2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/189633Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the $ν=1$ integer quantum Hall effect (IQHE) state on the fractional quantum Hall effect (FQHE) $ν=\frac{1}{2p+1}$ FQHE states. Using the hydrodynamical description of the quantum Hall fluid, it is shown that these transformations are generated by vector fields satisfying the Gauss law in the interacting non-commutative Chern-Simons gauge theory, and the corresponding field-theory Lagrangian is reconstructed. It is demonstrated that the geometric transformations induce quantum-mechanical non-unitary similarity transformations, establishing the interplay between integral and fractional QHEs.4 pages, no figures, minor corrections, accepted in Eur. Phys. J. CHigh Energy Physics - Theorytext