Electric Chern–Simons term, enlarged exotic Galilei symmetry and noncommutative plane

The extended exotic planar model for a charged particle is constructed. It includes a Chern–Simons-like term for a dynamical electric field, but produces usual equations of motion for the particle in background constant uniform electric and magnetic fields. The electric Chern–Simons term is responsible for the noncommutativity of the boost generators in the 10-dimensional enlarged exotic Galilei symmetry algebra of the extended system. The model admits two reduction schemes by the integrals of motion, one of which reproduces the usual formulation for the charged particle in external constant electric and magnetic fields with associated field-deformed Galilei symmetry, whose commuting boost generators are identified with the nonlocal in time Noether charges reduced on-shell. Another reduction scheme, in which electric field transmutes into the commuting space translation generators, extracts from the model a free particle on the noncommutative plane described by the twofold centrally extended Galilei group of the nonrelativistic anyons.