Jamila Douari

TITLE: Unified Particles Algebra and Non-Commutative Geometry

ABSTRACT: The non-commutative geometry is redefined by referring to statistical properties of our space. We consider an excitation operator on two-dimensional space. Thus excited annihilation and creation operators are defined and an excited particles algebra is given as unified symmetry of bosonic, fermionic and anyonic algebras, depending on the kind of statistics of the space in consideration. If the dimension of space d is greater than or equal to 3, and the statistical parameter \nu=0,1, we refind the bosonic and fermionc algebras respectively. For d=2, the unified algebra is the anyonic one, and it is constructed by considering an excitation applied to ordinary particles living in non-commutative 2-dimensional space. Then, the Heisenberg algebra is extended by a polynomial in some disorder operator denoted K_i and deformed in terms of the statistical parameter. By using the excited particles operators we realize a deformed \omega_\infty-algebra depending on the statistical parameter that we call in this work Excited Particles \omega_\infty-algebra. The Hamiltonian is given as a deformed "free particles" Hamiltonian extended by some statistical term.

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