Symmetry

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16 pages, 379 KiB

Open AccessArticle

Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

by Jerzy Lukierski

Symmetry 2021, 13(8), 1309; https://doi.org/10.3390/sym13081309 - 21 Jul 2021

Cited by 1 | Viewed by 2262

Abstract

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed

D = 4

quantum inhomogeneous conformal Hopf algebras

U_{θ} (s u (2, 2) ⋉ T^{4}

) and [...] Read more.

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed

D = 4

quantum inhomogeneous conformal Hopf algebras

U_{θ} (s u (2, 2) ⋉ T^{4}

) and

U_{\bar{θ}} (s u (2, 2) ⋉ {\bar{T}}^{4}

), where

T^{4}

describes complex twistor coordinates and

{\bar{T}}^{4}

the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of

D = 4

Heisenberg-conformal algebra (HCA)

s u (2, 2) ⋉ H_{ℏ}^{4, 4}

(

H_{ℏ}^{4, 4} = {\bar{T}}^{4} ⋉_{ℏ} T_{4}

is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length

λ_{p}

, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in

H_{ℏ}^{4, 4}

. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra

U_{θ} (s u (2, 2) ⋉ T^{4}) .

Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

57 pages, 776 KiB

Open AccessArticle

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

by Ivan Gutierrez-Sagredo and Francisco Jose Herranz

Symmetry 2021, 13(7), 1249; https://doi.org/10.3390/sym13071249 - 12 Jul 2021

Cited by 9 | Viewed by 1696

Abstract

The Cayley–Klein (CK) formalism is applied to the real algebra

so (5)

by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the [...] Read more.

The Cayley–Klein (CK) formalism is applied to the real algebra

so (5)

by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for

so (5)

together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter

so (4, 1)

and anti-de Sitter

so (3, 2)

as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature

Λ

and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all

κ

-deformations as particular cases. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

13 pages, 309 KiB

Open AccessArticle

Heisenberg Doubles for Snyder-Type Models

by Stjepan Meljanac and Anna Pachoł

Symmetry 2021, 13(6), 1055; https://doi.org/10.3390/sym13061055 - 11 Jun 2021

Cited by 17 | Viewed by 2230

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

8 pages, 488 KiB

Open AccessArticle

In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts

by Giovanni Amelino-Camelia, Giacomo D’Amico, Fabrizio Fiore, Simonetta Puccetti and Michele Ronco

Symmetry 2021, 13(4), 541; https://doi.org/10.3390/sym13040541 - 25 Mar 2021

Cited by 8 | Viewed by 1706

Abstract

Some recent studies exposed preliminary but rather intriguing statistical evidence of in vacuo dispersion-like spectral lags for gamma-ray bursts (GRBs), a linear correlation between time of observation and energy of GRB particles, which is expected in some models of quantum geometry. Those results [...] Read more.

Some recent studies exposed preliminary but rather intriguing statistical evidence of in vacuo dispersion-like spectral lags for gamma-ray bursts (GRBs), a linear correlation between time of observation and energy of GRB particles, which is expected in some models of quantum geometry. Those results focused on testing in vacuo dispersion for the most energetic GRB particles, and in particular only included photons with energy at emission greater than 40 GeV. We here extend the window of the statistical analysis down to 5 GeV and find results that are consistent with what had been previously noticed at higher energies. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

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50 pages, 584 KiB

Open AccessArticle

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

by Javier de Lucas and Daniel Wysocki

Symmetry 2021, 13(3), 465; https://doi.org/10.3390/sym13030465 - 12 Mar 2021

Cited by 2 | Viewed by 1529

Abstract

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. [...] Read more.

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

Review

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22 pages, 386 KiB

Open AccessReview

Interplay between Spacetime Curvature, Speed of Light and Quantum Deformations of Relativistic Symmetries

by Angel Ballesteros, Giulia Gubitosi and Flavio Mercati

Symmetry 2021, 13(11), 2099; https://doi.org/10.3390/sym13112099 - 5 Nov 2021

Cited by 17 | Viewed by 2597

Abstract

Recent work showed that

κ

-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter

Λ

, the [...] Read more.

Recent work showed that

κ

-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter

Λ

, the Planck scale quantum deformation parameter

κ

and the speed of light parameter c to move to the well-studied

κ

-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

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39 pages, 489 KiB

Open AccessReview

Poisson–Lie Groups and Gauge Theory

by Catherine Meusburger

Symmetry 2021, 13(8), 1324; https://doi.org/10.3390/sym13081324 - 22 Jul 2021

Cited by 2 | Viewed by 2618

Abstract

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices [...] Read more.

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

35 pages, 479 KiB

Open AccessReview

An Introduction to κ-Deformed Symmetries, Phase Spaces and Field Theory

by Michele Arzano and Jerzy Kowalski-Glikman

Symmetry 2021, 13(6), 946; https://doi.org/10.3390/sym13060946 - 26 May 2021

Cited by 8 | Viewed by 2188

Abstract

In this review, we give a basic introduction to the

κ

-deformed relativistic phase space and free quantum fields. After a review of the

κ

-Poincaré algebra, we illustrate the construction of the

κ

-deformed phase space of a classical relativistic particle using [...] Read more.

In this review, we give a basic introduction to the

κ

-deformed relativistic phase space and free quantum fields. After a review of the

κ

-Poincaré algebra, we illustrate the construction of the

κ

-deformed phase space of a classical relativistic particle using the tools of Lie bi-algebras and Poisson–Lie groups. We then discuss how to construct a free scalar field theory on the non-commutative

κ

-Minkowski space associated to the

κ

-Poincaré and illustrate how the group valued nature of momenta affects the field propagation. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

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17 pages, 332 KiB

Open AccessReview

Quantum Orbit Method in the Presence of Symmetries

by Nicola Ciccoli

Symmetry 2021, 13(4), 724; https://doi.org/10.3390/sym13040724 - 19 Apr 2021

Cited by 1 | Viewed by 1759

Abstract

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider

C^{*}

-algebra quantization obtained through groupoid techniques, and we try to put the [...] Read more.

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider

C^{*}

-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization. Full article

(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)

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