Maximal subalgebras of vector fields for equivariant quantizations (Q2774784)

The authors' interest in the present study comes from recent works about new equivariant quantizations [\textit{C. Duval}, \textit{P. Lecomte} and \textit{V. Ovsienko}, Ann. Inst. Fourier 49, 1999-2029 (1999; Zbl 0932.53048) and \textit{P. Lecomte} and \textit{V. Ovsienko}, Lett. Math. Phys. 49, 173-196 (1999; Zbl 0989.17015)]. The elaboration of new quantization methods has recently led to an interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables that are polynomial in the momenta and differential operators. Here, we determine which finite dimensional graded Lie subalgebras are maximal. In order to characterize these, we make use of results of Guillemin, Singer, and Sternberg and Kobayashi and Nagano.