SO q (N) covariant differential calculus on quantum space and quantum deformation of schr�dinger equation

Carow-Watamura, Ursula; Schlieker, Michael; Watamura, Satoshi

doi:10.1007/bf01549697

Cited by 92 publications

(123 citation statements)

References 14 publications

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“…The matrixB IJ KL satisfy the characteristic equation, 14) and also (−1)Î +KBIJ KL satisfy the same equation as (4.14), whereÎ andK denote the grassmann parities of the new coordinates. Using the characteristic equation (4.14), we can easily decompose the matrixB IJ KL into projection operators, as said in section two.…”

Section: Deformed Phase Spacementioning

confidence: 97%

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Differential calculus on the quantum superspace and deformation of phase space

Kobayashi

Uematsu

1992

Z. Phys. C - Particles and Fields

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Section: Deformed Phase Spacementioning

confidence: 97%

“…We now extend the differential calculus on quantum space developed by Wess and Zumino [10] and others [11][12][13][14] to this quantum superspace. Here we require that the exterior derivative d given by…”

Section: Quantum Superspacementioning

confidence: 99%

Differential calculus on the quantum superspace and deformation of phase space

Kobayashi

Uematsu

1992

Z. Phys. C - Particles and Fields

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“…For their definition we refer the reader to Appendix A. On each of these quantum spaces exist two differential calculi [32][33][34] with…”

Section: Basic Ideas On Q-deformed Quantum Symmetriesmentioning

confidence: 99%

-Deformed quantum Lie algebras

Schmidt

Wachter

2006

Journal of Geometry and Physics

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Attention is focused on q-deformed quantum algebras with physical importance, i.e. U q (su 2 ), U q (so 4 ) and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which shall serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as q-analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to the undeformed case. *

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“…Their interest lies in the fact that they relate the SO q (N )-covariant derivatives ∂ i ,∂ i [3,31], introduced following the approach of Woronowicz and Wess-Zumino [37,38,39,36], to the inner derivations e a ,ē a dual to the frame and which are defined using ordinary commutation relations, following the approach of Connes [11,12].…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…We give then a brief overview of the work of Carow-Watamura, Schlieker, Watamura [3] and Ogievetski [30] on the construction of two differential calculi on R N q , which are based on theR-matrix formalism and are covariant with respect to SO q (N ). They both yield the de Rham calculus in the commutative limit.…”

Section: Introductionmentioning

confidence: 99%

Geometrical Tools for Quantum Euclidean Spaces

Cerchiai

Fiore

Madore

2001

Commun. Math. Phys.

106

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We apply one of the formalisms of noncommutative geometry to R N q , the quantum space covariant under the quantum group SO q (N ). Over R N q there are two SO q (N )-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of R 3 q . As in the case N = 3, one has to slightly enlarge the algebra R N q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N = 3 there is a conformal ambiguity in the natural metrics on the differential calculi over R N q . While in our previous article the frame was found 'by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of R N q with U q so(N ) into R N q , an interesting result in itself.

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SO q (N) covariant differential calculus on quantum space and quantum deformation of schr�dinger equation

Cited by 92 publications

References 14 publications

Differential calculus on the quantum superspace and deformation of phase space

Differential calculus on the quantum superspace and deformation of phase space

-Deformed quantum Lie algebras

Geometrical Tools for Quantum Euclidean Spaces

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