Theq-calculus for genericq andq a root of unity

Int J of Quantum Chemistry

Kibler

2003

Two complementary approaches of ᏺ ϭ 2 fractional supersymmetrical quantum mechanics of order k are studied in this article. The first, based on a generalized Weyl-Heisenberg algebra W k (which comprises the affine quantum algebra U q (sl 2 ) with q k ϭ 1 as a special case), apparently contains solely one bosonic degree of freedom. The second uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetrical oscillator of order k.

Two approaches to fractional supersymmetrical quantum mechanics

Int J of Quantum Chemistry

Kibler

2003

On Supersymmetric Quantum Mechanics

Kibler

Fundamental World of Quantum Chemistry

2004

This paper constitutes a review on N = 2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W k. It is shown how a general Hamiltonian can be associated with the algebra W k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Pöschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k = 2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W k , of the N = 2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.

K-Fractional spin through Q-deformed (super)-algebras

Mansour

Reports on Mathematical Physics

Hassouni

1999

The splitting of a Q-deformed boson, in the Q ! q = e 2i k limit, is discussed. The equivalence between a Q-fermion and an ordinary one is established. The properties of quantum algebras U Q (sl(m)), the quantum superalgebras osp Q (1=2m) and U Q (sl(m=1)) and the deformed Virasoro algebra when their deformation parameter Q goes to a root of unity, are investigated. These properties are shown to be related to fractional supersymmetry and k-fermionic spin.