By Jean-Pierre Serre

ISBN-10: 0387965696

ISBN-13: 9780387965697

ISBN-10: 1475739109

ISBN-13: 9781475739107

These notes are a checklist of a direction given in Algiers from lOth to twenty first may well, 1965. Their contents are as follows. the 1st chapters are a precis, with out proofs, of the final homes of nilpotent, solvable, and semisimple Lie algebras. those are famous effects, for which the reader can seek advice from, for instance, bankruptcy I of Bourbaki or my Harvard notes. the idea of complicated semisimple algebras occupies Chapters III and IV. The proofs of the most theorems are basically entire; notwithstanding, i've got additionally discovered it helpful to say a few complementary effects with out facts. those are indicated via an asterisk, and the proofs are available in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. a last bankruptcy indicates, with out facts, how you can go from Lie algebras to Lie teams (complex-and additionally compact). it's only an advent, 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 likewise Mlle. Franr,:oise Pecha who used to be liable for the typing of the manuscript.

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**Example text**

Since tis invariant under the endomorphisms ad(H), HE~. we have t = t 11 ~ $ LileR t 11 gil. However,sincesilisisomorphic to sl 2 , t 11 sl 2 = Oandhenceafortiorit 11 gil= 0. We therefore have t c ~-By writing down the condition that tis invariant under ad(Xi), we see that t is orthogonal to each ai; hence t = 0, and g is semisimple. (I) ~ is a Cartan subalgebra of g, and R is the corresponding root system. For ~ is equal to its own normalizer. The fact that R is the corresponding root system is clear.

2p. ): gfl+kll -+ gfl+(k+1)110 -p~k~q-1 is an isomorphism follows from the structure of the irreducible sl 2 -modules. 47 3. 13. If IX e R, fJ e R, IX + fJ e R, we have [g", gil] = g"+ll. With the notation above we have q ~ 1. ,): gil -+ gil+" is an isomorphism, thus proving the result. The proof of Theorem 2 is now complete. Remark. Let W be the Weyl group associated with the root system R; let us identify W with a group of automorphisms of~- Then every element w e W is induced by an inner automorphism of g which leaves ~ invariant.

Let peR+ - {ex}. We have p= L yeS myy with my~ 0. Since R is reduced, and P=F ex, Pis not proportional to ex, and there exists some y =F ex such that my =F 0. (P) is equal to my. (P) e R +, proving the proposition. Corollary. Let p be half the sum of the positive roots. (p) = p- ex for all ex e s. Let Pa. be half the sum of the elements of R+ -{ex}. · On the other hand, p = Pa. + ex/2. (P) = p- ex. ) Proposition 7. Suppose that R is reduced. The set S* of inverse roots of the elements of S is a base for R*.

### Complex Semisimple Lie Algebras by Jean-Pierre Serre

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