New Perspectives in Lie Algebras, 2nd Edition

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 30 May 2026 | Viewed by 1654

Special Issue Editor


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Guest Editor
Instituto de Matemática Interdisciplinar, Departamento de Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid, Spain
Interests: real and complex lie algebras and groups; differential forms and distribution theory; contractions and deformations; Casimir invariants; symmetries in physics; representation theory; lie group analysis of differential equations; Lagrangian and Hamiltonian formalism in classical mechanics; integrable and superintegrable systems; symmetry-conditioned perturbation theory; inverse problems in dynamics; supersymmetry
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Special Issue Information

Dear Colleagues,

Lie algebras constitute some of the most versatile tools in modern mathematics, appearing in many different disciplines like mechanics, differential geometry, differential equations or quantum theories, as well as comprising a relevant technique in control theory or robotics, among fields.

We invite researchers to contribute either original papers or review articles concerning currently existing problems within the theory of Lie algebras/superalgebras, finite or infinite, and/or dealing with their applications in other disciplines, including the engineering sciences.

Articles written by authors with strong geometrical backgrounds or those describing new computational methods in representation or structural theory, as well as in applications, are especially welcome. Beyond the aforementioned topics, we are also open to receiving papers on other related concepts and applications.

Dr. Rutwig Campoamor-Stursberg
Guest Editor

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Keywords

  • structure theory of Lie algebras and superalgebras
  • classification problems
  • representation theory
  • realizations by vector fields or differential operators
  • computational methods
  • Lie algebras and symmetries of systems
  • applications to other natural sciences

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Published Papers (2 papers)

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Research

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30 pages, 389 KB  
Article
Nijenhuis Operators on 2D Pre-Lie Algebras and 3D Associative Algebras
by Xiaoguang Zou, Xiang Gao, Chuangchuang Kang and Jiafeng Lü
Axioms 2026, 15(1), 80; https://doi.org/10.3390/axioms15010080 - 22 Jan 2026
Viewed by 358
Abstract
In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions to the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras. Full article
(This article belongs to the Special Issue New Perspectives in Lie Algebras, 2nd Edition)

Review

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66 pages, 726 KB  
Review
New Perspectives on Kac–Moody Algebras Associated with Higher-Dimensional Manifolds
by Rutwig Campoamor-Stursberg, Alessio Marrani and Michel Rausch de Traubenberg
Axioms 2025, 14(11), 809; https://doi.org/10.3390/axioms14110809 - 31 Oct 2025
Viewed by 893
Abstract
In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle S1, we extend the approach to compact and non-compact group manifolds, [...] Read more.
In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle S1, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras g(M), their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on SU(2), SU(2)/U(1), and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications. Full article
(This article belongs to the Special Issue New Perspectives in Lie Algebras, 2nd Edition)
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