Lie groups and Lie algebras.Chapters 7-9 by N. Bourbaki

By N. Bourbaki


This is the English translation of Bourbaki's textual content Groupes et Algebres de Lie,

Chapters 7 to nine. It completes the formerly released translations of Chapters

1 to three (3-540-50218-1) and four to six (3-540-42650-7) via overlaying the constitution and illustration concept of

semi-simple Lie algebras and compact Lie teams.

Chapter 7 offers with Cartan subalgebras of Lie algebras, general components and

conjugacy theorems. bankruptcy eight starts with the constitution of break up semi-simple Lie

algebras and their root platforms. It is going directly to describe the finite-dimensional

modules for such algebras, together with the nature formulation of Hermann Weyl. It

concludes with the idea of Chevalley orders. bankruptcy nine is dedicated to the

theory of compact Lie teams, starting with a dialogue in their maximal tori,

root structures and Weyl teams. It is going directly to describe the illustration conception

of compact Lie teams, together with the applying of integration to set up

Weyl's formulation during this context. The bankruptcy concludes with a dialogue of the

actions of compact Lie teams on manifolds.

The 9 chapters jointly shape the main accomplished textual content on hand at the

theory of Lie teams and Lie algebras.

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Extra resources for Lie groups and Lie algebras.Chapters 7-9

Sample text

VII, §4, no. 2, Th. 1). Hence dimk ψ(k) = l, which completes the proof. 4. CONJUGACY OF CARTAN SUBALGEBRAS OF SOLVABLE LIE ALGEBRAS Let g be a solvable Lie algebra. Denote by C ∞ (g) the intersection of the terms of the descending central series of g (Chap. I, §1, no. 5). This is a characteristic ideal of g, and is the smallest ideal m of g such that g/m is nilpotent. Since C ∞ (g) ⊂ [g, g], C ∞ (g) is a nilpotent ideal of g (Chap. I, §5, no. 3, Cor. 5 of Th. 1). By Prop. 1 of no. 1, the set of ead x , for x ∈ C ∞ (g), is a subgroup of Aut(g) contained in the group of special automorphisms (Chap.

1). 1. REGULAR ELEMENTS FOR A LINEAR REPRESENTATION Lemma 1. Let M be an analytic manifold over k and a = (a0 , . . , an−1 , an = 1) a sequence of analytic functions on M. For all x ∈ M, let ra (x) be the upper bound of those i ∈ 0, n such that aj (x) = 0 for j < i and let ra0 (x) be the upper bound of those i ∈ 0, n such that aj is zero on a neighbourhood of x for j < i. (i) The function ra is upper semi-continuous. (ii) For all x ∈ M, ra0 (x) = lim inf y→x ra (y). (iii) The function ra0 is locally constant.

On the other hand, for s0 sufficiently small, the tangent linear map at 0 of the map s → exp ad s from s0 to End(g) is the map s → ad s from s to End(g). Thus T(s, 0) = [s, h0 ] for all s ∈ s. Now the map from g/h to g/h induced by ad h0 by passage to the quotient is bijective. It follows that T is bijective, hence (i). Since exp ad s = Ad exp s for all s ∈ s sufficiently close to 0, (iii) and the first assertion of (ii) follow. Every x ∈ Ω is of the form (Ad a)(h) with a ∈ G and h ∈ hr , so g0 (x) = (Ad a)(g0 (h)) = (Ad a)(h) is a subalgebra of g conjugate to h under Ad(G).

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