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Non-associative Structures and Their Applications in Physics and Geometry
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Non-associative Structures and Their Applications in Physics and Geometry

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 9426

Special Issue Editors

Dr. Ivan Kaygorodov

E-Mail Website
Guest Editor
Centro de Matemática e Aplicações, Universidade da Beira Interior, 6200-001 Covilhã, Portugal
Interests: non-associative algebras; superalgebras; n-ary algebras
Prof. Dr. Yunhe Sheng

E-Mail Website
Guest Editor
Department of Mathematics, Jilin University, Jilin 130012, China
Interests: mathematical physics; poisson geometry; higher-order Lie theory

Special Issue Information

Dear colleagues,

The modern development of geometry, mathematical physics, biology and so on brings new non-associative algebraic structures, such as Poisson algebras, n-ary algebras, bialgebras, dialgebras, quandles,  racks, and so on.

These needs fall into two broad groups: purely technological needs, and theoretical needs associated with developments in both applied algebra and other branches of mathematics. After all, it is not unreasonable to think that algebra is something like the "mathematics of mathematics".

There are many branches of algebra whose contributions solve problems posed by the scientific challenges arising from the advancement of technology. Two of them also stand out for their popularity in society: cryptography and coding theory.

Additionally, from the theoretical point of view, is remarkable the momentum that some disciplines have had in the last 20 years. Thus, axial algebras have been given a big push with the emergence of the connections with Moonshine theory. Additionally, the emergence of Hopf Algebras has made a huge impact on many branches of mathematics and physics. Additionally, of course, one cannot forget very active branches with immense applications at all times: module theory and quivers representation theory.

Thus, we present this Special Issue of Mathematics as a tool to show recent and interesting results on the branches of non-associative algebra and related structures.

Dr. Ivan Kaygorodov
Prof. Dr. Yunhe Sheng
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • non-associative algebras
  • superalgebras
  • n-ary algebras
  • Poisson algebras
  • quandles
  • braces
  • racks
  • bialgebras
  • differential equations

Published Papers (6 papers)

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Research

16 pages, 309 KiB  
Article
Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory
by Chuangchuang Kang, Guilai Liu, Zhuo Wang and Shizhuo Yu
Mathematics 2024, 12(3), 408; https://doi.org/10.3390/math12030408 - 26 Jan 2024
Viewed by 513
Abstract
A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras [...] Read more.
A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
20 pages, 343 KiB  
Article
Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras
by Jia Zhao and Yu Qiao
Mathematics 2024, 12(1), 166; https://doi.org/10.3390/math12010166 - 04 Jan 2024
Viewed by 657
Abstract
In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal [...] Read more.
In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
9 pages, 256 KiB  
Article
On Albert Problem and Irreducible Modules
by Elkin Oveimar Quintero Vanegas
Mathematics 2023, 11(18), 3866; https://doi.org/10.3390/math11183866 - 10 Sep 2023
Viewed by 658
Abstract
Motivated by the relation between Albert’s Problem and irreducible modules within the class of commutative power-associative algebras, in this paper, we show some equivalences to Albert’s Problem. Furthermore, we study some properties of irreducible modules for the zero algebra of dimension n and [...] Read more.
Motivated by the relation between Albert’s Problem and irreducible modules within the class of commutative power-associative algebras, in this paper, we show some equivalences to Albert’s Problem. Furthermore, we study some properties of irreducible modules for the zero algebra of dimension n and we concluded that there are no irreducible modules of dimension four. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
18 pages, 326 KiB  
Article
On Extendibility of Evolution Subalgebras Generated by Idempotents
by Farrukh Mukhamedov and Izzat Qaralleh
Mathematics 2023, 11(12), 2764; https://doi.org/10.3390/math11122764 - 19 Jun 2023
Cited by 1 | Viewed by 637
Abstract
In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was [...] Read more.
In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper’s findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
12 pages, 611 KiB  
Article
Non-Associative Structures and Their Applications in Differential Equations
by Yakov Krasnov
Mathematics 2023, 11(8), 1790; https://doi.org/10.3390/math11081790 - 09 Apr 2023
Cited by 1 | Viewed by 1391
Abstract
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between [...] Read more.
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
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49 pages, 469 KiB  
Article
The Algebraic Classification of Nilpotent Bicommutative Algebras
by Kobiljon Abdurasulov, Ivan Kaygorodov and Abror Khudoyberdiyev
Mathematics 2023, 11(3), 777; https://doi.org/10.3390/math11030777 - 03 Feb 2023
Viewed by 4787
Abstract
This paper is devoted to the complete algebraic classification of complex five-dimensional nilpotent bicommutative algebras. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
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