In this paper some properties of the irreducible multiplets of representation for the N = (p, q) -extended supersymmetry in one dimension are discussed. Essentially two results are here presented. At first a peculiar property of the one dimension is exhibited, namely that any multiplet containing 2d (d bosonic and d fermionic) particles in M different spin states, is equivalent to a {d, d} multiplet of just 2 spin states (all bosons and all fermions being grouped in the same spin). Later, it is shown that the classification of all multiplets of this kind carrying an irreducible representation of the N -extended supersymmetry is in one-to-one correspondence with the classification of real-valued Clifford Γ-matrices of Weyl type. In particular, p + q is mapped into D, the space-time dimensionality, while 2d is determined to be the dimensionality of the corresponding Γ-matrices. The implications of these results to the theory of spinning particles are analyzed. *

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.

Using the worldline SU (2|1) superfield approach, we construct N = 4 superconformally invariant actions for the d = 1 multiplets (1, 4, 3) and (2, 4, 2). The SU (2|1) superfield framework automatically implies the trigonometric realization of the superconformal symmetry and the harmonic oscillator term in the corresponding component actions. We deal with the general N = 4 superconformal algebra D(2, 1; α) and its central-extended α = 0 and α = −1 psu(1, 1|2) ⊕ su(2) descendants. We capitalize on the observation that D(2, 1; α) at α = 0 can be treated as a closure of its two su(2|1) subalgebras, one of which defines the superisometry of the SU (2|1) superspace, while the other is related to the first one through the reflection of µ, the parameter of contraction to the flat N = 4, d = 1 superspace. This closure property and its α = 0 analog suggest a simple criterion for the SU (2|1) invariant actions to be superconformal: they should be even functions of µ. We find that the superconformal actions of the multiplet (2, 4, 2) exist only at α = −1, 0 and are reduced to a sum of the free sigma-model type action and the conformal superpotential yielding, respectively, the oscillator potential ∼ µ 2 and the standard conformal inverse-square potential in the bosonic sector. The sigma-model action in this case can be constructed only on account of non-zero central charge in the superalgebra su(1, 1|2).

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.

Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the G 2 cosets over the Lorentz algebras, the identities satisfied by the higher-rank antisymmetric octonionic tensors and so on. Applications of these results range from the classification of octonionic generalized supersymmetries, the construction of octonionic superstrings, as well as the investigations concerning the recently discovered octonionic M-superalgebra and its superconformal extension.