Benkaddour, Ilham (2006): Matrix Formulation of Fractional Supersymmetry and qDeformation. Dissertation, LMU München: Faculty of Physics 

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Abstract
Supersymmetry, which is the only nontrivial $Z_{2}$ 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 qdeformation 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 fieldtheoretical methods, the fundamentals of twodimensional 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 qdeformed algebra and the qanalogues of the generalised KdV hierarchy. We construct in this part the algebra of qdeformed pseudodifferential operators, shown to be an essential step toward setting up a qdeformed integrability program. In fact, using the results of this qdeformed algebra, we derive the qanalogues of the generalised KdV hierarchy. We focus in particular on the first leading orders of this qdeformed hierarchy, namely the qKdV and qBoussinesq integrable systems. We also present the qgeneralisation 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 qcovariant 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.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  Fractional Supersymmetry, noncommutative geometry, integrable systems 
Subjects:  600 Natural sciences and mathematics > 530 Physics 600 Natural sciences and mathematics 
Faculties:  Faculty of Physics 
Language:  English 
Date Accepted:  27. April 2006 
1. Referee:  Wess, Julius 
Persistent Identifier (URN):  urn:nbn:de:bvb:1953038 
MD5 Checksum of the PDFfile:  2a1a558f51cfc253bf9c12600aa4d586 
Signature of the printed copy:  0001/UMC 15356 
ID Code:  5303 
Deposited On:  23. May 2006 
Last Modified:  16. Oct 2012 07:57 