On a classification of faithful representations of the Galilean lie algebra on the polynomial ring in three variables

We show a complete classification of faithful representations of the 2 + 1 space-times Galilean Lie algebra on the polynomial ring in three variables, where actions of the Galilean Lie algebra are given by derivations with coefficients of degree at most one. In particular, all such representations of the Galilean Lie algebra are explicitly constructed and classified by one parameter. In a more general setting we show that, with respect to a nonzero abelian ideal of a finite-dimensional Lie algebra, there is at most one such representation up to graded-equivalence.