Summary: We propose a heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformations of the classical symplectic diffeomorphisms. The area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. A 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.