Ebook: Complex Semisimple Lie Algebras

These notes are a record of a course given in Algiers from lOth to 21st May, 1965. Their contents are as follows. The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple Lie algebras. These are well-known results, for which the reader can refer to, for example, Chapter I of Bourbaki or my Harvard notes. The theory of complex semisimple algebras occupies Chapters III and IV. The proofs of the main theorems are essentially complete; however, I have also found it useful to mention some complementary results without proof. These are indicated by an asterisk, and the proofs can be found in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. A final chapter shows, without proof, how to pass from Lie algebras to Lie groups (complex-and also compact). It is just an introduction, aimed at guiding the reader towards the topology of Lie groups and the theory of algebraic groups. I am happy to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a first draft of these notes, and also Mlle. Franr,:oise Pecha who was responsible for the typing of the manuscript.

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These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study.

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These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study.
Content:
Front Matter....Pages i-ix
Nilpotent Lie Algebras and Solvable Lie Algebras....Pages 1-4
Semisimple Lie Algebras (General Theorems)....Pages 5-9
Cartan Subalgebras....Pages 10-16
The Algebra sl2 and Its Representations....Pages 17-23
Root Systems....Pages 24-42
Structure of Semisimple Lie Algebras....Pages 43-55
Linear Representations of Semisimple Lie Algebras....Pages 56-65
Complex Groups and Compact Groups....Pages 66-71
Back Matter....Pages 71-74

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These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study.
Content:
Front Matter....Pages i-ix
Nilpotent Lie Algebras and Solvable Lie Algebras....Pages 1-4
Semisimple Lie Algebras (General Theorems)....Pages 5-9
Cartan Subalgebras....Pages 10-16
The Algebra sl2 and Its Representations....Pages 17-23
Root Systems....Pages 24-42
Structure of Semisimple Lie Algebras....Pages 43-55
Linear Representations of Semisimple Lie Algebras....Pages 56-65
Complex Groups and Compact Groups....Pages 66-71
Back Matter....Pages 71-74
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