A relation between Z3-graded symmetry and shape invariant supersymmetric systems

Abstract

We study a notable property of shape invariant supersymmetric quantum mechanical systems. Particularly, we demonstrate that each shape invariant supersymmetric system can constitute a Z3-graded symmetric algebra. The latter is known from the literature to provide topological invariants which are generalizations of the Witten index. In addition, we relate the Z3-graded algebra to the generators of a deformed Lie algebra underlying each shape invariant system. We generalize the results to the case of sequential shape invariant systems, in which case we find a sequence of Z3-graded algebras. Finally, we present an example of shape invariant supersymmetric quantum system for which we express the elements of the Z3 in terms of the operators that constitute the supersymmetric algebra of the quantum system. In view of the fact that the shape invariance condition is somewhat an additional algebraic condition to supersymmetric quantum systems, with no origin to some concrete algebraic structure, our results might be useful towards this line of research.