Multiplicative δ-derivation in alternative algebras

2018 ◽  
Vol 11 (04) ◽  
pp. 1850051 ◽  
Author(s):  
M. V. L. Bharathi ◽  
K. Jayalakshmi
Keyword(s):  
Alternative Algebra ◽  
Inner Derivation ◽  
Complex Matrices ◽  
Complex Algebra ◽  
Alternative Algebras ◽  
Derivation Formula ◽  
Multiplicative Formula

Every multiplicative [Formula: see text]-derivation of an alternative algebra [Formula: see text] is additive if there exists an idempotent [Formula: see text] in [Formula: see text] satisfying the following conditions: (i) [Formula: see text] implies [Formula: see text]; (ii) [Formula: see text] implies [Formula: see text]; (iii) [Formula: see text] implies [Formula: see text] for [Formula: see text]. In particular, every [Formula: see text]-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for [Formula: see text]-derivations. As an application, we apply multiplicative [Formula: see text]-derivation to an alternative complex algebra [Formula: see text] of all [Formula: see text] complex matrices to see how it decomposes into a sum of [Formula: see text]-inner derivation and a [Formula: see text]-derivation on [Formula: see text] given by an additive derivation [Formula: see text] on [Formula: see text].

Derivations of matrix semirings

2020 ◽  
pp. 2150150
Author(s):  
Dimitrinka Vladeva
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Idempotent Semiring ◽  
Inner Derivation ◽  
Commutative Case ◽  
Linear Map ◽  
Algebra Ii ◽  
Matrix Calculus ◽  
Derivation Formula ◽  
Matrix Formula

It is well known that if [Formula: see text] is a derivation in semiring [Formula: see text], then in the semiring [Formula: see text] of [Formula: see text] matrices over [Formula: see text], the map [Formula: see text] such that [Formula: see text] for any matrix [Formula: see text] is a derivation. These derivations are used in matrix calculus, differential equations, statistics, physics and engineering and are called hereditary derivations. On the other hand (in sense of [Basic Algebra II (W. H. Freeman & Company, 1989)]) [Formula: see text]-derivation in matrix semiring [Formula: see text] is a [Formula: see text]-linear map [Formula: see text] such that [Formula: see text], where [Formula: see text]. We prove that if [Formula: see text] is a commutative additively idempotent semiring any [Formula: see text]-derivation is a hereditary derivation. Moreover, for an arbitrary derivation [Formula: see text] the derivation [Formula: see text] in [Formula: see text] is of a special type, called inner derivation (in additively, idempotent semiring). In the last section of the paper for a noncommutative semiring [Formula: see text] a concept of left (right) Ore elements in [Formula: see text] is introduced. Then we extend the center [Formula: see text] to the semiring LO[Formula: see text] of left Ore elements or to the semiring RO[Formula: see text] of right Ore elements in [Formula: see text]. We construct left (right) derivations in these semirings and generalize the result from the commutative case.

Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation

2020 ◽  
pp. 2150097
Author(s):  
Qinxiu Sun
Keyword(s):  
Alternative Algebra ◽  
Alternative Algebras ◽  
Alternative Formula

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative [Formula: see text]-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.

Action theory of alternative algebras

2019 ◽  
Vol 26 (2) ◽  
pp. 177-197
Author(s):  
José Manuel Casas ◽  
Tamar Datuashvili ◽  
Manuel Ladra
Keyword(s):  
Action Theory ◽  
Alternative Algebra ◽  
Alternative Algebras

Abstract We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an actor in the category of alternative algebras.

scholarly journals Right alternative algebras with commutators in a nucleus

1992 ◽  
Vol 46 (1) ◽  
pp. 81-90
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Linear Span ◽  
Alternative Algebra ◽  
Wedderburn Decomposition ◽  
Alternative Algebras ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
The Right ◽  
Nil Radical ◽  
Algebra Over A Field

Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

scholarly journals Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Sylvain Attan ◽  
A. Nourou Issa
Keyword(s):  
Triple System ◽  
Alternative Algebra ◽  
System Structure ◽  
Algebra Structure ◽  
Lie Triple System ◽  
Alternative Algebras

Every multiplicative Hom-Maltsev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Maltsev algebra and on every multiplicative right (or left) Hom-alternative algebra.

scholarly journals Central Extension of Left Alternative Algebras and the Second Cohomology Group

2021 ◽  
Vol 2021 ◽  
pp. 1-3
Author(s):  
Said Boulmane
Keyword(s):  
Central Extension ◽  
Cohomology Group ◽  
Alternative Algebra ◽  
Closed Field ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Alternative Algebras

The purpose of this paper is to prove that the second cohomology group H 2 A , F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A .

scholarly journals CMMSE: Combinatorial structures and finite-dimensional alternative algebras

2021 ◽  
Author(s):  
Manuel Ceballos
Keyword(s):  
Theoretical Study ◽  
Alternative Algebra ◽  
New Method ◽  
Combinatorial Structure ◽  
Combinatorial Structures ◽  
Isomorphism Classes ◽  
Alternative Algebras ◽  
Finite Dimensional

In this paper, the link between combinatorial structures and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one constructs and draws the (pseudo)digraph associated with a given alternative algebra and the second one tests if a given combinatorial structure is associated with some alternative algebra.

THE ROLE OF POLYETHYLENIMINE IN ENHANCING PERFORMANCE OF GLUTAMATE BIOSENSORS

2018 ◽  
Vol 8 (3) ◽  
pp. 36-41
Author(s):  
Diep Do Thi Hong ◽  
Duong Le Phuoc ◽  
Hoai Nguyen Thi ◽  
Serra Pier Andrea ◽  
Rocchitta Gaia
Keyword(s):  
First Generation ◽  
Good Reproducibility ◽  
Complex Matrices ◽  
Long Term Stability ◽  
In Vitro Characterization ◽  
Glutamate Oxidase

Background: The first biosensor was constructed more than fifty years ago. It was composed of the biorecognition element and transducer. The first-generation enzyme biosensors play important role in monitoring neurotransmitter and determine small quantities of substances in complex matrices of the samples Glutamate is important biochemicals involved in energetic metabolism and neurotransmission. Therefore, biosensors requires the development a new approach exhibiting high sensibility, good reproducibility and longterm stability. The first-generation enzyme biosensors play important role in monitoring neurotransmitter and determine small quantities of substances in complex matrices of the samples. The aims of this work: To find out which concentration of polyethylenimine (PEI) exhibiting the most high sensibility, good reproducibility and long-term stability. Methods: We designed and developed glutamate biosensor using different concentration of PEI ranging from 0% to 5% at Day 1 and Day 8. Results: After Glutamate biosensors in-vitro characterization, several PEI concentrations, ranging from 0.5% to 1% seem to be the best in terms of VMAX, the KM; while PEI content ranging from 0.5% to 1% resulted stable, PEI 1% displayed an excellent stability. Conclusions: In the result, PEI 1% perfomed high sensibility, good stability and blocking interference. Furthermore, we expect to develop and characterize an implantable biosensor capable of detecting glutamate, glucose in vivo. Key words: Glutamate biosensors, PEi (Polyethylenimine) enhances glutamate oxidase, glutamate oxidase biosensors

scholarly journals On the Perron–Frobenius theory for complex matrices

2012 ◽  
Vol 437 (4) ◽  
pp. 1071-1088 ◽  
Author(s):  
Dimitrios Noutsos ◽  
Richard S. Varga
Keyword(s):  
Complex Matrices
Sign in / Sign up

Export Citation Format

Share Document