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Unicity of the Lie product

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Unicity of the Lie product COMPOSITIO MATHEMATICA SEBASTIAN J. VAN STRIEN Unicity of the Lie product Compositio Mathematica, tome 40, no 1 (1980), p. 79-85. <> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 79 UNICITY OF THE LIE PRODUCT Sebastian J. van Strien COMPOSITIO MATHEMATICA, Vol. 40, Fasc. 1, 1980, pag. 79-85 @ 1980 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands 1. Statement of the result For a C°° manifold M, .!(M) denotes the linear space of Coo vectorfields on M. Let X: lF(M) x 1’(M)-+1’(M) be a bilinear opera- tor, defined for every n dimensional manifold M. This operator is called natural if for every smooth open embedding f : N --&#x3E;M the following diagram commutes: where M, N are Coo manifolds and f * is the composition iF(M) r X(f(N» - lF(N), r the restriction operator, i.e. f*X(x) = df(x)-I(X(f(x») for X E lf(M). In this note 1 shall prove that the Lie-product ([X, Y] = X - Y - Y - X for X, Y e X(M» is charac- terised by this property: THEOREM: Let X be a bilinear natural operator in the above sense, then there exists a constant À E R such that X(X, Y) = A . [X, Y], for all X, Y E X(M). Palais and others [3], [4], [5] prove analogous results for operations on differential forms. Peetre [6] has a similar characterisation of linear (not bilinear) differential operators. The formal techniques are similar to those in [7]. 1 am i

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