Heisenberg doubles of quantized Poincaré algebras

Borowiec, A.; Pachoł, Anna

doi:10.1007/s11232-011-0139-2

Cited by 13 publications

(19 citation statements)

References 18 publications

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“…In this paper, we investigated three Heisenberg doubles related to the two types of noncommutative Snyder models. The Heisenberg double construction was widely investigated for other noncommutative spacetimes [1][2][3][4][5], but not yet (to our knowledge) for the Snyder space. Therefore, this work offers the first study on Heisenberg doubles for the Snyder model, as well as for the extended Snyder model.…”

Section: Discussionmentioning

confidence: 99%

“…It is worth noting that in the limit of β → 0, the coproducts for the momenta ( 5) and (10) reduce to ∆p i = 1 ⊗ p j + p i ⊗ 1. For β = 0 (classical case), there exists full-duality between algebra A generators x i , M ij and group elements p i , Λ ij (where Λ ij are the matrix elements of Lorentz matrices; see, e.g., [1][2][3][4][5]).…”

Section: Different Realization For Coproducts Of Momentamentioning

confidence: 99%

“…The Heisenberg double, although interesting from a mathematical point of view, can also be seen as a generalization of the quantum mechanics phase space (Heisenberg algebra). Such constructions have been of interest especially for the κ-Minkowski noncommutative spacetime and its deformed relativistic symmetry κ-Poincaré quantum group [1][2][3][4][5], as well as for the θ-deformation [5] and for the general Lie algebra-type noncommutative spaces [6].…”

Section: Introductionmentioning

confidence: 99%

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Heisenberg Doubles for Snyder-Type Models

Meljanac

Pachoł

2021

Symmetry

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A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

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Section: Discussionmentioning

confidence: 99%

Section: Different Realization For Coproducts Of Momentamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Heisenberg Doubles for Snyder-Type Models

Meljanac

Pachoł

2021

Symmetry

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“…The Lie algebra an(n), when its generators are identified with space-time coordinates, is known in the literature as the n + 1-dimensional κ-Minkowski space. For further technical details on κ-deformations the interested reader can consult [37][38][39][40][41][42][43][44][45][46]. The main goal of this Section will be to show that starting from minimal ingredients, namely the structures constants the Lie algebra an(n),…”

Section: Deforming Momentum Space To the An (N) Groupmentioning

confidence: 99%

Deformed phase spaces with group valued momenta

Arzano

Nettel

2016

Phys. Rev. D

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We introduce a general framework for describing deformed phase spaces with group valued momenta. Using techniques from the theory of Poisson-Lie groups and Lie bi-algebras we develop tools for constructing Poisson structures on the deformed phase space starting from the minimal input of the algebraic structure of the generators of the momentum Lie group. The tools developed are used to derive Poisson structures on examples of group momentum space much studied in the literature such as the n-dimensional generalization of the κ-deformed momentum space and the SL(2, Ê) momentum space in three space-time dimensions. We discuss classical momentum observables associated to multi-particle systems and argue that these combine according the usual four-vector addition despite the non-abelian group structure of momentum space.

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“…Later in [9], using a formalism developed in [10], it was shown that this construction can be extended to include also a κ-deformation, by passing from an algebra so(1, N ) to an algebra so(1, N ; g) with generatorŝ X µν , in which the metric η µν is changed into a nondiagonal metric g µν . In that paper, only the Weyl realization of the Hopf algebra was considered and the coproduct and twist were calculated only in that case.…”

Section: Introductionmentioning

confidence: 99%

Associative realizations of the extended Snyder model

Meljanac

Mignemi

2020

Phys. Rev. D

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Usually, the realizations of the noncommutative Snyder model lead to a nonassociative star product. However, it has been shown that this problem can be avoided by adding to the spacetime coordinates new tensorial degrees of freedom. The model so obtained, called extended Snyder model, can be subject to a κ-deformation, giving rise to a unification of the Snyder and the κ-Poincaré algebras in the formalism of extended spacetime.In this paper we review this construction and consider the generic realizations of the κ-deformed extended Snyder model, calculating the associated star product, coproduct and twist in a perturbative setting. We also introduce a representation of the Lorentz algebra in the extended space and discuss a possible interpretation of the tensorial degrees of freedom.

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Heisenberg doubles of quantized Poincaré algebras

Abstract: We briefly analyze some general questions concerning the twist deformation of the Heisenberg double. We reconsider Heisenberg doubles based on quantized Poincaré (Hopf ) algebras as illustrative examples.

Cited by 13 publications

References 18 publications

Heisenberg Doubles for Snyder-Type Models

Heisenberg Doubles for Snyder-Type Models

Deformed phase spaces with group valued momenta

Associative realizations of the extended Snyder model

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