Generalized Transvectants Rankin–Cohen Brackets

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Generalized Transvectants-Rankin–Cohen Brackets V. OVSIENKO1 and P. REDOU2 1CNRS, Institut de Girard Desargues, URA CNRS 746, Universite Claude Bernard - Lyon I, 43 bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. e-mail: 2Ecole Nationale d'Ingenieurs de Brest Technopole Brest Iroise, Parvis Blaise Pascal, BP 30815, 29608 Brest Cedex, France. e-mail: (Received: 30 July 2002) Abstract. We introduce o? p? 1; q? 1?-invariant bilinear differential operators on the space of tensor densities on Rn generalizing the well-known bilinear sl2-invariant differential opera- tors in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear o? p? 1; q? 1?-invariant differential operators given by powers of the Laplacian. Mathematics Subject Classifications (2000). 53A30, 53A55. Key words. conformal structures, invariant differential operators, modules of differential operators, tensor densities. 1. Introduction In the one-dimensional case, the problem of classification of SL2-invariant (bi)linear differential operators was treated in the classical literature. Consider the following action of SL?2; R? on the space of (smooth) functions in one variable, for instance, on R;S1, or holomorphic functions on the upper half- plane: f ?x? 7! f ax? b cx? d ?cx? d?2l; ?1:1?

  • dl?rn? ? jlnt

  • lie algebra

  • invariant differential

  • generalized transvectants-rankin–cohen

  • bilinear conformally invariant

  • differential operators

  • rxx ?

  • invariant linear

  • operators


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Published 01 November 1918
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Letters in Mathematical Physics63:19–28, 2003. #2003Kluwer Academic Publishers. Printed in the Netherlands.
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Generalized Transvectants-Rankin–Cohen Brackets
1 2 V. OVSIENKO and P. REDOU 1 CNRS, Institut de Girard Desargues, URA CNRS 746, Universite´ Claude Bernard - Lyon I, 43bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. e-mail: valentin.ovsienko@cpt.univ-mrs.fr 2 Ecole Nationale d’Ingenieurs de Brest Technopole Brest Iroise, Parvis Blaise Pascal, BP 30815, 29608 Brest Cedex, France. e-mail: redou@enib.fr
(Received: 30 July 2002)
Abstract.We introduce oðpþ1;qþ1Þ-invariant bilinear differential operators on the space n of tensor densities onRgeneralizing the well-known bilinear sl2-invariant differential opera-tors in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear oðpþ1;qþ1Þ-invariant differential operators given by powers of the Laplacian.
Mathematics Subject Classifications (2000).53A30, 53A55.
Key words.conformal structures, invariant differential operators, modules of differential operators, tensor densities.
1. Introduction In the one-dimensional case, the problem of classification of SL2-invariant (bi)linear differential operators was treated in the classical literature. Consider the following action of SLð2;RÞon the space of (smooth) functions in 1 one variable, for instance, onR;S, or holomorphic functions on the upper half-plane:   axþb 2l fðxÞ 7 !fðcxþdÞ;ð1:1Þ cxþd
wherelis a parameterl2R(orC). This SLð2;RÞ-module of functions is called the space ofl-densities and denotedFl. The classification of SLð2;RÞ-invariant linear differential operators fromFltoFm (i.e. of the operators commuting with the action (1.1)) was obtained in classical works on projective differential geometry, namely, for everyk¼1;2;. . .;there exists a unique (up to a constant) SLð2;RÞ-invariant linear differential operator of orderk:
Ak:F!F: 1k1þk 2 2
k k It is given byAkðfÞ ¼df=dx.
ð1:2Þ