Irreps and Off-shell Invariant Actions of the N-extended Supersymmetric Quantum Mechanics∗

The complete classification of the irreducible representations of theN-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmet-ric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson’s doublings of algebras in association with the N-extended su-persymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N = 8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechan-ical systems to the dimensional reduction of higher-dimensional supersymmetric theories