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Integrability is an important property of Hamiltonian systems from the point of view of such ap­ plications. 4 are integrable. Little is known about criteria of integrability of more general Hamiltonian systems. 1. Let Q be the four-dimensional space-time Ę manifold of general relativity with coordinates (÷ ). Let g V and g^ denote the components of the covariant and the contravar- iant metric tensors respectively and let symbols. be the Christoffel Κ The cotangent bundle Ρ = T*Q with coordinates (χ ,p ) represents the space-time-momentum-energy manifold of a relativistic particle.

Are null on the constants. For such weak star2r+1 Deformations and Quantization 53 products, the uniqueness theorem is valid again. Moreover the ar­ gument of § 6,b shows that we have : Theorem - Each weak star-product of (W3F) is equivalent to a weak Vey star-product. For b (W) = 0, S. Gutt has proved : Proposition - If b^(W) - 03 all the formal Lie algebras of the 3 form (8-3) for which C^ - Q /3! are weakly equivalent. 9 - LIE ALGEBRAS GENERATED BY A WEAK TWISTED PRODUCT AND VEY LIE ALGEBRAS a) Consider, on an arbitrary manifold, a formal Lie algebra : Ă [u,v ] 2= P(u,v) + (9-D V C ae rn uno iet where the 2r+1 ^^ * Ó ν r=l C 2 r( u +,1v ) constants.

V : b) Consider a formal series in oo (3-5) Ô V = Σ v s=0 S S oo Ô s = Id,T+ Í Σ . s=l v where the T g (s > 1) are endomorphisms of N ; Ô s acts naturally on E(N;V). Consider also another bilinear map Í x Í responding to the formal series : Γ f (3-6) u *^ í = uv + Σ r=l ν C (u,v) E(N;v) cor­ 36 ANDRE LICHNEROWICZ where the are differential 2-cochains again. Suppose that (3-5) is such that we have formally the identity (3-7) Ô (u *' í) V = V T u * Ô í ν í í By means of universal formulas, we can prove : Proposition - The deformation ( 3 - 1 ) of ( N , .

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