Maximal subalgebras of vector fields for equivariant quantizations

The elaboration of new quantization methods has recently developed the 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.