Hom-Lie Algebras Research Articles

Modular structure theory on Hom-Lie algebras

Abstract The aim of this paper is to transfer the restrictedness theory to Hom-Lie algebras. The concept of restricted Hom-Lie algebras, which is introduced in [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No. 4955], will be used in this paper. First, the existence and uniqueness of p-structures on a Hom-Lie algebra is studied. Then the definition of a restrictable Hom-Lie algebra is given and the equivalence relation between restrictable Hom-Lie algebras and restricted Hom-Lie algebras is constructed. Finally, the p-envelopes of a Hom-Lie algebra are defined and studied.

Generalized Hom–Lie–Rinehart algebras

The first-order Hom-differential operators on a commutative Hom-associative algebra with unit are introduced. We describe Hom–Lie algebra associated with a module of first-order Hom-differential operators. We give the notion of a generalized Hom–Lie–Rinehart algebra and we show the characterization of generalized Hom–Lie–Rinehart algebras in terms of Hom–Gerstenhaber–Jacobi algebras. We prove the necessary and sufficient condition to obtain a generalized Hom–Lie–Rinehart algebra structure on a Hom–Lie algebroid.

Biderivations of Hom-Lie Algebras and Superalgebras

Biderivations of Hom-Lie Algebras and Superalgebras

Hom-Lie Algebras with Derivations

Hom-Lie Algebras with Derivations

On quadratic Hom-Lie algebras with twist maps in their centroids and their relationship with quadratic Lie algebras

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhlouf we call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided.

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study K\"{a}hler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

Nijenhuis-operator on Hom-Lie conformal algebras

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom-NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom-NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators.

Hom-Lie Superalgebras in Characteristic 2

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and αk-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology.

Relative Rota-Baxter operators on Hom-Lie triple systems

In this paper, we first introduce the notion of relative Rota-Baxter operators on Hom-Lie triple systems and define a cohomology of relative Rota-Baxter operators on Hom-Lie triple systems with coefficients in a suitable representation. Furthermore, we introduce and study Hom-pre-Lie triple systems as the underlying structure of relative Rota-Baxter operators on Hom-Lie triple systems. Finally, we provide relationships between relative Rota-Baxter operators on Hom-Lie algebras and associated Hom-Lie triple systems.

Differential geometry of Hom-Lie algebras and Hom-Lie algebroids

In this paper, the geometry of Hom-Lie algebras and Hom-Lie algebroids are studied. In this direction, the effects of connections on Hom-Lie algebras and Hom-Lie algebroids are first investigated. By finding the relations among hom-curvature operator, torsion and connection on Hom-Lie algebras, Bianchi-like identities for hom-curvature operator and torsion are found. Then, with the help of these found properties, the relationships among Hom-Lie algebras, Hom-admissible algebras, Hom-Flexible algebras, Hom-Pre-Lie algebras and Hom-Post algebras are examined. These relationships have been also carried to the set of vector fields of the manifold with Otsuki connection. Then, conditions of being Hom-Lie-Admissible algebroid, Hom-Pre-Lie algebroid and Hom-Post-Lie algebroid for a Hom-bundle are obtained.

Cohomology and deformations of compatible Hom-Lie algebras

In this paper, we consider compatible Hom-Lie algebras as a twisted version of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-Lie algebras generalizing the recent work of Liu, Sheng and Bai. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-Lie algebras.

Realization of representations of the Hom-Lie algebra 𝔰𝔩 q (2, ℂ) of Jackson

The weight modules of the Lie algebra are well known. In the first part of this paper we deal with a realization of weight modules of in the space , where is the algebra of Laurent polynomials. In the second part, we consider the Hom-Lie algebra of Jackson where . The q-analogue of the above realization in the space is considered. We obtain two kinds of q-modules. The regular q-modules which have limits the modules obtained in the classical realization when q goes to 1 . The other q-modules have no limits when q goes to 1 and they are called singular modules.

Fuzzy Hom–Lie Ideals of Hom–Lie Algebras

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras.

Twisted Rota-Baxter operators on 3-Hom-Lie algebras

–In this paper, we first introduce the notion of twisted Rota-Baxter operators on 3-Hom-Lie algebras and define a cohomology of twisted Rota-Baxter operators on 3-Hom-Lie algebras with coefficients in a suitable representation. Furthermore, we introduce and study 3-Hom-NS-Lie-algebras as the underlying structure of twisted Rota-Baxter operators on 3-Hom-Lie algebras. Finally, we investigate twisted Rota-Baxter operators on 3-Hom-Lie algebras induced by Hom-Lie algebras.

Transposed Poisson structures on Witt type algebras

We describe 12-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras V(f), where f:Γ→C is non-trivial and f(0)=0. More precisely, if |f(Γ)|≥4, then all the transposed Poisson algebra structures on V(f) are mutations of the group algebra structure (V(f),⋅) on V(f). If |f(Γ)|=3, then we obtain the direct sum of 3 subspaces of V(f), corresponding to cosets of Γ0 in Γ, with multiplications being different mutations of ⋅. The case |f(Γ)|=2 is more complicated, but also deals with certain mutations of ⋅. As a consequence, new Lie algebras that admit non-trivial Hom-Lie algebra structures are found.

Twisted Rota-Baxter operators on Hom-Lie algebras

<abstract><p>Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.</p></abstract>

Extraction Algorithm of HOM–LIE Algebras Based on Solvable and Nilpotent Groups

Hom–Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom–Lie algebras are studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well‐developed. We establish a theory of the solvable and nilpotent Hom–Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research.Dedicated to Al Farouk School & Kinder garten-Irbid-Jordan

Cohomology of nonabelian embedding tensors on Hom-Lie algebras

<abstract><p>In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.</p></abstract>

HOMOLOGY THEORY OF MULTIPLICATIVE HOM-LIE ALGEBRAS

In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe

On the classification of 3-dimensional complex hom-Lie algebras

Let hom-Ln(C) be the algebraic set of complex n-dimensional hom-Lie algebras. The group GL(n,C) acts on it via change of basis. We classify hom-Lie structures with nilpotent twisting map on 3-dimensional complex Lie algebras, up to isomorphism, and we give the classification of orbit closures in such family. For this purpose, we introduce some invariants of hom-Lie algebras to study both problems. The ideas and techniques presented here can be easily extrapolated to study similar problems on other linear algebraic structures.