Planar Carrollean dynamics, and the Carroll quantum equation

We expand on the known result that Carroll algebra in $2+1$ dimensions admits two non-trivial central extensions by computing associated Lie group, which we call extended group. The symplectic geometry to this group is then computed describe motion of planar elementary particles, free case, when coupled an electromagnetic field, and a gravitational field. compare motions particles $3+1$ same conditions, also give dynamics with spin. In background, differ from ones due 2 new Casimir invariants, turn out be non-trivial. coupling field leaves trivial, however. Finally, obtain quantum equation obeyed wave functions \textit{via} geometric quantization.