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