MDPI - Publisher of Open Access Journals

21 pages, 324 KiB

Open AccessArticle

(Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Groups

by Esmaeil Peyghan, Leila Nourmohammadifar, Akram Ali and Ion Mihai

Axioms 2024, 13(10), 693; https://doi.org/10.3390/axioms13100693 - 4 Oct 2024

Viewed by 775

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional

sl (2, R)

and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is [...] Read more.

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional

sl (2, R)

and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field

ζ

is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

16 pages, 303 KiB

Open AccessArticle

Generalized Reynolds Operators on Hom-Lie Triple Systems

by Yunpeng Xiao, Wen Teng and Fengshan Long

Symmetry 2024, 16(3), 262; https://doi.org/10.3390/sym16030262 - 21 Feb 2024

Viewed by 1723

Abstract

In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop a cohomology of generalized Reynolds operators on Hom-Lie triple systems. As applications, we use the first cohomology group to classify linear deformations and we study the obstruction class of an extendable order n deformation. Finally, we introduce and investigate Hom-NS-Lie triple system as the underlying structure of generalized Reynolds operators on Hom-Lie triple systems. Full article

(This article belongs to the Section Mathematics)

13 pages, 1588 KiB

Open AccessArticle

Dynamics of Fricke–Painlevé VI Surfaces

by Michel Planat, David Chester and Klee Irwin

Dynamics 2024, 4(1), 1-13; https://doi.org/10.3390/dynamics4010001 - 2 Jan 2024

Cited by 2 | Viewed by 1911

Abstract

The symmetries of a Riemann surface

Σ ∖ {a_{i}}

with n punctures

a_{i}

are encoded in its fundamental group

π_{1} (Σ)

. Further structure may be described through representations (homomorphisms) of

π_{1}

over a Lie [...] Read more.

The symmetries of a Riemann surface

Σ ∖ {a_{i}}

with n punctures

a_{i}

are encoded in its fundamental group

π_{1} (Σ)

. Further structure may be described through representations (homomorphisms) of

π_{1}

over a Lie group G as globalized by the character variety

C = Hom (π_{1}, G) / G

. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere

Σ = S_{2}^{(4)}

and the ‘space-time-spin’ group

G = S L_{2} (C)

. In such a situation,

C

possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface

V_{a, b, c, d} (x, y, z)

with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or

P_{V I}

); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of

P_{V I}

. In this paper, we feature the parametric representation of some solutions of

P_{V I}

: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups

f_{p}

encountered in TQC or DNA/RNA sequences are proposed. Full article

► Show Figures

Figure 1

33 pages, 532 KiB

Open AccessReview

Nonassociative Algebras, Rings and Modules over Them

by Sergey Victor Ludkowski

Mathematics 2023, 11(7), 1714; https://doi.org/10.3390/math11071714 - 3 Apr 2023

Viewed by 2652

Abstract

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI