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Publisher:

Springer

Publication Date:

2005

Number of Pages:

653

Format:

Hardcover

Series:

Springer Monographs in Mathematics

Price:

89.95

ISBN:

3-540-24170-1

Category:

Monograph

[Reviewed by , on ]

Duncan Melville

03/26/2006

The stated goal of the authors is to provide a "foundation for the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero" in a self-contained work that will be useful to "both graduate students and mathematicians working in this area." The book is thus aimed squarely at the research community. The clarity and lucidity of the statement of their goal is indicative of the text as a whole, which is admirably well organized and presented in order to achieve its central aims. The reader should also be aware that the delimitation of its goal shows what the book is not: it does not aim to give a comprehensive treatment of Lie algebras over fields of prime characteristic (although the authors do indicate in some places where proofs may be generalized); it does not concern itself with Kac-Moody algebras, nor superalgebras, nor quantum groups.

The approach taken is geometric. After introducing Lie algebras and establishing their basic properties and the classification of semi-simple algebras via root systems, and doing the same for algebraic groups, the authors immediately spend a couple of chapters establishing the correspondences between the two classes of objects and the connections between properties of Lie algebras and their corresponding algebraic groups and vice versa. With this correspondence in hand, the authors turn to the core of the book, deeper properties of Lie algebras and algebraic groups, working back and forth from the two different perspectives. They cover quotients by subgroups as algebraic varieties, solvable and reductive groups, Borel and parabolic subgroups and subalgebras, and Cartan subgroups and subalgebras. There is a chapter on S-triples, and polarizations are introduced to obtain Richardson's Theorem on orbits; a subsequent chapter analyzes adjoint orbits. There are results on sigma-root systems and symmetric Lie algebras and their invariants. Deploying their orbit machinery, they study sheets of Lie algebras.

As the summary (which covers only a part of the contents) above indicates, the book contains a wealth of detail and takes the reader from the basic classical concepts to the modern borders of this still-active area. Complete proofs are given and the authors present their material clearly and concisely throughout.

The authors also desired their book to be as self-contained as possible. The interweaving of the deeper properties of Lie algebras and algebraic groups rests upon a vast base of algebra and geometry. The results needed are widely scattered, appearing in many different forms, having been introduced for many disparate purposes. This situation acts as a barrier to entry for the beginner, and so the authors have rectified the problem by assembling the necessary background material in the early chapters, where they are able to present only those results needed later and can ensure a uniform notation. Thus, while the book comprises 40 chapters in some 650 pages, Lie algebras themselves do not make an appearance until Chapter 19, with algebraic groups entering on Chapter 21. The first 18 chapters provide background.

The book is written in a terse, focused style: definition, theorem and proof with little discursiveness, few examples and no exercises. The reader must bring his or her own motivation, but would certainly find extensive engagement with the book amply rewarded. This is a rich resource and reference, gathering together material not otherwise to be found in a single place, though probably not a book that should be read linearly from cover to cover.

Duncan Melville is Peterson Professor of mathematics at St. Lawrence University in Canton, NY. His main research interests are in Lie Algebras and the history of Mesopotamian mathematics.

