李代数和表示论导论 内容简介

本书是一部优秀的李群及其表示论研究生教材。深受数学专业和物理专业的研究生好评。

李代数和表示论导论 本书特色

本书是一部优秀的李群及其表示论研究生教材，深受数学专业和物理专业的研究生好评。本书初版于1972年，以后经过多次修订重印，本书是1997年的第7次修订重印版。书中对一些问题的处理很有特色，立足点较高，但叙述十分清晰，如线性变换的Jordan-Chevalley分解、Cartan子代数的共轭定理、同构定理的证明、根系统的公理化处理、Weyl特征子公式、Chevalley群的基本结构等。

李代数和表示论导论李代数和表示论导论前言

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 incoro -porate some of them here and to provide easier access to the subject fornon.specialists. For the specialist, the following features should be noted:
(1) The Jordan-Chevalley decompositionS'of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case.
(2) The conjugacy theorem for Caftan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.
(3) The isomorphism theorem is proved first in an elementary way (Theorem 14.2), but later obtained again as a corollary of Serre's Theorem(18.3), which gives a presentation by generators and relations.
(4) From the outset, the simple algebras of types A, B, C, D are emphasized in the text and exercises.
(5) Root systems are treated axiomatically (Chapter III), along with some of the theory of weights.
(6) A conceptual approach to Weyl's character formula, based on Harish-Chandra's theory of "characters" and independent of Freudenthal's multiplicity formula (22.3), is presented in §23 and §24. This is inspired by D.-N. Verma's thesis, and recent work of I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand. ..
(7) The basic constructions in the theory of Chevalley groups are given in Chapter VII, following lecture notes of R. Steinberg.
I have had to omit many standard topics (most of which I feel are better suited to a second course), e.g., cohomology, theorems of Levi and Mal'cev, theorems of Ado and lwasawa, classification over non-algebraically closed fields, Lie algebras in prime characteristic, I hope the reader will be stimulated to pursue these topics in the books and articles listed under References, especially Jacobson [1], Bourbaki [1], [2], Winter [1], Seligman [1].
A few words about mechanics: Terminology is mostly traditional, and notation has been kept to a minimum, to facilitate skipping back and forth in the text. After Chapters I-III, the remaining chapters can be read in almost any order if the reader is willing to follow up a few references (except that VII depends on §20 and §21, while VI depends on §17). A reference to Theorem 14.2 indicates the (unique) theorem in subsection 14.2 (of §14). Notes following some sections indicate nonstandard sources or further reading, but I have not tried to give a history of each theorem (for historical remarks, cf. Bourbaki [2] and Freudenthai-deVries [1]): The reference list consists largely of items mentioned explicitly; for more extensive bibliographies, consult Jacobson [1], Seligman [1]. Some 240 exercises, of all shades of difficulty, have been included; a few of the easier ones are needed in the text.
This text grew out of lectures which I gave at the N.S.F. Advanced Science Seminar on Algebraic Groups at Bowdoin College in 1968; my intention then was to enlarge on J.-P. Serre's excellent but incomplete lecture notes [2]. My other literary debts (to the books and lecture notes of N. Bourbaki, N. Jacobson, R. Steinberg, D. J. Winter, and others) will be obvious. Less obvious is my personal debt to my teachers, George Seligman and Nathan Jacobson, who first aroused my interest in Lie algebras. I am grateful to David J. Winter for giving me pre-publication access to his book, to Robert L. Wilson for making many helpful criticisms of an earlier version of the manuscript, to Connie Engle for her help in preparing the final manuscript, and to Michael J. DeRise for moral support. Financial assistance from the Courant Institute of Mathematical Sciences and the National Science Foundation is also gratefully acknowledged. ...
New York, April 4, 1972
J.E. Humphreys

