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.

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.

The Frattini Subsystem of a Lie Triple System

2009 ◽  
Vol 37 (10) ◽  
pp. 3750-3759 ◽  
Author(s):  
Zhixue Zhang ◽  
Liangyun Chen ◽  
Wenli Liu ◽  
Ximei Bai
Keyword(s):  
Triple System ◽  
Lie Triple System

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 The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

2018 ◽  
Vol 2019 (20) ◽  
pp. 6179-6215 ◽  
Author(s):  
Jie Du ◽  
Qiang Fu
Keyword(s):  
Canonical Basis ◽  
Alternative Algebra ◽  
Integral Form ◽  
Loop Algebra ◽  
Algebra Structure ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Weyl Duality ◽  
Integral Quantum

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

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

Lie triple systems and Leibniz algebras

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Revaz Kurdiani
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Triple System ◽  
Leibniz Algebra ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Lie Triple System ◽  
Triple Systems ◽  
Leibniz Algebras ◽  
Universal Central Extensions

AbstractThe present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

scholarly journals Jordan–Lie Super Algebra and Jordan–Lie Triple System

1997 ◽  
Vol 198 (2) ◽  
pp. 388-411 ◽  
Author(s):  
Susumu Okubo ◽  
Noriaki Kamiya
Keyword(s):  
Triple System ◽  
Lie Triple System
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