Log In
Or create an account ->
Imperial Library
Home
About
News
Upload
Forum
Help
Login/SignUp
Index
Cover
Title Page
Copyright Page
Preface
Contents
CHAPTER I: Basic Concepts
1. Definition and construction of Lie and associative algebras
2. Algebras of linear transformations. Derivations
3. Inner derivations of associative and Lie algebras
4. Determination of the Lie algebras of low dimensionalitie
5. Representations and modules
6. Some basic module operations
7. Ideals, solvability, nilpotency
8. Extension of the base field
CHAPTER II: Solvable and Nilpotent Lie Algebras
1. Weakly closed subsets of an associative algebra
2. Nil weakly closed sets
3. Engel’s theorem
4. Primary components. Weight spaces
5. Lie algebras with semi-simple enveloping associative algebras
6. Lie’s theorems
7. Applications to abstract Lie algebras. Some counter examples
CHAPTER III: Cartan’s Criterion and Its Consequences
1. Cartan subalgebras
2. Products of weight spaces
3. An example
4. Cartan’s criteria
5. Structure of semi-simple algebras
6. Derivations
7. Complete reducibility of the representations of semi-simple algebra
8. Representations of the split three-dimensional simple Lie algebra
9. The theorems of Levi and Malcev-Harish-Chandra
10. Cohomology groups of a Lie algebra
11. More on complete reducibility
CHAPTER IV: Split Semi-simple Lie Algebras
1. Properties of roots and root spaces
2. A basic theorem on representations and its consequences for the structure theory
3. Simple systems of roots
4. The isomorphism theorem. Simplicity
5. The determination of the Cartan matrices
6. Construction of the algebras
7. Compact forms
CHAPTER V: Universal Enveloping Algebras
1. Definition and basic properties
2. The Poincaré-Birkhoff-Witt theorem
3. Filtration and graded algebra
4. Free Lie algebras
5. The Campbell-Hausdorff formula
6. Cohomology of Lie algebras. The standard complex.
7. Restricted Lie algebras of characteristic p
8. Abelian restricted Lie algebras
CHAPTER VI: The Theorem of Ado-Iwasawa
1. Preliminary results
2. The characteristic zero case
3. The characteristic p case
CHAPTER VII: Classification of Irreducible Modules
1. Definition of certain Lie algebras
2. On certain cyclic modules for 2
3. Finite-dimensional irreducible modules
4. Existence theorem and isomorphism theorem for semi-simple Lie algebras
5. Existence of E7 and E8
6. Basic irreducible modules
CHAPTER VIII: Characters of the Irreducible Modules
1. Some properties of the Weyl group
2. Freudenthal’s formula
3. Weyl’s character formula
4. Some examples
5. Applications and further results
CHAPTER IX: Automorphisms
1. Lemmas from algebraic geometry
2. Conjugacy of Cartan subalgebras
3. Non-isomorphism of the split simple Lie algebras
4. Automorphisms of semi-simple Lie algebras over an algebraically closed field
5. Explicit determination of the automorphisms for the simple Lie algebras
CHAPTER X: Simple Lie Algebras over an Arbitrary Field
1. Multiplication algebra and centroid of a non-associative algebra
2. Isomorphism of extension algebras
3. Simple Lie algebras of types A-D
4. Conditions for isomorphism
5. Completeness theorems
6. A closer look at the isomorphism conditions
7. Central simple real Lie algebras
Bibliography
Index
← Prev
Back
Next →
← Prev
Back
Next →