Transitive Lie Algebras of Nilpotent Vector Fields and their Tanaka Prolongations

Transitive nilpotent local Lie algebras of vector fields can be easily constructed from dilations h of Rn with positive weights (give me a sequence of n positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on Rn ) as the Lie algebras g<0(h) of the polynomial vector fields of negative weights with respect to h. We provide a condition for the dilation h such that the Lie algebras of polynomial vectors defined by h are exactly the Tanaka prolongations of the corresponding nilpotent Lie algebras g<0(h) . However, in some cases of dilations h we can find some ‘strange’ elements of the Tanaka prolongations of g<0(h) , which we describe in detail. In particular, we give a complete description of derivations of degree 0 for the Lie algebra g<0(h) .