Introduction to Lie Algebras and Representation Theory / by J.E. Humphreys.

Format:

Book

Author/Creator:

Humphreys, James E., author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 2197-5612 ; 9.

Graduate Texts in Mathematics, 2197-5612 ; 9

Language:

English

Subjects (All):

Algebra.

Local Subjects:

Algebra.

Physical Description:

1 online resource (XIII, 173 pages).

Edition:

First edition 1972.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1972.

System Details:

text file PDF

Summary:

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor- porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Contents:

I. Basic Concepts

1. Definitions and first examples

2. Ideals and homomorphisms

3. Solvable and nilpotent Lie algebras

II. Semisimple Lie Algebras

4. Theorems of Lie and Cartan

5. Killing form

6. Complete reducibility of representations

7. Representations of sl (2, F)

8. Root space decomposition

III. Root Systems

9. Axiomatics

10. Simple roots and Weyl group

11. Classification

12. Construction of root systems and automorphisms

13. Abstract theory of weights

IV. Isomorphism and Conjugacy Theorems

14. Isomorphism theorem

15. Cartan subalgebras

16. Conjugacy theorems

V. Existence Theorem

17. Universal enveloping algebras

18. The simple algebras

VI. Representation Theory

20. Weights and maximal vectors

21. Finite dimensional modules

22. Multiplicity formula

23. Characters

24. Formulas of Weyl, Kostant, and Steinberg

VII. Chevalley Algebras and Groups

25. Chevalley basis of L

26. Kostant's Theorem

27. Admissible lattices

References

Afterword (1994)

Index of Terminology

Index of Symbols.

Other Format:

Printed edition:

ISBN:

9781461263982

Access Restriction:

Restricted for use by site license.