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Cartan connections and natural and projectively equivariant quantizations

P. Mathonet·F. Radoux
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Abstract

In this paper, we analyse the question of existence of a natural and projectively equivariant symbol calculus, using the theory of projective Cartan connections. We establish a close relationship between the existence of such a natural symbol calculus and the existence of an \sl(m+1,\R)-equivariant calculus over \R^{m} in the sense of [15,1]. Moreover we show that the formulae that hold in the non-critical situations over \R^{m} 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.

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P. Mathonet, F. Radoux. 2006-06-22. Cartan connections and natural and projectively equivariant quantizations. https://arxiv.org/abs/math/0606556

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