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Alors : (i) Il existe une structure d 'algebre de Poisson sur H, provenant d 'une structure de Poisson canonique sur C[XI , X 2 , X 3] ; (ii) Pour cette structure de Poisson, Htois(H) s'identifie Ii l'algebre H' = C[XI , X2, X3l! (F{, F~, F~). (H), ou H designe l'hypersurface definie par F. 31 DEFORMATIONS DES SURFACES DE KLEIN Preuve. Prouvons (i). Pour P, Q E C[X1, X 2, X 3], posons {P, Q} = dP 1\ dQ 1\ dF/ dX 1 1\ dX 2 1\ dX 3, ce qui definit un crochet de Poisson dans C[X1, X 2 , X 3 ] pour lequel I'ideal (F) est un ideal de Poisson.

Xn ) by [XO, X;] = Xi+1, i = [Xi, X n - i ] = (_l)i X n , + I)-dimensional 1, ... , n - 1; i = 1, ... , k. This is a filiform Lie algebra. In the basis (Zo, Zl, ... ,Zn), where Zo = Xo + Xl, Zi = Xi, i =, ... ,n, this Lie algebra is defined by [ZO, Zi] = Zi+ll [Zi, Zn-i] i = 1, ... ,n - 2; = (_l)i Zn, i = 1, ... , k. Consider the algebraic Lie algebra Derg of all derivations of g. Clearly, the central descending sequence of g is a flag that is invariant under all derivations. As we have supposed that dimg ~ 6, it follows that Derg is solvable.

Case (2) Analogously to case (1), by using the description of DerQn given in section 1, we have adsIads2=ads2adsl. 0 Lemma 6 Let 9 be a faithful non-decomposable Lie algebra with a filiform nilradical n. Then dim 9 - dim n = 1. NILPOTENT LIE ALGEBRAS 57 Proof Let s a complementary to n subspace in 9. We suppose dim s 2: 2. ,;, (Xo, Xl, ... , X n ) is the basis of L n , (Zo, Zt, ... ,Zn) is the basis of Qn, d l and d 2 are the elements of maximal torus T of derivation of n, described in the section 1.

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