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2020 | Book

Isomorphisms of Four Dimensional Perfect Non-simple Evolution Algebras Read chapter Read first chapter

Associative and Non-Associative Algebras and Applications

3rd MAMAA, Chefchaouen, Morocco, April 12-14, 2018

Editors: Mercedes Siles Molina, Dr. Laiachi El Kaoutit, Mohamed Louzari, L'Moufadal Ben Yakoub, Mohamed Benslimane

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

Table of Contents

Frontmatter

Algebraic and Analytic Methods in Associative and Non-associative Structures. Applications

Frontmatter
Isomorphisms of Four Dimensional Perfect Non-simple Evolution Algebras
Abstract
In this paper we complete the classification of four dimensional perfect non-simple evolution algebras (under mild conditions on the based field), started in Casado et al. (Linear Algebra and its Applications, [1]). We consider the different parametric families of evolution algebras appearing in the classification and study which algebras in the same family are isomorphic.
Antonio Behn, Yolanda Cabrera Casado, Mercedes Siles Molina
Power-Associative Evolution Algebras
Abstract
The paper is devoted to the study of evolution algebras that are power-associative algebras. We give the Wedderburn decomposition of evolution algebras that are power-associative algebras and we prove that these algebras are Jordan algebras. Finally, we use this decomposition to classify these algebras up to dimension six.
Moussa Ouattara, Souleymane Savadogo
A Survey on Isometries Between Lipschitz Spaces
Abstract
The famous Banach–Stone theorem, which characterizes surjective linear isometries between C(X) spaces as certain weighted composition operators, has motivated the study of isometries defined on different function spaces (see [33, 34]). The research on surjective linear isometries between spaces of Lipschitz functions is a subject of long tradition which goes back to the sixties with the works of de Leeuw [61] and Roy [81], and followed by those by Mayer-Wolf [67], Weaver [97], Araujo and Dubarbie [3], and Botelho, Fleming and Jamison [8]. This topic continues to attract the attention of some authors (see [44, 52, 62]). In the setting of Lipschitz spaces, we present a survey on non-necessarily surjective linear isometries and codimension 1 linear isometries [55], vector-valued linear isometries [56], local isometries and generalized bi-circular projections [54], 2-local isometries [52, 57], projections and averages of isometries [12] and hermitian operators [13, 14]. We also raise some open problems on bilinear isometries and approximate isometries in the same context.
M. G. Cabrera-Padilla, A. Jiménez-Vargas, Moisés Villegas-Vallecillos
The Principal Eigenvalue for a Class of Singular Quasilinear Elliptic Operators and Applications
Abstract
We characterize the principal eigenvalue associated to the singular quasilinear elliptic operator \(-\Delta u - \mu (x) \frac{|\nabla u|^q}{u^{q-1}}\) in a bounded smooth domain \(\Omega \subset \mathrm{I\!R}^N\) with zero Dirichlet boundary conditions. Here, \(1<q\le 2\) and \(0\le \mu \in L^\infty (\Omega )\). As applications we derive some existence of solutions results (as well as uniqueness, nonexistence and homogenization results) to a problem whose model is
$$ \left\{ \begin{array}{l} \displaystyle -\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) \quad \text { in }\,\, \Omega , \\ u= 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad \,\!\!\!\!\!\!\!\!\!\! \text { on} \,\, \partial \Omega , \end{array} \right. $$
where \(\lambda \in \mathrm{I\!R}\) and \(f\in L^p(\Omega )\) for some \(p>\frac{N}{2}\).
José Carmona, Salvador López-Martínez, Pedro J. Martínez-Aparicio
Non-commutative Poisson Algebras Admitting a Multiplicative Basis
Abstract
A non-commutative Poisson algebra is a Lie algebra endowed with a, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we consider a non-commutative Poisson algebra \({\mathfrak P}\) of arbitrary dimension, over an arbitrary base field \({\mathbb F}\), a basis \({\mathfrak B}=\{e_{i}\}_{i \in I}\) of \({\mathfrak P}\) is called multiplicative if for any \(i,j \in I\) we have that \(e_ie_j \in {\mathbb F} e_r\) and \([e_i,e_j]\in {\mathbb F} e_s\) for some \(r,s \in I\). We show that if \({\mathfrak P}\) admits a multiplicative basis then it decomposes as the direct sum of well-described ideals admitting each one a multiplicative basis. Also the minimality of \({\mathfrak P}\) is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the previous decomposition is the direct sum of the family of its minimal ideals admitting a multiplicative basis.