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李代数和表示论导论 目录

preface .
i. basic concepts
1. definitions and first examples
1.1 the notion of lie algebra
1.2 linear lie algebras
1.3 lie algebras of derivations
1.4 abstract lie algebras
2. ideals and homomorphisms
2.1 ideals
2.2 homomorphisms and representations
2.3 automorphisms
3. solvable and nilpotent lie algebras
3.1 solvability:
3.2 nilpotency
3.3 proof of engers theorem
ii. semisimple lie algebras
4. theorems of lie and caftan
4.1 lie's theorem
4.2 jordan-chevalley decomposition
4.3 cartan's criterion.
.5. killing form
5.1 criterion for semisimplicity
5.2 simple ideals of l
5.3 inner derivations
5.4 abstract jordan decomposition
6. complete reducibility of representations
6.1 modules
6.2 casimir element of a representation
6.3 weyl's theorem
6.4 preservation of jordan decomposition
7. representations of si (2, f)
7.1 weights and maximal vectors
7.2 classification of irreducible modules
8. root space decomposition
8.1 maximal total subalgebras and roots
8.2 centralizer of h
8.3 orthogonality properties
8.4 integrality properties
8.5 rationality properties. summary
iii. root systems
9. axiomatics
9.1 reflections in a euclidean space
9.2 root systems
9.3 examples
9.4 pairs of roots
10. simple roots and weyl group
10.1 bases and weyl chambers
10.2 lemmas on simple roots
10.3 the weyl group
10.4 irreducible root systems
11. classification
11.1 cartan matrix of
11.2 coxeter graphs and dynkin diagrams
11.3 irreducible components
11.4 classification theorem
12. construction of root systems and automorphisms
12.1 construction of types a-g
12.2 automorphisms of/b
13. abstract theory of weights
13.1 weights
13.2 dominant weights
13.3 the weight δ
13.4 saturated sets of weights
iv. isomorphism and conjugacy theorems ..
14. isomorphism theorem
14.1 reduction to the simple case
14.2 isomorphism theorem
14.3 automorphisms
15. cartan subaigebras
15.1 decomposition of l relative to ad x
15.2 engel subalgebras
15.3 cartan subalgebras
15.4 functorial properties
16. conjugacy theorems
16.1 the group ξ(l)
16.2 conjugacy of csa's (solvable case)
16.3 borel subalgebras
16.4 conjugacy of borel subalgebras
16.5 automorphism groups
v. existence theorem
17. universal enveloping algebras
17.1 tensor and symmetric algebras
17.2 construction of ц(l)
17.3 pbw theorem and consequences
17.4 proof of pbw theorem
table of contents
17.5 free lie algebras
18. generators and relations
18.1 relations satisfied by l
18.2 consequences of (s1)-(s3)
18.3 serre's theorem
18.4 application: existence and uniqueness theorems
19. the simple algebras
19.1 criterion for semisimplicity
19.2 the classical algebras
19.3 the algebra g2
vi. representation theory
20. weights and maximal vectors
20.1 weight spaces
20.2 standard cyclic modules
20.3 existence and uniqueness theorems
21. finite dimensional modules
21.1 necessary condition for finite dimension
21.2 sufficient condition for finite dimension
21.3 weight strings and weight diagrams
21.4 generators and relations for ν(λ)
22. multiplicity formula
22.1 a universal casimir element
22.2 traces on weight spaces
22.3 freudenthal's formula
22.4 examples
22.5 formal characters.
23. characters
23.1 lnvariant polynomial functions
23.2 standard cyclic modules and characters
23.3 harish-chandra's theorem
appendix
24. formulas of weyi, kostant, and steinberg
24.1 some functions on h*
24.2 kostant's multiplicity formula
24.3 weyl's formulas
24.4 steinberg's formula
appendix
vii. chevalley algebras and groups
25. chevalley basis of l
25.1 pairs of roots
25.2 existence of a cheva!!ey basis
25.3 uniqueness questions
25.4 reduction modulo a prime
25.5 construction of cheva!ley groups (adjoint type)
26. kostant's theorem
26.1 a combinatorial lemma
26.2 special case: si (2, f)
26.3 lemmas on commutation
26.4 proof of kostant's theorem
27. admissible lattices
27.1 existence of admissible lattices
27.2 stabilizer of an admissible lattice
27.3 variation of admissible lattice
27.4 passage to an arbitrary field
27.5 survey of related results
references
afterword (1994)
index of terminology
index of symbols ...

