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Symmetry
Special Issues
Symmetry and Lie Algebras
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Symmetry and Lie Algebras

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

Deadline for manuscript submissions: 30 November 2025 | Viewed by 1337

Special Issue Editors

Prof. Dr. María Alejandra Alvarez

E-Mail Website
Guest Editor
Departamento de Matemáticas, Universidad de Antofagasta, Antofagasta 1240000, Chile
Interests: Lie algebra; representation theory; homological algebra
Dr. María del Carmen Rodríguez-Vallarte

E-Mail Website
Guest Editor
Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico
Interests: Lie theory; Lie algebra

Special Issue Information

Dear Colleagues,

Symmetry is a fundamental concept in physics and mathematics, representing invariance under certain transformations. A Lie algebra is an algebraic structure defined on a vector space through a binary operation called the Lie bracket, satisfying certain properties. Symmetry and Lie algebras are deeply connected: every continuous symmetry of a physical system is associated with a Lie algebra. This association is established through the concept of infinitesimal transformations, which are small, local changes that preserve the symmetry of a system. Lie algebras provide a way to represent these infinitesimal transformations. Each element of the Lie algebra corresponds to a generator of a symmetry transformation. Lie algebras offer a rich and versatile mathematical framework with applications across a wide range of scientific disciplines. Their ability to capture the symmetries of systems makes them an indispensable tool for both theoretical and applied research.

The aim of this Special Issue is to bring together new results and contributions regarding the relevance of symmetry in Lie algebras, their applications, and related topics. Potential topics include, but are no limited to, the following:

  • algebraic computing structures;
  • applied mathematics;
  • cryptography;
  • differential equations;
  • differential geometry;
  • mathematical physics;
  • representation theory;
  • supersymmetry;
  • topology.

Prof. Dr. María Alejandra Alvarez
Dr. María del Carmen Rodríguez-Vallarte
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 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

  • algebraic computing structures
  • applied mathematics
  • cryptography
  • differential equations
  • differential geometry
  • mathematical physics
  • representation theory
  • supersymmetry
  • topology

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

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17 pages, 295 KiB  
Article
Discrete-Time Dynamical Systems on Structured State Spaces: State-Transition Laws in Finite-Dimensional Lie Algebras
by Simone Fiori
Symmetry 2025, 17(3), 463; https://doi.org/10.3390/sym17030463 - 19 Mar 2025
Cited by 1 | Viewed by 226
Abstract
The present paper elaborates on the development of a theory of discrete-time dynamical systems on finite-dimensional structured state spaces. Dynamical systems on structured state spaces possess well-known applications to solving differential equations in physics, and it was shown that discrete-time systems on finite- [...] Read more.
The present paper elaborates on the development of a theory of discrete-time dynamical systems on finite-dimensional structured state spaces. Dynamical systems on structured state spaces possess well-known applications to solving differential equations in physics, and it was shown that discrete-time systems on finite- (albeit high-) dimensional structured state spaces possess solid applications to structured signal processing and nonlinear system identification, modeling and control. With reference to the state-space representation of dynamical systems, the present contribution tackles the core system-theoretic problem of determining suitable laws to express a system’s state transition. In particular, the present contribution aims at formulating a fairly general class of state-transition laws over the Lie algebra associated to a Lie group and at extending some properties of classical dynamical systems to process Lie-algebra-valued state signals. Full article
(This article belongs to the Special Issue Symmetry and Lie Algebras)
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18 pages, 354 KiB  
Article
Brauer Analysis of Some Time–Memory Trade-Off Attacks and Its Application to the Solution of the Yang–Baxter Equation
by Agustín Moreno Cañadas, Ismael Gutierrez, Odette M. Mendez, Andrés Sarrazola-Alzate and Jesus Antonio Zuluaga-Moreno
Symmetry 2025, 17(3), 391; https://doi.org/10.3390/sym17030391 - 4 Mar 2025
Viewed by 461
Abstract
This paper is focused on some algebraic and combinatorial properties of a TMTO (Time–Memory Trade-Off) for a chosen plaintext attack against a cryptosystem with a perfect secrecy property. TMTO attacks aim to retrieve the preimage of a given one-way function more efficiently than [...] Read more.
This paper is focused on some algebraic and combinatorial properties of a TMTO (Time–Memory Trade-Off) for a chosen plaintext attack against a cryptosystem with a perfect secrecy property. TMTO attacks aim to retrieve the preimage of a given one-way function more efficiently than an exhaustive search and with less memory than a dictionary attack. TMTOs for chosen plaintext attacks against cryptosystems with a perfect secrecy property are associated with some directed graphs, which can be defined by suitable collections of multisets called Brauer configurations. Such configurations induce so-called Brauer configuration algebras, the algebraic and combinatorial invariant analysis of which is said to be a Brauer analysis. In this line, this paper proposes formulas for dimensions of Brauer configuration algebras (and their centers) induced by directed graphs defined by TMTO attacks. These results are used to provide some set-theoretical solutions for the Yang–Baxter equation. Full article
(This article belongs to the Special Issue Symmetry and Lie Algebras)
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Review

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26 pages, 334 KiB  
Review
Conjectures on the Stability of Linear Control Systems on Matrix Lie Groups
by Víctor Ayala, María Torreblanca and William Valdivia
Symmetry 2025, 17(4), 593; https://doi.org/10.3390/sym17040593 - 14 Apr 2025
Viewed by 261
Abstract
Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, [...] Read more.
Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures. Full article
(This article belongs to the Special Issue Symmetry and Lie Algebras)
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