A Novel Generalisation of Supersymmetry: Quantum ${\mathbb{Z}}_2^2$-Oscillators and their `superisation'

We propose a very simple toy model of a ${\mathbb{Z}}_2^2$-supersymmetric quantum system and show, via Klein’s construction, how to understand the system as being an $N=2$ supersymmetric system with an extra ${\mathbb{Z}}_2^2$-grading. That is, the commutation/anticommutation rules are defined via the standard boson/fermion rules, but the system still has an underlying ${\mathbb{Z}}_2^2$-grading that needs to be taken into account.