Common aspects of q-deformed Lie algebras and fractional calculus

Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the first time allows a smooth transition between different Lie algebras. The corresponding fractional q-number is derived for a fractional harmonic oscillator. It is shown that the resulting energy spectrum is an appropriate tool to describe, for example, the ground-state spectra of even–even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the Bα(E2) values for the fractional q-deformed symmetric rotor are calculated. A first interpretation of half-integer representations of the fractional rotation group is given in terms of a description of K=1/2− band spectra of odd–even nuclei.