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
Solvable and Nilpotent Lie Algebras
1. Weakly closed subsets of an associative algebra
4. Primary components. Weight spaces
5. Lie algebras with semi-simple enveloping associative algebras
7. Applications to abstract Lie algebras. Some counter examples
Cartan’s Criterion and Its Consequences
5. Structure of semi-simple algebras
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
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
4. The isomorphism theorem. Simplicity
5. The determination of the Cartan matrices
6. Construction of the algebras
1. Definition and basic properties
2. The Poincaré-Birkhoff-Witt theorem
3. Filtration and graded algebra
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
2. The characteristic zero case
Classification of Irreducible Modules
1. Definition of certain Lie algebras
2. On certain cyclic modules for
3. Finite-dimensional irreducible modules
4. Existence theorem and isomorphism theorem for semi-simple Lie algebras
Characters of the Irreducible Modules
1. Some properties of the Weyl group
5. Applications and further results
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
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
6. A closer look at the isomorphism conditions