ON FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS

M. Daoud

Current address: 
Laboratoire de Physique de la Mati\`ere Condens\'ee, 
Facult\'e des Sciences, Universit\'e Ibn Zohr, 
BP 28/S, Agadir, Morocco., M. Kibler
 
Institut de Physique Nucl\'eaire de Lyon, IN2P3-CNRS et 
Universit\'e Claude Bernard, 
Villeurbanne Cedex, France

Two approaches of ${\cal N} = 2$ fractional
supersymmetric quantum mechanics of order $k$ are studied in
a complementary way. The first one, based on a generalized
Weyl--Heisenberg algebra $W_k$ (that comprizes the affine
quantum algebra $U_q(sl_2)$ with $q^k = 1$ as
a special case), apparently contains solely one 
bosonic degree of freedom. The second one uses
generalized bosonic and
$k$-fermionic degrees of freedom. As an illustration, a
particular emphasis is put on the fractional supersymmetric
oscillator of order $k$.