  We propose an heuristic rule for the area transformation on the
non-commutative plane. The non-commutative area preserving transformations are
quantum deformation of the classical symplectic diffeomorphisms. Area
preservation condition is formulated as a field equation in the non-commutative
Chern-Simons gauge theory. The higher dimensional generalization is suggested
and the corresponding algebraic structure - the infinite dimensional $\sin$-Lie
algebra is extracted. As an illustrative example the second-quantized
formulation for electrons in the lowest Landau level is considered.
