A non-abelian tensor product of Hom-Lie algebras

Journal of Algebra and Its Applications, 2014

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

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.

Journal of Pure and Applied Algebra

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat 1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

2015

We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we review the main results. We discuss the envelopment problem, operads, and the Diamond Lemma; the usual tools have to be adapted to this new situation. Moreover we study Hilbert series for the hom-associative operad and free algebra, and describe them up to total degree equal 8 and 9 respectively.

2022

In this paper, we show the relation among the relative central extensions in an isoclinism family of a particular relative central extension of Hom-Lie algebras. We define the notion of isoclinism on the central relative extensions of a pair of Hom-Lie algebras. Then, we figure out the concept of isomorphism in the equivalence class of isoclinisms on the central relative extensions of a pair of Hom-Lie algebras.

Operads and Universal Algebra - Proceedings of the International Conference, 2012

The aim of this paper is to introduce and study Rota-Baxter Hom-algebras. Moreover we introduce a generalization of the dendriform algebras and tridendriform algebras by twisting the identities by mean of a linear map. Then we explore the connections between these categories of Hom-algebras.

Linear and Multilinear Algebra

We construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal (α)-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zero-th and first homology with trivial coefficients. When n = 2 we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with universal central extensions. A generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras is developed.

2020

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.

2018

In this work, the hom-center-symmetric algebras are constructed and discussed. Their bimodules, dual bimodules and matched pairs are defined. The relation between the dual bimodules of hom-center-symmetric algebras and the matched pairs of hom-Lie algebras is established. Furthermore, the Manin triple of hom-center-symmetric algebras is given. Finally, a theorem linking the matched pairs of hom-center-symmetric algebras, the hom-center-symmetric bialgebras and the matched pairs of sub-adjacent hom-Lie algebras is provided.

2021

The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of Poincaré-Birkhoff-Witt theorem for involutive Hom-Lie algebras, we provide an embedding theorem.

Linear and Multilinear Algebra, 2017

We construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal (α)-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zero-th and first homology with trivial coefficients. When n = 2 we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with universal central extensions. A generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras is developed.

2017

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.

Glasgow Mathematical Journal, 2004

Non-abelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical Chevalley-Eilenberg homology of Lie algebras. The relation of cyclic homology with Milnor cyclic homology of non-commutative associative algebras is established in terms of the long exact non-abelian homology sequence of Lie algebras. Some explicit formulas for the second and the third non-abelian homology of Lie algebras are obtained. Using the generalised notion of the Lie algebra of derivations, we introduce the second non-abelian cohomology of Lie algebras with coefficients in crossed modules and extend the seven-term exact non-abelian cohomology sequence of Guin to nine-term exact sequence.

Communications in Algebra, 2019

In the theories of groups and Lie algebras, investigations of the properties of the non-abelian tensor product and their relations to the second homology groups are worthwhile. It is the purpose of the present paper to exhibit such investigations about the non-abelian tensor product of Leibniz algebras. The isomorphism between the non-abelian tensor square and non-abelian exterior square of a Lie algebra L, will enable us to set a simple connection between HL 2 ðLÞ and H 2 ðLÞ. Furthermore, we shall relate the concepts of capability and solvability of a Leibniz algebra to its tensor square. Finally, we give an upper bound for the dimension of the nonabelian tensor square and the second homology of a nilpotent Leibniz algebra in terms of the dimension of its center and derived subalgebra.