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Abdenacer Makhlouf

Publications and source records attributed to Abdenacer Makhlouf.

108 records · Page 6Linked to original sources
arXiv2011

Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras

We present a procedure to construct (n+1)-Hom-Nambu-Lie algebras from n-Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n+k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

math.RA↗
arXiv2011

Hom-dendriform algebras and Rota-Baxter Hom-algebras

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.

math.RA↗
arXiv2010

Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms

The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the double extension theory to Hom-Lie algebras. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. We establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution. Centerless involutive quadratic Hom-Lie algebras are characterized. Also elements of a representation theory for Hom-Lie algebras, including adjoint and coadjoint representations are supplied with application to quadratic Hom-Lie algebras.

math.RA↗
arXiv2010

Deformations of Hom-Alternative and Hom-Malcev algebras

The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give a procedure to provide deformations of alternative algebras (resp. Malcev algebras) into Hom-alternative algebras (resp. Hom-Malcev algebras). Then it is used to supply examples for which we compute some cohomology invariants.

math.RA↗
arXiv2010

Cohomology and Deformations of Hom-algebras

The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Also we enhance the deformations theory of this Hom-algebras by studying the obstructions.

math.RA↗
arXiv2010

Paradigm of Nonassociative Hom-algebras and Hom-superalgebras

The aim of this paper is to give a survey of nonassociative Hom-algebra and Hom-superalgebra structures. The main feature of these algebras is that the identities defining the structures are twisted by homomorphisms. We discuss Hom-associative algebras, Hom-Flexible algebras, Hom-Lie algebras, $G$-hom-associative algebras, Hom-Poisson algebras, Hom-alternative algebras and Hom-Jordan algebras and $\mathbb{Z}_2$-graded versions. We give an overview of the development of Hom-algebras structures which have been intensively investigated recently.

math.RA↗
arXiv2010

Kaplansky's Construction Type and Classification of Weak bialgebras and Weak Hopf algebras

In this paper, we study weak bialgebras and weak Hopf algebras. These algebras form a class wider than bialgebras respectively Hopf algebras. The main results of this paper are Kaplansky's constructions type which lead to weak bialgebras or weak Hopf algebras starting from a regular algebra or a bialgebra. Also we provide a classification of 2-dimensional and 3-dimensional weak bialgebras and weak Hopf algebras. We determine then the stabilizer group and the representative of these classes, the action being that of the linear group.

math.RA↗
arXiv2009

Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras

The purpose of this paper is to investigate ternary multiplications constructed from a binary multiplication, linear twisting maps and a trace function. We provide a construction of ternary Hom-Nambu and Hom-Nambu-Lie algebras starting from a binary multiplication of a Hom-Lie algebra and a trace function satisfying certain compatibility conditions involving twisting maps. We show that mutual position of kernels of twisting maps and the trace play important role in this context, and provide examples of Hom-Nambu-Lie algebras obtained using this construction.

math.RA↗
arXiv2009

Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the action of the Galois group, and HH.(B) are shown to be isomorphic.

math.KT↗
arXiv2009

Hom-alternative algebras and Hom-Jordan algebras

The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.

math.RA↗
arXiv2009

Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras

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 $\mathbb{Z}_2$-graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction of a q-deformed Witt superalgebra.

math.RA↗
arXiv2008

Hom-Algebras and Hom-Coalgebras

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 Hom-algebras 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.

math.RA↗
arXiv2008

Bialgebra structures of 2-associative algebras

This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative algebra and the classification of these structures in low dimensions.

math.RA↗
arXiv2007

Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

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 $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.

math.RA↗
arXiv2007

Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras

The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.

math.RA↗
arXiv2006

A comparison of deformations and geometric study of associative algebras varieties

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. We describe in each case the corresponding notions of degeneration and rigidity. We illustrate these notions with examples and give some general properties. The last part of this work shows how these notions help in the study of associative algebras varieties. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory of Schlessinger and was developed by A. Fialowski in order to deform infinite dimensional nilpotent Lie algebras. In a Nonstandard framework, M.Goze introduced the notion of perturbation for studying the rigidity of finite dimensional complex Lie algebras. All these approaches has in common the fact that we make an "extension" of the field. These theories may be applied to any multilinear structure. We shall be concerned in this paper with the category of associative algebras.

math.RA↗
arXiv2002

Degeneration, rigidity and irreducible components of Hopf algebras

The aim of this work is to discuss the concepts of degeneration, deformation and rigidity, and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf algebras and the irreducible components for n=p^2, p prime and for n<14.

math.RA↗