CARTAN CALCULUS OF Z3-GRADED DIFFERENTIAL CALCULUS ON THE QUANTUM PLANE

2012 ◽  
Vol 09 (05) ◽  
pp. 1250040
Author(s):  
ERDOĜAN MEHMET ÖZKAN
Keyword(s):  
Differential Calculus ◽  
Quantum Plane ◽  
Lie Derivatives ◽  
Noncommutative Differential Calculus

To give a Z3-graded Cartan calculus on the extended quantum plane, the noncommutative differential calculus on the extended quantum plane is extended by introducing inner derivations and Lie derivatives.

DEFORMED HARMONIC OSCILLATOR AND NONLINEAR COHERENT STATES: NONCOMMUTATIVE QUANTUM SPACE APPROACH

2009 ◽  
Vol 24 (10) ◽  
pp. 1963-1986 ◽  
Author(s):  
MOHAMMAD HOSSEIN NADERI ◽  
MAHMOOD SOLTANOLKOTABI ◽  
RASOUL ROKNIZADEH
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Differential Calculus ◽  
Quantum Plane ◽  
Quantum Harmonic Oscillator ◽  
Commutation Relations ◽  
Noncommutative Differential Calculus ◽  
Two Parameter ◽  
Noncommutative Quantum ◽  
Space Approach

In this paper, by using the Wess–Zumino formalism of noncommutative differential calculus, we show that the concept of nonlinear coherent states originates from noncommutative geometry. For this purpose, we first formulate the differential calculus on a GL p, q(2) quantum plane. By using the commutation relations between coordinates and their interior derivatives, we then construct the two-parameter (p, q)-deformed quantum phase space together with the associated deformed Heisenberg commutation relations. Finally, by applying the obtained results for the quantum harmonic oscillator we construct the associated coherent states, which can be identified as nonlinear coherent states. Furthermore, we show that some of the well-known deformed (nonlinear) coherent states, such as two-parameter (p, q)-deformed coherent states, Maths-type q-deformed coherent states, Phys-type q-deformed coherent states and Quesne deformed coherent states, can be easily obtained from our treatment.

scholarly journals n=3 differential calculus and gauge theory on a reduced quantum plane

2003 ◽  
Vol 44 (10) ◽  
pp. 4784 ◽  
Author(s):  
M. El Baz ◽  
A. El Hassouni ◽  
Y. Hassouni ◽  
E. H. Zakkari
Keyword(s):  
Gauge Theory ◽  
Differential Calculus ◽  
Quantum Plane

scholarly journals The Two-Parameter Higher-Order Differential Calculus and Curvature on a Quantum Plane

2007 ◽  
Vol 17 (4) ◽  
pp. 651-662 ◽  
Author(s):  
M. El Baz ◽  
A. El Hassouni ◽  
Y. Hassouni ◽  
E. H. Zakkari
Keyword(s):  
Differential Calculus ◽  
Higher Order ◽  
Quantum Plane ◽  
Two Parameter

scholarly journals DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE AT A CUBIC ROOT OF UNITY

2000 ◽  
Vol 12 (02) ◽  
pp. 227-285 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA ◽  
R. TRINCHERO
Keyword(s):  
Representation Theory ◽  
Quantum Group ◽  
Lorentz Group ◽  
Differential Calculus ◽  
Differential Algebra ◽  
Space Time ◽  
Quantum Plane ◽  
Root Of Unity ◽  
Finite Dimensional

We consider the algebra of N×N matrices as a reduced quantum plane on which a finite-dimensional quantum group ℋ acts. This quantum group is a quotient of [Formula: see text], q being an Nth root of unity. Most of the time we shall take N=3; in that case dim(ℋ)=27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group ℋ also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of ℋ. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

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