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Journal of the London Mathematical Society Advance Access published online on August 23, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm030
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© 2007 London Mathematical Society

Cartan connections and natural and projectively equivariant quantizations

P. Mathonet and F. Radoux

University of Liége
Institute of Mathematics
Grande Traverse 12 - B37
B-4000 Liége
Belgium
Fabian.Radoux{at}uni.lu

In this paper, the question of existence of a natural and projectively equivariant symbol calculus is analysed using the theory of projective Cartan connections. A close relationship is established between the existence of such a natural symbol calculus and the existence of an sl(m + 1, R)-equivariant calculus over Rm. Moreover, it is shown that the formulae that hold in the non-critical situations over Rm for the sl(m + 1,R)-equivariant calculus can be directly generalized to an arbitrary manifold by simply replacing the partial derivatives by invariant differentiations with respect to a Cartan connection.

p.mathonet{at}ulg.ac.be

2000 Mathematics Subject Classification 53B10, 53C10, 22E46.

The second author thanks the Belgian FRIA for his Research Fellowship.

Received January 26, 2006; revised September 20, 2006;
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