Group theory in physics: An introduction by Cornwell J.F.

By Cornwell J.F.

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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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