Preface

1. Results on topological spaces

1.1 Irreducible sets and spaces

1.2 Dimension

1.3 Noetherian spaces

1.4 Constructible sets

1.5 Gluing topological spaces

2. Rings and modules

2.1 Ideals

2.2 Prime and maximal ideals

2.3 Rings of fractions and localization

2.4 Localization of modules

2.5 Radical of an ideal

2.6 Local rings

2.7 Noetherian rings and modules

2.8 Derivations

2.9 Module of differentials

3. Integral extensions

3.1 Integral dependence

3.2 Integrally closed rings

3.3 Extensions of prime ideals

4. Factorial rings

4.1 Generalities

4.2 Unique factorization

4.3 Principal ideal domains and Euclidean domains

4.4 Polynomial and factorial rings

4.5 Symmetric polynomials

4.6 Resultant and discriminant

5. Field extensions

5.1 Extensions

5.2 Algebraic and transcendental elements

5.3 Algebraic extensions

5.4 Transcendence basis

5.5 Norm and trace

5.6 Theorem of the primitive element

5.7 Going Down Theorem

5.8 Fields and derivations

5.9 Conductor

6. Finitely generated algebras

6.1 Dimension

6.2 Noether’s Normalization Theorem

6.3 Krull’s Principal Ideal Theorem

6.4 Maximal ideals

6.5 Zariski topology

7. Gradings and filtrations

7.1 Graded rings and graded modules

7.2 Graded submodules

7.3 Applications

7.4 Filtrations

7.5 Grading associated to a filtration

8. Inductive limits

8.1 Generalities

8.2 Inductive systems of maps

8.3 Inductive systems of magmas, groups and rings

8.4 An example

8.5 Inductive systems of algebras

9. Sheaves of functions

9.1 Sheaves

9.2 Morphisms

9.3 Sheaf associated to a presheaf

9.4 Gluing

9.5 Ringed space

10. Jordan decomposition and some basic results on groups

10.1 Jordan decomposition

10.2 Generalities on groups

10.3 Commutators

10.4 Solvable groups

10.5 Nilpotent groups

10.6 Group actions

10.7 Generalities on representations

10.8 Examples

11. Algebraic sets

11.1 Affine algebraic sets

11.2 Zariski topology

11.3 Regular functions

11.4 Morphisms

11.5 Examples of morphisms

11.6 Abstract algebraic sets

11.7 Principal open subsets

11.8 Products of algebraic sets

12. Prevarieties and varieties

12.1 Structure sheaf

12.2 Algebraic prevarieties

12.3 Morphisms of prevarieties

12.4 Products of prevarieties

12.5 Algebraic varieties

12.6 Gluing

12.7 Rational functions

12.8 Local rings of a variety

13. Projective varieties

13.1 Projective spaces

13.2 Projective spaces and varieties

13.3 Cones and projective varieties

13.4 Complete varieties

13.5 Products

13.6 Grassmannian variety

14. Dimension

14.1 Dimension of varieties

14.2 Dimension and the number of equations

14.3 System of parameters

14.4 Counterexamples

15. Morphisms and dimension

15.1 Criterion of affineness

15.2 Affine morphisms

15.3 Finite morphisms

15.4 Factorization and applications

15.5 Dimension of fibres of a morphism

15.6 An example

16. Tangent spaces

16.1 A first approach

16.2 Zariski tangent space

16.3 Differential of a morphism

16.4 Some lemmas

16.5 Smooth points

17. Normal varieties

17.1 Normal varieties

17.2 Normalization

17.3 Products of normal varieties

17.4 Properties of normal varieties

18. Root systems

18.1 Reflections

18.2 Root systems

18.3 Root systems and bilinear forms

18.4 Passage to the field of real numbers

18.5 Relation between two roots

18.6 Base of a root system

18.7 Weyl chambers

18.8 Highest root

18.9 Closed subsets of roots

18.10 Weights

18.11 Graphs

18.12 Dynkin diagrams

18.13 Classification of root systems

19. Lie algebras

19.1 Generalities on Lie algebras

19.2 Representations

19.3 Nilpotent Lie algebras

19.4 Solvable Lie algebras

19.5 Radical and the largest nilpotent ideal

19.6 Nilpotent radical

19.7 Regular linear forms

19.8 Cartan subalgebras

20. Semisimple and reductive Lie algebras

20.1 Semisimple Lie algebras

20.2 Examples

20.3 Semisimplicity of representations

20.4 Semisimple and nilpotent elements

20.5 Reductive Lie algebras

20.6 Results on the structure of semisimple Lie algebras

20.7 Subalgebras of semisimple Lie algebras

20.8 Parabolic subalgebras

21. Algebraic groups

21.1 Generalities

21.2 Subgroups and morphisms

