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Benkaddour, Ilham (2006): Matrix Formulation of Fractional Supersymmetry and q-Deformation. Dissertation, LMU München: Faculty of Physics
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Abstract

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.