Symmetry, Algebraic Methods and Applications

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

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 5495

Special Issue Editor

Institute of Scietific Research and Development, Tomsk State Pedagogical University, 60 Kievskaya St., 634041 Tomsk, Russia
Interests: algebraic methods in mathematical physics and in the theory of gravitation

Special Issue Information

Dear Colleagues,

Symmetry is the fundamental law of nature. All known fundamental interactions and all existing and hypothetical models of these interactions obey it. Therefore, symmetry is explicit or implicit in all fundamental equations of any realistic model describing real physical processes.

Symmetry theory aims to find, study, and use these symmetries. The main tools of the theory are algebraic and group methods used in theoretical and mathematical physics. The main goal of this issue is to present papers dealing with analytical research in this direction of scientific activity, as well as with the theory of partial differential equations (PDE).

Topics include but are not limited to:

  1. The algebras of symmetry operators and the problem of exact solutions of the main equations of mathematical physics;
  2. The separation of variables theory;
  3. Algebraic methods and the problem of exact solutions of field equations in the theory of gravitation;
  4. Applications of the methods of symmetry theory to cosmology and the theory of gravitation;
  5. Algebraic methods in field theory;
  6. The method of semi-algebraic sets in PDE and mathematical physics;
  7. Lie symmetry for nonlinear equations;
  8. The method of the inverse scaling problem;
  9. hidden symmetry.

Dr. Valery V. Obukhov
Guest Editor

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 submissions that pass pre-check are 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 2400 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

  • symmetry operators
  • the lie symmetry method
  • differential equations
  • dynamical systems
  • theory of gravitation
  • classical and quantum field theory
  • hamilton–jacobi equation
  • maxwell equations
  • klein–gordon–fock equation
  • dirac equation

Published Papers (5 papers)

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Research

12 pages, 289 KiB  
Article
Exact Solutions of Maxwell Equations in Homogeneous Spaces with the Group of Motions G3(VIII)
by Valeriy V. Obukhov
Symmetry 2023, 15(3), 648; https://doi.org/10.3390/sym15030648 - 04 Mar 2023
Cited by 6 | Viewed by 799
Abstract
The problem of the classification of the exact solutions to Maxwell’s vacuum equations for admissible electromagnetic fields and homogeneous space-time with the group of motions G3(VIII) according to the Bianchi classification is considered. All non-equivalent solutions [...] Read more.
The problem of the classification of the exact solutions to Maxwell’s vacuum equations for admissible electromagnetic fields and homogeneous space-time with the group of motions G3(VIII) according to the Bianchi classification is considered. All non-equivalent solutions are found. The classification problem for the remaining groups of motion, G3(N), has already been solved in other papers. All non-equivalent solutions of empty Maxwell equations for all homogeneous spaces with admissible electromagnetic fields are now known. Full article
(This article belongs to the Special Issue Symmetry, Algebraic Methods and Applications)
11 pages, 309 KiB  
Article
Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory
by Alexander Shapovalov and Alexander Breev
Symmetry 2023, 15(2), 282; https://doi.org/10.3390/sym15020282 - 19 Jan 2023
Cited by 3 | Viewed by 938
Abstract
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on [...] Read more.
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation. Full article
(This article belongs to the Special Issue Symmetry, Algebraic Methods and Applications)
22 pages, 329 KiB  
Article
The Canonical Isomorphisms in the Yetter-Drinfeld Categories for Dual Quasi-Hopf Algebras
by Yan Ning, Daowei Lu and Xiaofan Zhao
Symmetry 2022, 14(11), 2358; https://doi.org/10.3390/sym14112358 - 09 Nov 2022
Viewed by 839
Abstract
Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan [...] Read more.
Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A. Balan firstly introduced the notion of right-right Yetter-Drinfeld modules over H and studied its Galois extension. As a continuation, the aim of this paper is to introduce more properties of Yetter-Drinfeld modules. First, we will describe all the other three kinds of Yetter-Drinfeld modules over H, and the monoidal and braided structure of the categories of Yetter-Drinfeld modules explicitly. Furthermore, we will prove that the category HHYDfd of finite dimensional left-left Yetter-Drinfeld modules is rigid. Then we will compute explicitly the canonical isomorphisms in HHYDfd. Finally, as an application, we will rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra. Full article
(This article belongs to the Special Issue Symmetry, Algebraic Methods and Applications)
19 pages, 292 KiB  
Article
The Mei Symmetries for the Lagrangian Corresponding to the Schwarzschild Metric and the Kerr Black Hole Metric
by Nimra Sher Asghar, Kinza Iftikhar and Tooba Feroze
Symmetry 2022, 14(10), 2079; https://doi.org/10.3390/sym14102079 - 06 Oct 2022
Cited by 1 | Viewed by 1302
Abstract
In this paper, the Mei symmetries for the Lagrangians corresponding to the spherically and axially symmetric metrics are investigated. For this purpose, the Schwarzschild and Kerr black hole metrics are considered. Using the Mei symmetries criterion, we obtained four Mei symmetries for the [...] Read more.
In this paper, the Mei symmetries for the Lagrangians corresponding to the spherically and axially symmetric metrics are investigated. For this purpose, the Schwarzschild and Kerr black hole metrics are considered. Using the Mei symmetries criterion, we obtained four Mei symmetries for the Lagrangian of Schwarzschild and Kerr black hole metrics. The results reveal that, in the case of the Schwarzschild metric, the obtained Mei symmetries are a subset of the Lie point symmetries of equations of motion (geodesic equations), while in the case of the Kerr black hole metric, the Noether symmetry set is a subset of the Mei symmetry set and that Mei symmetries and the Lie point symmetries of the equations of motion are same. Full article
(This article belongs to the Special Issue Symmetry, Algebraic Methods and Applications)
12 pages, 293 KiB  
Article
Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model
by Andronikos Paliathanasis
Symmetry 2022, 14(8), 1679; https://doi.org/10.3390/sym14081679 - 12 Aug 2022
Viewed by 962
Abstract
We present the Lie symmetry analysis for a hyperbolic partial differential system known as the one-dimensional Saint-Venant-Exner model. The model describes shallow-water systems with bed evolution given by the Exner terms. The sediment flux is considered to be a power-law function of the [...] Read more.
We present the Lie symmetry analysis for a hyperbolic partial differential system known as the one-dimensional Saint-Venant-Exner model. The model describes shallow-water systems with bed evolution given by the Exner terms. The sediment flux is considered to be a power-law function of the velocity of the fluid. The admitted Lie symmetries are classified according to the power index of the sediment flux. Furthermore, the one-dimensional optimal system is determined in all cases. From the Lie symmetries we derive similarity transformations which are applied to reduce the hyperbolic system into a set of ordinary differential equations. Closed-form exact solutions, which have not been presented before in the literature, are presented. Finally, the initial value problem for the similarity solutions is discussed. Full article
(This article belongs to the Special Issue Symmetry, Algebraic Methods and Applications)
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