By Jean-Pierre Serre
These notes are a checklist of a direction given in Algiers from lOth to twenty first may possibly, 1965. Their contents are as follows. the 1st chapters are a precis, with out proofs, of the overall homes of nilpotent, solvable, and semisimple Lie algebras. those are famous effects, for which the reader can discuss with, for instance, bankruptcy I of Bourbaki or my Harvard notes. the idea of advanced semisimple algebras occupies Chapters III and IV. The proofs of the most theorems are primarily whole; despite the fact that, i've got additionally stumbled on it worthwhile to say a few complementary effects with out evidence. those are indicated through an asterisk, and the proofs are available in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. a last bankruptcy exhibits, with no evidence, the right way to move from Lie algebras to Lie teams (complex-and additionally compact). it's only an creation, geared toward guiding the reader in the direction of the topology of Lie teams and the idea of algebraic teams. i'm satisfied to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a primary draft of those notes, and in addition Mlle. Franr,:oise Pecha who was once accountable for the typing of the manuscript.
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Extra info for Complex Semisimple Lie Algebras
X;), and where l) has the elements H; as a basis. Moreover, l) (resp. x) can be identified with the free Lie algebra generated by the elements Yj (resp. X;). ) Now let and Ojj = ad(Yj)-n(i,J)+t(lj). We have Ou ex, Oij e 1). We denote by u (resp. u-) the ideal of x (resp. of l)) generated by the elements Ou (resp. Ojj) fori ¥ j. Lett = u E9 u-. (a) u, u-, and t are ideals ofa. Let U a be the universal enveloping algebra of a. The adjoint representation ad: a--. End(a) defines a Ua-module structure on a.
If R is oftype G2 , the algebra n has a presentation consisting of two generators X 1 , X 2 and of two relations: We now give an application of Theorem 7: Corollary. There is an automorphism q of g which is equal to - 1 on which sends Xi to - Y;, li to -Xi, for all i. One has q 2 = 1. ~. and Let us put Hi= -H1, Xi= -Y;, Y;' = -Xi. Clearly the elements x;, Y;', Hi satisfy the Weyl relations and the relations 911 , Ojj. Hence by (ii) there is a homomorphism q: g--+ g mapping X" Y;, H 1 to x;, Y;', Hi.
On the other hand, we have just seen that its transpose maps a* to a~ for each a E R. But, by Prop. 2, the elements a~ form a root system in V0*, and in particular they span V0*. It follows that ti is surjective, and hence i is injective, giving (b). D Finally, (c) follows from the facts proved above. Theorem 5 reduces the theory of complex root systems to that of real root systems. All the definitions and results of the preceding sections are therefore applicable in the complex case. CHAPTER VI Structure of Semisimple Lie Algebras Throughout this chapter, g denotes a complex semisimple Lie algebra, and a Cartan subalgebra of g (cf.