van Nostrand, Princeton, New Jersey, 1953;

Jacobson, Nathan, "Lie Algebras," No. 10 (Interscience Tracts in Pure and Applied Mathematics). Wiley (Interscience), London and New York, 1962;

Zariski, Oscar, and Samuel, Pierre, "Commutative Algebra," Vol. I. Van Nostrand, Princeton, New Jersey, 1958.

References are also given to some of the research papers listed in the bibliography at the end. It is my hope that this book will serve to make more of the papers cited there accessible to the interested reader.

Completion of this manuscript was partially supported by National Science Foundation Grant GP 2496. I am grateful for this support, and happy to acknowledge it.

RICHARD D. SCHAFER

Professor of Mathematics, Emeritus Massachusetts Institute of Technology Cambridge, Massachusetts

September, 1966

(Continues…)

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Excerpted from "An Introduction to Nonassociative Algebras"
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Copyright © 1994 Richard D. Schafer.
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Table of Contents

"An important addition to the mathematical literature … contains very interesting results not available in other books; written in a plain and clear style, it reads very smoothly."—Bulletin of the American Mathematical Society
This concise study was the first book to bring together material on the theory of nonassociative algebras, which had previously been scattered throughout the literature. It emphasizes algebras that are, for the most part, finite-dimensional over a field. Written as an introduction for graduate students and other mathematicians meeting the subject for the first time, the treatment's prerequisites include an acquaintance with the fundamentals of abstract and linear algebra.
After an introductory chapter, the book explores arbitrary nonassociative algebras and alternative algebras. Subsequent chapters concentrate on Jordan algebras and power-associative algebras. Throughout, an effort has been made to present the basic ideas, techniques, and flavor of what happens when the associative law is not assumed. Many of the proofs are given in complete detail.
Dover unabridged republication of the edition published by Academic Press, New York, 1966.
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