Title:

A braided geometric view of fractional supersymmetry

The simplest examples, braided lines and planes, yield the full supersymmetry transformation and superPoincaré 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 qdeformed bosonic oscillator algebra in the limit when q 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 q = 1). The corresponding Fock space decomposition is also studied. Using these results and the Schwinger realisation of U_{q}(sl(2)), we obtain a similar decomposition for this algebra. Motivated by this results we study the complete U_{q}(sl(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 U_{q}(sl(2)), and also to a novel point of view on the origin of intrinsic spin. The properties of the quantum hermitian matrices L_{q}(2) when q is a root of unity are also discussed, as are those for the braided form of Lq(2). These also have novel structure, and can be interpreted as providing (fractionally) supersymmetric analogues of GL_{q}(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.
