Special Issue "Non-associative Structures and Other Related Structures"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (20 December 2019).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Special Issue Information

Dear Colleagues,

Non-associative algebras are currently a fashionable research direction. There are two important classes of non-associative structures: Lie structures and Jordan structures. Various Jordan structures play an important role in quantum group theory and in fundamental physical theories.

In recent years, several attempts to unify non-associative structures have led to interesting results. The UJLA structures are not the only structures which realize such a unification.

Associative algebras and Lie algebras can be unified at the level of Yang–Baxter structures. Several papers published in the open access journal Axioms deal with the Yang–Baxter equation.

The Yang–Baxter equation can be interpreted in terms of logical circuits and, in logic, it represents a kind of compatibility condition when working with many logical sentences in the same time. This equation is also related to the theory of universal quantum gates and to quantum computers. It has many applications in quantum groups and knot theory.

Contributions related to non-associative structures, various aspects of the Yang–Baxter Equation, and their applications are invited.

Dr. Florin Felix Nichita
Guest Editor

Manuscript Submission Information

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Keywords

  • Yang–Baxter equation
  • Non-associative algebras
  • Lie structures
  • Jordan structures
  • Associative algebras
  • Unification theories
  • Braces
  • Noncommutative algebras
  • Applications in physics
  • Duality
  • Knot invariants

Published Papers (8 papers)

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Editorial

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Open AccessEditorial
Non-Associative Structures and Other Related Structures
Axioms 2020, 9(2), 40; https://doi.org/10.3390/axioms9020040 - 13 Apr 2020
Abstract
In January 2019, MDPI published a book titled Hopf Algebras, Quantum Groups and Yang–Baxter Equations, based on a successful special issue [...] Full article

Research

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Open AccessArticle
Separability of Nonassociative Algebras with Metagroup Relations
Axioms 2019, 8(4), 139; https://doi.org/10.3390/axioms8040139 - 12 Dec 2019
Cited by 1
Abstract
This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found [...] Read more.
This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found under which such algebras are separable. Algebras satisfying these conditions are described. Full article
Open AccessArticle
Smashed and Twisted Wreath Products of Metagroups
Axioms 2019, 8(4), 127; https://doi.org/10.3390/axioms8040127 - 11 Nov 2019
Cited by 3
Abstract
In this article, nonassociative metagroups are studied. Different types of smashed products and smashed twisted wreath products are scrutinized. Extensions of central metagroups are studied. Full article
Open AccessArticle
Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras
Axioms 2019, 8(3), 82; https://doi.org/10.3390/axioms8030082 - 19 Jul 2019
Cited by 1
Abstract
We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions [...] Read more.
We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights. Full article
Open AccessArticle
Cohomology Theory of Nonassociative Algebras with Metagroup Relations
Axioms 2019, 8(3), 78; https://doi.org/10.3390/axioms8030078 - 04 Jul 2019
Cited by 4
Abstract
Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups [...] Read more.
Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups are investigated. Necessary structural properties of metagroups are studied. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley–Dickson algebras. Full article
Open AccessArticle
Dual Numbers and Operational Umbral Methods
Axioms 2019, 8(3), 77; https://doi.org/10.3390/axioms8030077 - 02 Jul 2019
Cited by 3
Abstract
Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a [...] Read more.
Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus. Full article
Open AccessArticle
Unification Theories: New Results and Examples
Axioms 2019, 8(2), 60; https://doi.org/10.3390/axioms8020060 - 18 May 2019
Cited by 4
Abstract
This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We [...] Read more.
This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a “modified” Yang–Baxter type equation, which unifies several problems in mathematics. Full article
Open AccessArticle
Unification Theories: Examples and Applications
Axioms 2018, 7(4), 85; https://doi.org/10.3390/axioms7040085 - 16 Nov 2018
Cited by 4
Abstract
We consider several unification problems in mathematics. We refer to transcendental numbers. Furthermore, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras. Full article
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