We present an algorithmic classification of the irreps of the N -extended onedimensional supersymmetry algebra linearly realized on a finite number of fields. Our work is based on the 1-to-1 [1] correspondence between Weyl-type Clifford algebras (whose irreps are fully classified) and classes of irreps of the N -extended 1D supersymmetry. The complete classification of irreps is presented up to N ≤ 10. The fields of an irrep are accommodated in l different spin states. N = 10 is the minimal value admitting length l > 4 irreps. The classification of length-4 irreps of the N = 12 and real N = 11 extended supersymmetries is also explicitly presented.Tensoring irreps allows us to systematically construct manifestly (N -extended) supersymmetric multi-linear invariants without introducing a superspace formalism. Multi-linear invariants can be constructed both for unconstrained and multilinearly constrained fields. A whole class of off-shell invariant actions are produced in association with each irreducible representation. The explicit example of the N = 8 off-shell action of the (1, 8, 7) multiplet is presented.Tensoring zero-energy irreps leads us to the notion of the fusion algebra of the 1D N -extended supersymmetric vacua.

The linear finite irreducible representations of the algebra of the 1D N -Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to N ≤ 8.

The first-order differential Lévy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of (1 + d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural Z 2 -graded Lie symmetry.In this paper we show that, for a certain class of supersymmetric PDE's, a natural Z 2 ×Z 2graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the (1 + 1)-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a Z 2 × Z 2 -graded Lie superalgebra, realized by first and secondorder differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the Z 2 -and Z 2 × Z 2 -graded extensions are no longer allowed.The construction of the Z 2 × Z 2 -graded Lie symmetry of the (1 + 2)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra. The fact that a Lie superalgebra appears even in a purely bosonic setting is not so surprising. Indeed, for the harmonic oscillator, the old results of [20] can be expressed, in modern language, by stating that the Fock vacuum of creation/annihilation operators can be replaced by a lowest weight representation of an osp(1|2) spectrum-generating superalgebra.

We furnish an algebraic understanding of the inequivalent connectivities (computed up to N ≤ 10) of the graphs associated to the irreducible supermultiplets of the N -extended Supersymmetric Quantum Mechanics. We prove that the inequivalent connectivities of the N = 5 and N = 9 irreducible supermultiplets are due to inequivalent decompositions into two sets of N = 4 (respectively, N = 8) supermultiplets. "Oxido-reduction" diagrams linking the irreducible supermultiplets of the N = 5, 6, 7, 8 supersymmetries are presented. We briefly discuss these results and their possible applications.

We propose to use the second Hopf map for the reduction (via SU (2) group action) of the eight-dimensional N = 8 supersymmetric mechanics to five-dimensional supersymmetric systems specified by the presence of an SU (2) Yang monopole. For our purpose we develop the relevant reduction procedure. The reduced system is characterized by its invariance under the N = 5 or N = 4 supersymmetry generators (with or without an additional conserved BRST charge operator) which commute with the su(2) generators.