q-graded Heisenberg algebras and deformed supersymmetries

The notion of q-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for q complex number in the unit disk. Within this formulation, we consider the extension of the notion of supersymmetry in the enveloping algebra. We recover the ordinary Z2 grading or Grassmann parity for associative superalgebra and a modified version of the usual supersymmetry. As a specific problem, we focus on the interesting limit q→−1 for which the Arik and Coon deformation [J. Math. Phys. 17, 524 (1976)] of the Heisenberg algebra allows one to map fermionic modes to bosonic ones in a modified sense. Different algebraic consequences are discussed.