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Lie Algebras and Algebraic Groups

P. Tauvel and R. W. T. Yu
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Duncan Melville
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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.


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


Index of notations

Index of terminologies