David J Winter - Abstract Lie Algebras

DAVID J. WINTER

Department of Mathematics
The University of Michigan

DOVER PUBLICATIONS, INC.
Mineola, New York

Copyright

Copyright 1972 by The Massachusetts Institute of Technology All rights reserved.

Bibliographical Note

This Dover edition, first published in 2008, is an unabridged republication of the work originally published in 1972 by The MIT Press, Cambridge, Massachusetts. This edition is published by special arrangement with The MIT Press, 55 Hayward Street, Cambridge, MA 02142.

Library of Congress Cataloging-in-Publication Data

Winter, David J.

Abstract lie algebras / David J. Winter.Dover ed. p. cm.

Includes index.

This Dover edition, first published in 2008 is an unabridged republication of the work originally published in 1972 by The MIT Press, Cambridge, Massachusetts.

eISBN-13: 978-0-486-78346-8

1. Lie algebras. I. Title.

QA251.W68 2008
512.482dc22

2007032515

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

This book grew out of a one-semester course on Lie algebras given at the University of Michigan in 19681969. Aside from basic algebra and linear algebra, the material is self-contained. The first three chapters may be regarded as a solid introduction to the theory of Lie algebras. They may also be of interest to those who are already familiar with Lie algebras, since the development of the theory and the proofs quite often are not along standard lines. The fourth chapter consists mainly of material of fairly recent origin, including some unpublished material, on the general structure of Lie algebras of arbitrary characteristic.

, the distribution of Cartan subalgebras and maximal tori in a Lie p-algebra is discussed.

In developing parts of .

David J. Winter

Ann Arbor, Michigan

April 1971

ABSTRACT LIE ALGEBRAS

In this chapter, we introduce the language of modules in a form designed for the material developed later in the book. Throughout, is a set and k a field.

We begin with some basic definitions and properties of modules.

An -module over k is a vector space over k together with a mapping , denoted , such that (m + n)s = (ms) + (ns) for , and .

Let be an -module over k, . Then TM is the linear transformation of defined by mTM = mT for .

The direct sum of -modules over k is the -module with underlying vector space the direct sum of the vector spaces together with the mapping defined by

For an -module over k and , we let be the subspace of generated by the set .

An -submodule of an -module is a subspace of such that . Such an is proper if and

If is an -submodule of an -module , the quotient-module of by is the vector space quotient together with the mapping