Antonio J. Calderón Martín, Boubacar Dieme, Francisco J. Navarro Izquierdo
Multiplication Algebras: Algebraic and Analytic Aspects
Abstract
Applications of multiplication algebras to the algebraic and analytic strengthenings of primeness and semiprimeness of (possibly non-associative) algebras are fully surveyed, and complete normed complex algebras whose closed multiplication algebras satisfy the von Neumann inequality are studied in detail.
Miguel Cabrera García, Ángel Rodríguez Palacios
Generalized Drazin Inverse and Commuting Riesz Perturbations
Abstract
In this note, we provide necessary and sufficient conditions for the stability of generalized Drazin invertible operators under commuting Riesz perturbation. We also focus on the commuting perturbation class of meromorphic operators.
Mourad Oudghiri, Khalid Souilah
Generalized Rigid Modules and Their Polynomial Extensions
Abstract
Let R be a ring with unity, \(\sigma \) an endomorphism of R and \(M_R\) a right R-module. In this paper, we study some connections between rigid, \(\sigma \)-rigid, semicommutative, \(\sigma \)-semicommutative, abelian and \(\sigma \)-reduced modules. Also, we show that the class of \(\sigma \)-rigid modules is not closed under homomorphic images and some module extensions. Moreover, we examine the transfer of \(\sigma \)-reducibly, \({\sigma }\)-semicommutative and \(\sigma \)-rigidness from a module \(M_R\) to its extensions of the form \(M[x]/M[x](x^n)\) where \(n\ge 2\) is an integer, and vice versa.
Mohamed Louzari, Armando Reyes
n-Ary k-Actions Between Sets and Their Applications
Abstract
We consider families F of n-ary k-actions
$$f : \overset{k)}{\mathfrak A\times \cdots \times \mathfrak A} \times \overset{n-k)}{\mathfrak B\times \cdots \times \mathfrak B} \rightarrow \mathfrak A$$
between arbitrary non-empty sets \(\mathfrak A\) and \(\mathfrak B\) and show that if every \(f\in F\) fixes some element in \(\mathfrak A\), then this family induces an adequate decomposition of \(\mathfrak A\) as the (orthogonal) disjoint-pointed union of well-described F-invariant subsets (F-submodules). If \(\mathfrak A\) is furthermore a division F-module, it is shown that the above decomposition is by means of the family of its pointed simple F-submodules. The obtained results are applied to the structure theory of arbitrarily graded n-linear k-modules by stating a second Wedderburn type theorem for the class of n-linear k-modules with an arbitrary division grading.
Antonio J. Calderón Martín, Babacar Gaye, Francisco J. Navarro Izquierdo
Primary Group Rings
Abstract
A ring R is said to be primary if the Jacobson radical J(R) is nilpotent and the factor ring R/J(R) is simple artinian. The main result of this note is the characterization of the primary group rings of not necessary abelian groups. This generalizes the work of Chin and Qua (Rendiconti del Seminario Matematico della Università di Padova 137:223–228 2017, [1]) in which the author characterizes the primary group rings of abelian groups.
Mohammed El Badry, Mostafa Alaoui Abdallaoui, Abdelfattah Haily
Semi-ring Based Gröbner–Shirshov Bases over a Noetherian Valuation Ring
Abstract
Commutative and non commutative Gröbner–Shirshov bases were first studied over fields and after extended to some particular rings. In theses works, the monomials are in a monoid. Recently, Bokut and al. gave a new extension of Gröbner–Shirshov bases over a field by choosing the monomials in a semi-ring rather in a monoid. In this paper, we study Gröbner–Shirshov bases where the monomials are in a semi-ring and the coefficients are in a noetherian valuation ring and we establish the relation between weak and strong Gröbner bases.
Yatma Diop, Laila Mesmoudi, Djiby Sow
The 4-Rank of the Class Group of Some Real Pure Quartic Number Fields
Abstract
Let https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-35256-1_12/477717_1_En_12_IEq1_HTML.gif be a real pure quartic number field and https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-35256-1_12/477717_1_En_12_IEq2_HTML.gif its real quadratic subfield, where \(p\equiv 5 \pmod 8 \) and \( q\equiv 1 \pmod 4\) are two different odd prime numbers such that \((\frac{p}{q})=1\). In this work, we are interested in studying the 2-rank and the 4-rank of the class group of K.
Mbarek Haynou, Mohammed Taous

