Reality in the differential calculus on q-Euclidean spaces

Ogievetsky, O.; Zumino, Bruno

doi:10.1007/bf00398308

Cited by 40 publications

(75 citation statements)

References 6 publications

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“…Thus the factors ρ −1 i and ρ −i −1 in (41) provide deforming maps and we have the following Theorem 5.1 The q-differential algebra Diff c SOq (N) is isomorphic to the q-difference calculus of the same dimension α∈I Diff q k(i) (x i ) with k(i) = 2 for i > 0, k(i) = −2 for i < 0, and k(0) = 1 as outlined in proposition 4.1. ✷ We note that this result is not covered by the treatment in [17]. As mentioned above in the differential calculus on a commutative space obtained in that paper is not consistent with an involution on the coordinate algebra.…”

Section: Q-differential Algebras Coming From Orthogonal Quantum Groupsmentioning

confidence: 71%

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On the deformability of Heisenberg algebras

Pillin

1996

Commun.Math. Phys.

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Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.

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Section: Q-differential Algebras Coming From Orthogonal Quantum Groupsmentioning

confidence: 71%

“…This element can be expressed in terms of the generators of V q (N) and the partial derivatives [18]. A straightforward calculation gives the following almost commutative relations:…”

Section: Q-differential Algebras Coming From Orthogonal Quantum Groupsmentioning

confidence: 99%

On the deformability of Heisenberg algebras

Pillin

1996

Commun.Math. Phys.

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“…In this section we construct the phase space of (V, r V ), where V := R n , for a standard r-matrix on so(n, R), using methods which are completely analogous to those used in [24] for the investigation of a real differential calculus on quantum Euclidean spaces.…”

Section: Crossed Product Phase Spaces and Quasitriangularitymentioning

confidence: 99%

Phase spaces related to standard classicalr-matrices

Zakrzewski¹

1996

J. Phys. A: Math. Gen.

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Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding phase spaces (symplectic groupoids). IntroductionThe recent development of noncommutative geometry and, in particular, the theory of quantum groups, raises the question what happens with known models of physical systems when we pass from usual configurations to non-commutative ones. For the classical mechanical systems, this means that we allow the configuration space to be a Poisson manifold (positions need not commute). The phase space corresponding to usual configuration manifold (Poisson structure equal zero) is its cotangent bundle. For a general Poisson manifold, the role of the phase space plays the corresponding symplectic groupoid (if such exists, it is unique -if one restricts to connected and simply connected fibers).It is natural to consider first mechanical systems with symmetry. In the Poisson case a symmetry is described by a Poisson action (of a Poisson group). This requirement imposes a reasonable limitation on a choice of the Poisson structure and actually leads to a construction of it.In this paper we construct Poisson structures on real finite-dimensional vector spaces (the configuration spaces), such that the action of a chosen linear simple Poisson group becomes a Poisson action (the Poisson structure on the group is typically given by a standard classical r-matrix). We construct also the corresponding phase spaces.

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“…Much work has been done recently to explore the SO q (3)-symmetric quantum mechanics developed in [1] and [2]. In particular, a lot is known about the qdeformations of the harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

“…and h (2) [q]l (x) is just the complex conjugate of h (1) [q]l (x). Since (−x) l ( 1 x D) l is a linear operator, it follows that h (1) [q]l should satisfy…”

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confidence: 99%