| Summary: | The simplest examples, braided lines and planes, yield the full supersymmetry transformation and super-Poincaré algebra in one and two dimensions, as well as the generalisation of this to the case of fractional supersymmetry. In terms of this deformed geometry these algebras/transformations have straightforward interpretations. For example, in one dimension, (fractional) supersymmetry transformations arise as translations along the braided line in the limit as its deformation parameter goes to a root of unity, so that a theory is (fractionally) supersymmetric if it is invariant under such translations. The various structures of which one makes use when working with supersymmetry, for example the supercharge, covariant derivative, Berezin integral and superspace, arise naturally in the context of this deformed geometry, as do their fractional analogues. The properties of the <I>q</I>-deformed bosonic oscillator algebra in the limit when <I>q</I> goes to a root of unity are also discussed, and this algebra is found to decompose into the direct product of an ordinary bosonic oscillator algebra and an anyonic oscillator algebra (fermionic when <I>q</I> = -1). The corresponding Fock space decomposition is also studied. Using these results and the Schwinger realisation of <I>U<SUB>q</SUB></I>(<I>sl</I>(2)), we obtain a similar decomposition for this algebra. Motivated by this results we study the complete <I>U<SUB>q</SUB></I>(<I>sl</I>(2)) Hopf algebra in the limit as its deformation parameter goes to a root of unity. This leads to new Hopf algebras which are (fractionally) supersymmetric analogues of <I>U<SUB>q</SUB></I>(<I>sl</I>(2)), and also to a novel point of view on the origin of intrinsic spin. The properties of the quantum hermitian matrices<I> L<SUB>q</SUB></I>(2) when <I>q</I> is a root of unity are also discussed, as are those for the braided form of <I>Lq</I>(2). These also have novel structure, and can be interpreted as providing (fractionally) supersymmetric analogues of <I>GL</I><SUB>q</SUB>(2). The physical interpretation of these results suggests that both four dimensional supersymmetry and the four dimensional Dirac equation have their origin in deformed geometry.
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