Special Issue "Quantum Group Symmetry and Quantum Geometry"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics and Symmetry".

Deadline for manuscript submissions: 31 May 2021.

Special Issue Editors

Prof. Dr. Ángel Ballesteros
E-Mail Website
Guest Editor
Physics Department, University of Burgos, Burgos, Spain
Interests: mathematical physics; classical and quantum integrable systems; quantum groups; noncommutative geometry; quantum gravity
Dr. Giulia Gubitosi
E-Mail Website
Guest Editor
Physics Department, University of Burgos, Burgos, Spain
Interests: quantum gravity phenomenology; deformed relativistic symmetries; primordial cosmology; quantum groups; noncommutative geometry
Prof. Dr. Francisco J. Herranz
E-Mail Website
Guest Editor
Physics Department, University of Burgos, Burgos, Spain
Interests: mathematical physics; quantum groups; integrable systems; noncommutative geometry; quantum gravity

Special Issue Information

Dear Colleagues,

Quantum groups appeared during the eighties as the underlying algebraic symmetries of several two-dimensional integrable models. They are noncommutative generalizations of Lie groups endowed with a Hopf algebra structure, and the possibility of defining noncommutative spaces that are covariant under quantum group (co)actions soon provided a fruitful link with noncommutative geometry. At the same time, when quantum group analogues of the Lie groups of spacetime symmetries (Galilei, Poincaré and (anti-) de Sitter) were constructed, they attracted the attention of quantum gravity researchers. In fact, they provided a possible mathematical framework to model the "quantum" geometry of space–time and the quantum deformations of its kinematical symmetries at the Planck scale, where nontrivial features are expected to arise because of the interplay between gravity and quantum theory.

This Special Issue is open to contributions dealing with any of the many facets of quantum group symmetry and their generalizations. On the more formal side, possible topics include the theory of Poisson–Lie groups and Poisson homogeneous spaces as the associated semiclassical objects; Hopf algebras; the classification of quantum groups and spaces, their representation theory and its connections with q-special functions; the construction of noncommutative differential calculi; and the theory of quantum bundles. On application side, possible topics are: classical and quantum integrable models with quantum group invariance; the applications of quantum groups in different (2+1) quantum gravity contexts (like combinatorial quantisation, state sum models or spin foams); and quantum kinematical groups and their noncommutative spacetimes in connection with deformed special relativity and quantum gravity phenomenology.

Prof. Dr. Ángel Ballesteros
Dr. Giulia Gubitosi
Prof. Dr. Francisco J. Herranz
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • quantum groups
  • Poisson–Lie groups
  • Lie bialgebras and r-matrices
  • Poisson homogeneous spaces
  • non-commutative differential calculi
  • integrable models
  • quantum gravity
  • deformed symmetries
  • noncommutative spacetimes

Published Papers (3 papers)

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Research

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Open AccessArticle
In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts
Symmetry 2021, 13(4), 541; https://doi.org/10.3390/sym13040541 - 25 Mar 2021
Viewed by 256
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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Open AccessArticle
Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras
Symmetry 2021, 13(3), 465; https://doi.org/10.3390/sym13030465 - 12 Mar 2021
Viewed by 257
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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Open AccessReview
Quantum Orbit Method in the Presence of Symmetries
Symmetry 2021, 13(4), 724; https://doi.org/10.3390/sym13040724 - 19 Apr 2021
Viewed by 276
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 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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