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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. <http://www.numdam.org/item?id=CM_1980__40_1_79_0> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). 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 http://www.numdam.org/ 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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