Matrix Formulation of Fractional Supersymmetry and q-Deformation

Supersymmetry, which is the only non-trivial $Z_{2}$ extension of
the Poincar\'e algebra, can be generalized to fractional
supersymmetry, when the space time is smaller than 3. Since
symmetries play an important role in physics; the principal task of
quantum groups consist in extanding these standard symmetries to
the deformed ones, which might be used in physics as well. This two
aspects will be the main focus of this thesis. In this work, we
discuss the matrix formulation of fractional supersymmetry, the
q-deformation of KdV hierarchy systems and noncommutative geometry.
In the first part fractional supersymmetry generated by more than
one charge operator and those which can be described as a matrix
model are studied. Using parafermionic field-theoretical methods,
the fundamentals of two-dimensional fractional supersymmetry
$Q^{k}=P$ are set up. Known difficulties induced by methods based
on the $U_{q}(sl(2))$ quantum group representations and
noncommutative geometry are avoided in the parafermionic approach.
Moreover, we find that fractional supersymmetric algebras are
naturally realized as matrix models. The $k=3$ case is studied in
detail. In the second part we will study the q-deformed algebra and
the q-analogues of the generalised KdV hierarchy. We construct in
this part the algebra of q-deformed pseudo-differential operators,
shown to be an essential step toward setting up a q-deformed
integrability program. In fact, using the results of this
q-deformed algebra, we derive the q-analogues of the generalised
KdV hierarchy. We focus in particular on the first leading orders
of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq
integrable systems. We also present the q-generalisation of the
conformal transformations of the currents $u_{n}$, $n\geq 2$, and
discuss the primary condition of the fields $w_{n}$, $n\geq 2$, by
using the Volterra gauge group transformations for the q-covariant
Lax operators. In the last part we will discuss quantum groups and
noncommutative space. All studies in this part are based on the
idea of replacing the ordinary coordinates with non commuting
operators. We will also formulate some aspects of noncommutative
geometry mathematically and we will be mainly concerned with
quantum algebra and quantum spaces.