21.3 Connectedness

21.4 Actions of an algebraic group

21.5 Modules

21.6 Group closure

22. Affine algebraic groups

22.1 Translations of functions

22.2 Jordan decomposition

22.3 Unipotent groups

22.4 Characters and weights

22.5 Tori and diagonalizable groups

22.6 Groups of dimension one

23. Lie algebra of an algebraic group

23.1 An associative algebra

23.2 Lie algebras

23.3 Examples

23.4 Computing differentials

23.5 Adjoint representation

23.6 Jordan decomposition

24. Correspondence between groups and Lie algebras

24.1 Notations

24.2 An algebraic subgroup

24.3 Invariants

24.4 Functorial properties

24.5 Algebraic Lie subalgebras

24.6 A particular case

24.7 Examples

24.8 Algebraic adjoint group

25. Homogeneous spaces and quotients

25.1 Homogeneous spaces

25.2 Some remarks

25.3 Geometric quotients

25.4 Quotient by a subgroup

25.5 The case of finite groups

26. Solvable groups

26.1 Conjugacy classes

26.2 Actions of diagonalizable groups

26.3 Fixed points

26.4 Properties of solvable groups

26.5 Structure of solvable groups

27. Reductive groups

27.1 Radical and unipotent radical

27.2 Semisimple and reductive groups

27.3 Representations

27.4 Finiteness properties

27.5 Algebraic quotients

27.6 Characters

28. Borel subgroups, parabolic subgroups and Cartan subgroups

28.1 Borel subgroups

28.2 Theorems of density

28.3 Centralizers and tori

28.4 Properties of parabolic subgroups

28.5 Cartan subgroups

29. Cartan subalgebras, Borel subalgebras and parabolic subalgebras

29.1 Generalities

29.2 Cartan subalgebras

29.3 Application to semisimple Lie algebras

29.4 Borel subalgebras

29.5 Properties of parabolic subalgebras

29.6 More on reductive Lie algebras

29.7 Other applications

29.8 Maximal subalgebras

30. Representations of semisimple Lie algebras

30.1 Enveloping algebra

30.2 Weights and primitive elements

30.3 Finite-dimensional modules

30.4 Verma modules

30.5 Results on existence and uniqueness

30.6 A property of the Weyl group

31. Symmetric invariants

31.1 Invariants of finite groups

31.2 Invariant polynomial functions

31.3 A free module

32. S-triples

32.1 Theorem of Jacobson-Morosov

32.2 Some lemmas

32.3 Conjugation of S-triples

32.4 Characteristic

32.5 Regular and principal elements

33. Polarizations

33.1 Definition of polarizations

33.2 Polarizations in the semisimple case

33.3 A non-polarizable element

33.4 Polarizable elements

33.5 Theorem of Richardson

34. Results on orbits

34.1 Notations

34.2 Some lemmas

34.3 Generalities on orbits

34.4 Minimal nilpotent orbit

34.5 Subregular nilpotent orbit

34.6 Dimension of nilpotent orbits

34.7 Prehomogeneous spaces of parabolic type

35. Centralizers

35.1 Distinguished elements

35.2 Distinguished parabolic subalgebras

35.3 Double centralizers

35.4 Normalizers

35.5 A semisimple Lie subalgebra

35.6 Centralizers and regular elements

36. s -root systems

36.1 Definition

36.2 Restricted root systems

36.3 Restriction of a root

37. Symmetric Lie algebras

37.1 Primary subalgebras

37.2 Definition of symmetric Lie algebras

37.3 Natural subalgebras

37.4 Cartan subspaces

37.5 The case of reductive Lie algebras

37.6 Linear forms

38. Semisimple symmetric Lie algebras

38.1 Notations

38.2 Iwasawa decomposition

38.3 Coroots

38.4 Centralizers

38.5 S-triples

38.6 Orbits

38.7 Symmetric invariants

38.8 Double centralizers

38.9 Normalizers

38.10 Distinguished elements

39. Sheets of Lie algebras

39.1 Jordan classes

39.2 Topology of Jordan classes

39.3 Sheets

39.4 Dixmier sheets

39.5 Jordan classes in the symmetric case

39.6 Sheets in the symmetric case

40. Index and linear forms

40.1 Stable linear forms

40.2 Index of a representation

40.3 Some useful inequalities

40.4 Index and semi-direct products

40.5 Heisenberg algebras in semisimple Lie algebras

40.6 Index of Lie subalgebras of Borel subalgebras

40.7 Seaweed Lie algebras

40.8 An upper bound for the index

40.9 Cases where the bound is exact

40.10 On the index of parabolic subalgebras

References

Index of notations

Index of terminologies

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