Homological and Categorical Methods in Algebra

Frontmatter
Frobenius Monoidal Algebras and Related Topics
Abstract
We survey results on Frobenius algebras and illustrate their importance to the structure of some generalizations of the notion of Hopf algebra, as well as their connections to topics like monoidal categories, 2-categories, functors, topological quantum field theories, etc.
Daniel Bulacu, Blas Torrecillas
The Functor and Its Relationship with Homological Functors and
Abstract
In this paper we construct a functor that we call localization functor defined in the category of complexes of left A-modules where A is a not necessarily commutative ring and we study some of its properties. Besides, we study its relationship with the homological functors \(Tor_n\) and \(\overline{EXT}^n\).
Bassirou Dembele, Mohamed Ben Faraj ben Maaouia, Mamadou Sanghare
BOCSES over Small Linear Categories and Corings
Abstract
This note does not claim anything new, since the material exposed here is somehow folkloric. We provide the main steps in showing the equivalence of categories between the category of BOCSES over a small linear category and the category of corings over the associated ring with enough orthogonal idempotents.
Laiachi El Kaoutit
Quadratic Color Hom-Lie Algebras
Abstract
The purpose of this paper is to study quadratic color Hom-Lie algebras. We present some constructions of quadratic color Hom-Lie algebras which we use to provide several examples. We describe \(T^*\)-extensions and central extensions of color Hom-Lie algebras and establish some cohomological characterizations.
Faouzi Ammar, Imen Ayadi, Sami Mabrouk, Abdenacer Makhlouf
The Extension Property in the Category of Direct Sum of Cyclic Torsion-Free Modules over a BFD
Abstract
It is known that, in the category of vector space, all automorphisms satisfy the extension property. However, Schupp (Proc. A.M.S. 101(2):226–228, 1987 [11]) proved that the automorphisms having the property of extension in the category of groups, characterize the inner automorphisms. Ben Yakoub (Port. Math. 51(2):231–233, 1994, [5]) proved that this result is not true in the category of algebras. But there are some very important results in the category of groups. Then, in order to generalize the result of Schupp, Abdelalim and Essannouni (Port. Math. (Nova) 59(3):325–333, 2002, [1]) characterized the automorphisms having the property of extension, in the category of abelian groups. Let A be an integral bounded factorization domain and M a direct sum of cyclic torsion-free modules over A. This work aims to prove that the automorphisms of M that satisfy the property of the extension are none other than the homotheties of invertible ratio.
Seddik Abdelalim, Abdelhak Chaichaa, Mostafa El garn

History of Mathematics

Frontmatter
Arabic Scientific and Technical Heritage in Morocco
Abstract
During the first centuries of the birth of the Muslim empire, many scientific disciplines were well developed. Subsequently, several new areas of knowledge had emerged. Thus, there was the birth of algebra and combinatorial analysis in mathematics and the birth of cryptanalysis in cryptology. There were some very important discoveries in several scientific fields such as in physics, in chemistry and in medicine. In this paper, we will overfly the development of certain scientific disciplines in Morocco by highlighting the application of the results of these disciplines in the socio-economic world.
Abdelmalek Azizi
Metadata
Title
Associative and Non-Associative Algebras and Applications
Editors
Mercedes Siles Molina
Dr. Laiachi El Kaoutit
Mohamed Louzari
L'Moufadal Ben Yakoub
Mohamed Benslimane
Copyright Year
2020
Electronic ISBN
978-3-030-35256-1
Print ISBN
978-3-030-35255-4
DOI
https://doi.org/10.1007/978-3-030-35256-1

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