Algebre: Exercises avec solutions - download pdf or read online

By E Ramis

ISBN-10: 2225813140

ISBN-13: 9782225813146

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K, avec k ;::: o. 2. Soit X un CW-complexe fini; on munit l'anneau 0) de la filtration naturelle: k F(X) (oli Fest U ou C. p-l designe l'homomorphisme induit par }'inclusion de (p-1)-ieme squelette X p- 1 dans X. En utilisant cette filtration, a k F* (X) est associee une suite spectrale - dite d'Atiyah-Hirzebruch -dont Ie terme E z et de la forme: et Ie terme E oo est defini par: Dans Ie cas de la 1<- theorie complexe, la differentielle d 3 de la suite spectrale d'Atiyah-Hirzebruch est don nee par Ie carre de Steenrod Sq 3; en KO -theorie, la differentielle d z est don nee par les formules suivantes: d~,8k-l ~,8k = SqZ : E~,8k-l(X) = HP(X; Zz) = SqZ 0 p : E~,8k(X) -+ E~+Z,8k-Z(X) = HP+z(X; Zz) = W(X; Z) - + E~+Z,8k-l(X) = HP+z(X; Zz) ou Sqz designe Ie carre de Steenrod et p : H*(X; Z) --4 H*(X; Zz) est I'homomorphisme de restriction induit par la projection canonique Z --4 Zz.

In fact, if t = 0, then ads is nilpotent on n. But [s, s] en. It follows that s is a nilpotent element of g. Hence s = O. Thus dim s ::; dim t. 0 Corollary 1 If exists a non-decomposable faithful Lie algebra 9 with dim s = 2, then n = L n or Qn. Lemma 5 Let 9 be a faithful Lie algebra (decomposable or not) with a filiform nilradical n and let dimg - dim n = 2. Then we may choose a complementary subspace s to n in 9 such that ads is a maximal torus of derivations of g. KHAKIMDJANOV 56 Proof We study the two following possible cases: (1) g=sGJL n , (2) g=sGJQn.

NILPOTENT LIE ALGEBRAS 55 Case 2. dim s = 1 or 2. 2) and from Mal'cev theorem [10], we have Theorem 2 Let 9 be a faithful decomposable Lie algebra with a filiform nilradical of codimension 2: 1. Then 9 is isomorphic to the algebra 9 = TEBidn where B~+l (a1, ... , at), n is one of the following; L n , Qn, A~+1 (a1"'" ad, Cn + 1 (a1, ... , at) and T is a subalgebra of the maximal torus of derivations of n described before. The Lie algebras gl = T 1 EBid n g2 = T2 EBid n are isomorphic if and only if T 1 = T2.

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Algebre: Exercises avec solutions by E Ramis

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