Equivariant quantizations and Cartan connections (Q998097)

This is mostly a review article of results of the contribution of P. Mathonet and his collaborators in the field of \(G\)-equivariant quantization. After reviewing the results referring to projectively equivariant quantization [\textit{P. B. A. Lecomte} and \textit{V. Yu. Ovsienko}, Lett. Math. Phys. 49, 173--196 (1999; Zbl 0989.17015)], P. Mathonet recalls some facts about equivariant quantization with respect to irreducible filtered finite dimensional transitive algebras [\textit{F. Boniver} and \textit{P. Mathonet}, J. Math. Phys. 42, 582--589 (2001; Zbl 1032.17041); \textit{F. Boniver} and \textit{P. Mathonet}, J. Geom. Phys. 56, 712--730 (2006; Zbl 1145.53069)]. The new examples concern the equivariant quantization associated with symplectic and orthogonal algebras. A small chapter is devoted to projectively equivalent connections. After recalling Bordeman's existence theorem connected with Thomas-Whithead connections, P. Mathonet summarizes the results obtained in collaboration with F. Radoux [\textit{P. Mathonet} and \textit{F. Radoux}, Lett. Math. Phys. 72, No. 3, 183--196 (2005; Zbl 1091.53006)]. The main theorem is an explicit formula for quantization in terms of the Cartan connection associated to a projective class of connections.