Abdenacer Makhlouf
Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms.
Thu, 09/23/2010 - 15:01 — SomNambulAuthors: Abdenacer Makhlouf, Saïd Benayadi
Subjects: Rings and Algebras
AbstractThe 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.Cohomology and Deformations of Hom-algebras.
Wed, 05/05/2010 - 15:01 — SomNambulAuthors: Abdenacer Makhlouf, Faouzi Ammar, Zeyneb Ejbehi
Subjects: Rings and Algebras
AbstractThe 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.Paradigm of Nonassociative Hom-algebras and Hom-superalgebras.
Tue, 01/26/2010 - 16:01 — SomNambulAuthors: Abdenacer Makhlouf
Subjects: Rings and Algebras
AbstractThe 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.Kaplansky's Construction Type and Classification of Weak bialgebras and Weak Hopf algebras.
Thu, 01/14/2010 - 16:01 — SomNambulAuthors: Abdenacer Makhlouf, Zoheir Chebel
Subjects: Rings and Algebras
AbstractIn 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.Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras.
Wed, 12/02/2009 - 16:01 — SomNambulAuthors: Abdenacer Makhlouf, Joakim Arnlind, Sergei Silvestrov
Subjects: Rings and Algebras
AbstractThe 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.Hom-alternative algebras and Hom-Jordan algebras.
Thu, 09/03/2009 - 17:38 — SomNambulAuthors: Abdenacer Makhlouf
Subjects: Rings and Algebras
AbstractThe 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.
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