Journal of Generalized Lie Theory and Applications

… in Hopf Algebras and …, 2007

Main constructions and examples of quasi-deformations of Lie algebras via twisted derivations leading to quasi-Lie algebras are reviewed. Among quasi-Lie algebras presented are quasi-Hom-Lie deformations of sl 2 enveloping algebra, color Lie algebras and quasi-Hom-Lie and Hom-Lie algebras deformations of infinite-dimensional Lie algebras of Witt and Virasoro type.

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).

2010

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Homalgebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group. 1 2 ABDENACER MAKHLOUF AND SERGEI SILVESTROV six terms in Jacobi identity of the quasi-Lie or of the quasi-Hom-Lie algebras can be combined pairwise in a suitable way. That possibility depends deeply on how the twisting maps interact with each other and with the bracket multiplication.

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).

arXiv: Rings and Algebras, 2020

Representations of color Hom-Lie algebras are reviewed, and it is shown that there exist a series of coboundary operators. We also introduce the notion of a color omni-Hom-Lie algebra associated to a vector space and an even invertible linear map. We show how regular color Hom-Lie algebra structures on a vector space can be characterized. Moreover, it is shown that the underlying algebraic structure of the color omni-Hom-Lie algebra is a color Hom-Leibniz algebra.

2010

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).

Journal of Algebra, 2005

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras.

2021

Several recent results concerning Hom-Leibniz algebra are reviewed, the notion of symmetric Hom-Leibniz superalgebra is introduced and some properties are obtained. Classification of 2-dimensional Hom-Leibniz algebras is provided. Centroids and derivations of multiplicative Hom-Leibniz algebras are considered including the detailed study of 2-dimensional Hom-Leibniz algebras. Introduction Hom-Lie algebras and quasi-Hom-Lie algebras were introduced first in 2003 in [45] where a general method for construction of deformations and discretizations of Lie algebras of vector fields based on twisted derivations obeying twisted Leibniz rule was developped motivated by the examples of q-deformed Jacobi identities in q-deformations of Witt and Visaroro and in related q-deformed algebras discovered in 1990’th in string theory, vertex models of conformal field theory, quantum field theory and quantum mechanics, and also in development of qdeformed differential calculi and q-deformed homological...

Israel Journal of Mathematics, 1996

We study the structure of Lie algebras in the category HAd of H-comodules for a cotriangular bialgebra (H, ( I )) and in particular the H-Lie structure of an algebra A in HA//. We show that if A is a sum of two H-commutative subrings, then the H-commutator ideal of A is nilpotent; thus if A is also semiprime, A is H-commutative. We show an analogous result for arbitrary H-Lie algebras when H is cocommutative. We next discuss the H-Lie ideal structure of A. We show that if A is H-simple *

Journal of Algebra, 2010

The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γ-graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a Z 2 -graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction of a q-deformed Witt superalgebra.