李代数和表示论导论 目录

preface .
i. basic concepts
1. definitions and first examples
1.1 the notion of lie algebra
1.2 linear lie algebras
1.3 lie algebras of derivations
1.4 abstract lie algebras
2. ideals and homomorphisms
2.1 ideals
2.2 homomorphisms and representations
2.3 automorphisms
3. solvable and nilpotent lie algebras
3.1 solvability:
3.2 nilpotency
3.3 proof of engers theorem
ii. semisimple lie algebras
4. theorems of lie and caftan
4.1 lie's theorem
4.2 jordan-chevalley decomposition
4.3 cartan's criterion.
.5. killing form
5.1 criterion for semisimplicity
5.2 simple ideals of l
5.3 inner derivations
5.4 abstract jordan decomposition
6. complete reducibility of representations
6.1 modules
6.2 casimir element of a representation
6.3 weyl's theorem
6.4 preservation of jordan decomposition
7. representations of si (2, f)
7.1 weights and maximal vectors
7.2 classification of irreducible modules
8. root space decomposition
8.1 maximal total subalgebras and roots
8.2 centralizer of h
8.3 orthogonality properties
8.4 integrality properties
8.5 rationality properties. summary
iii. root systems
9. axiomatics
9.1 reflections in a euclidean space
9.2 root systems
9.3 examples
9.4 pairs of roots
10. simple roots and weyl group
10.1 bases and weyl chambers
10.2 lemmas on simple roots
10.3 the weyl group
10.4 irreducible root systems
11. classification
11.1 cartan matrix of
11.2 coxeter graphs and dynkin diagrams
11.3 irreducible components
11.4 classification theorem
12. construction of root systems and automorphisms
12.1 construction of types a-g
12.2 automorphisms of/b
13. abstract theory of weights
13.1 weights
13.2 dominant weights
13.3 the weight δ
13.4 saturated sets of weights
iv. isomorphism and conjugacy theorems ..
14. isomorphism theorem
14.1 reduction to the simple case
14.2 isomorphism theorem
14.3 automorphisms
15. cartan subaigebras
15.1 decomposition of l relative to ad x
15.2 engel subalgebras
15.3 cartan subalgebras
15.4 functorial properties
16. conjugacy theorems
16.1 the group ξ(l)
16.2 conjugacy of csa's (solvable case)
16.3 borel subalgebras
16.4 conjugacy of borel subalgebras
16.5 automorphism groups
v. existence theorem
17. universal enveloping algebras
17.1 tensor and symmetric algebras
17.2 construction of ц(l)
17.3 pbw theorem and consequences
17.4 proof of pbw theorem
table of contents
17.5 free lie algebras
18. generators and relations
18.1 relations satisfied by l
18.2 consequences of (s1)-(s3)
18.3 serre's theorem
18.4 application: existence and uniqueness theorems
19. the simple algebras
19.1 criterion for semisimplicity
19.2 the classical algebras
19.3 the algebra g2
vi. representation theory
20. weights and maximal vectors
20.1 weight spaces
20.2 standard cyclic modules
20.3 existence and uniqueness theorems
21. finite dimensional modules
21.1 necessary condition for finite dimension
21.2 sufficient condition for finite dimension
21.3 weight strings and weight diagrams
21.4 generators and relations for ν(λ)
22. multiplicity formula
22.1 a universal casimir element
22.2 traces on weight spaces
22.3 freudenthal's formula
22.4 examples
22.5 formal characters.
23. characters
23.1 lnvariant polynomial functions
23.2 standard cyclic modules and characters
23.3 harish-chandra's theorem
appendix
24. formulas of weyi, kostant, and steinberg
24.1 some functions on h*
24.2 kostant's multiplicity formula
24.3 weyl's formulas
24.4 steinberg's formula
appendix
vii. chevalley algebras and groups
25. chevalley basis of l
25.1 pairs of roots
25.2 existence of a cheva!!ey basis
25.3 uniqueness questions
25.4 reduction modulo a prime
25.5 construction of cheva!ley groups (adjoint type)
26. kostant's theorem
26.1 a combinatorial lemma
26.2 special case: si (2, f)
26.3 lemmas on commutation
26.4 proof of kostant's theorem
27. admissible lattices
27.1 existence of admissible lattices
27.2 stabilizer of an admissible lattice
27.3 variation of admissible lattice
27.4 passage to an arbitrary field
27.5 survey of related results
references
afterword (1994)
index of terminology
index of symbols ...