General one-loop renormalization group evolutions and electroweak symmetry breaking in the (M+1)SSM

We study analytically the general features of electroweak symmetry breaking in the context of the Minimal Supersymmetric Standard Model extended by one Higgs singlet. The exact analytical forms of the renormalization group evolutions of the Yukawa couplings and of the soft supersymmetry breaking parameters are derived to one-loop order. They allow on one hand controllable approximations in closed analytical form, and on the other a precise study of the behaviour of infrared quasi fixed point regimes which we carry out. Some of these regimes are shown to be phenomenologically inconsistent, leading to too small an effective $\mu$-parameter. The remaining ones serve as a suitable benchmark to understand analytically some salient aspects, often noticed numerically in the literature, in relation to the electroweak symmetry breaking in this model. The study does not need any specific assumption on $\tan \beta$ or on boundary conditions for the soft supersymmetry breaking parameters, thus allowing a general insight into the sensitivity of the low energy physics to high energy assumptions.Comment: Latex, 41 pages, 7 figure

Fractional Supersymmetry and Infinite Dimensional Lie Algebras

In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation \D of any Lie algebra \g. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of \g this infinite dimensional Lie algebra, containing the Lie algebra \g as a sub-algebra, is explicitly constructed.Comment: 8 pages, D.V.Volkov Memorial Conference ``Supersymmetry and Quantum Field Theory'', Kharkov, July 25-29, 2000), two figure

Local Fractional Supersymmetry for Alternative Statistics

A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in $1D$ supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry {\it i.e.} a fractional supergravity which is then quantized {\it \`a la Dirac} to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline.Comment: 15 pages, latex, no figur

Hopf algebras for ternary algebras

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order three. It turns out that these variables are the variables which generate the three-exterior algebra.Comment: 21 page