QUANTUM SUPERSPACE, Q-EXTENDED SUPERSYMMETRY AND PARASUPERSYMMETRIC QUANTUM-MECHANICS

We describe q-deformation of the extended supersymmetry and construct q-extended supersymmetric Hamiltonian. For this purpose we formulate q -superspace formalism and construct q-supertransformation group. On th is basis q-extended supersymmetric Lagrangian is built. The canonical quantization of this system is considered. The connection with multi-d imensional matrix representations of the parasupersymmetric quantum me chanics is discussed and q-extended supersymmetric harmonic oscillator is considered as a simplest example of the described constructions. W e show that extended supersymmetric Hamiltonians obey not only extende d SUSY but also the whole family of symmetries (q-extended supersymmet ry) which is parametrized by continuous parameter q on the unit circle .