- 李代数和表示论导论(影印版)(英文版)
- - 作者：(美)J.E.汉弗雷斯
  - 出版社：世图出版公司
  - ISBN：9787519255862
  - 出版日期：2019/03/01
  - 页数：171
- 售价：23.6

内容大纲
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。
作者介绍
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 Engel's Theorem
Ⅱ.SEMISIMPLE LIE ALGEBRAS
  4.Theorems of Lie and Cartan
    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 reducibifity 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 sl (2,F)
    7.1  Weights and maximal vectors
    7.2  Classification of irreducible modules
  8.Root space decomposition
    8.1  Maximal toral 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
  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 subalgebras
    15.1  Decomposition of L relative to ad x
    15.2  Engel subalgebras
    15.3  Caftan subalgebras
    15.4  Functorial properties
  16.Conjugacy theorems
    16.1  The group E(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 U(L)
    17.3  PBW Theorem and consequences
    17.4  Proof of PBW Theorem
    17.5  Free Lie algebras
  18.Generators and relations
    18.1  Relations satisfied by L
    18.2  Consequences of (S1)-($3)
    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 V(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  Invariant polynomial functions
    23.2  Standard cyclic modules and characters
    23.3  Harish-Chandra's Theorem Appendix
  24.Formulas of Weyl, 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 Chevalley basis
    25.3  Uniqueness questions
    25.4  Reduction modulo a prime
    25.5  Construction of Chevalley groups (adjoint type)
  26.Kostant's Theorem
    26.1  A combinatorial lemma
    26.2  Special case: sl (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

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