Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

100identify X and X + X and say by some abuse of language that X itself isan infinitesimal CR automorphism.In the algebraic category, the main drawback of infinitesimal CR automorphismis that they do not have algebraic flow. For instance, the complexdilatation vector field X = iz ∂ z has transcendent flow, even if it is an infinitesimalCR automorphism of every algebraic hypersurface in C 2 whoseequation is of the form v = ϕ(z¯z), even if the coefficient of X is algebraic.Thus instead of infinitesimal CR automorphisms which generate oneparametergroups of biholomorphic transformations of M, we shall studyalgebraically dependent one-parameter families of biholomorphic transformations(not necessarily making a one parameter group). To begin with, weneed to introduce some precise definitions about local algebraic Lie transformationgroups.2.3. Local Lie group actions in the K-algebraic category. Often in real orin complex analytic geometry, the interest cannot be focalized on global Lietransformation groups, but only on local transformations which are close tothe identity. For instance, the transformation group of a small piece of a realanalytic CR manifold in C n which is not contained in a global, large or compactCR manifold is almost never a true, global transformation group. Consequentlythe usual axioms of Lie transformation groups must be localized.Philosophically speaking, the local point of view is often the most adequateand the richest one, because a given analytico-geometric object often possessesmuch more local invariant than global invariants, if any. Historicallyspeaking, the local Lie transformation groups were first studied, before theintroduction of the now classical notion of global Lie group. Especially, inhis first masterpiece work [Lie1880] on the subject, Sophus Lie essentiallydealt with local “Lie” groups: he classified all continuous local transformationgroups acting on an open subset of C 2 . This general classificationprovided afterwards in the years 1880–1890 many applications to the localstudy of differential equations: local normal forms, local solvability, etc.In this paragraph we define precisely local actions of local Lie groups andwe focus especially on the K-algebraic category.Let c ∈ N ∗ , let g = (g 1 , . . .,g c ) ∈ K c and let two positive numberssatisfy 0 < δ 2 < δ 1 . We formulate the desired definition by means of thetwo precise polydiscs ∆ c (δ 2 ) ⊂ ∆ c (δ 1 ) ⊂ K c . A local K-algebraic Liegroup of dimension c consists of the following data:(1) A K-algebraic multiplication mapping µ : ∆ c (δ 2 ) × ∆ c (δ 2 ) → ∆ c (δ 1 )which is locally associative (µ(g, µ(g ′ , g ′′ )) = µ(µ(g, g ′ ), g ′′ )), wheneverµ(g ′ , g ′′ ) ∈ ∆ c (δ 2 ), µ(g, g ′ ) ∈ ∆ c (δ 2 ) and which satisfiesµ(0, g) = µ(g, 0) = g, where the origin 0 ∈ K c corresponds to theidentity element in the group structure.