scholarly journals Snyder Space-Time: K-Loop and Lie Triple System

2010 ◽  
Author(s):  
Florian Girelli
Keyword(s):  
Triple System ◽  
Space Time ◽  
Lie Triple System

Differential equations of smooth loops related to some space-time models: Integrability conditions and geometry

2018 ◽  
Vol 27 (07) ◽  
pp. 1841004
Author(s):  
L. Sbitneva
Keyword(s):  
Differential Equations ◽  
Homogeneous Spaces ◽  
Space Time ◽  
Lie Triple System ◽  
Lie Group Theory ◽  
Moufang Loops ◽  
Integrability Conditions ◽  
Curvature And Torsion ◽  
The Right ◽  
Bol Loops

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory. A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops. We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration. The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

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

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

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.

Some subsystems of a lie triple system closely related to its Frattini subsystem

2013 ◽  
Vol 34 (5) ◽  
pp. 791-800
Author(s):  
Liangyun Chen ◽  
Dong Liu ◽  
Xiaoning Xu
Keyword(s):  
Triple System ◽  
Lie Triple System

scholarly journals Generalized derivations of Lie triple systems

2016 ◽  
Vol 14 (1) ◽  
pp. 260-271 ◽  
Author(s):  
Jia Zhou ◽  
Liangyun Chen ◽  
Yao Ma
Keyword(s):  
Triple System ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
Lie Triple System ◽  
Triple Systems ◽  
Basic Properties ◽  
Derivation Algebra ◽  
The Relationship

AbstractIn this paper, we present some basic properties concerning the derivation algebra Der (T), the quasiderivation algebra QDer (T) and the generalized derivation algebra GDer (T) of a Lie triple system T, with the relationship Der (T) ⊆ QDer (T) ⊆ GDer (T) ⊆ End (T). Furthermore, we completely determine those Lie triple systems T with condition QDer (T) = End (T). We also show that the quasiderivations of T can be embedded as derivations in a larger Lie triple system.

The Steinberg Lie Algebra st2(S)

2016 ◽  
Vol 23 (01) ◽  
pp. 129-136
Author(s):  
Yongjie Wang ◽  
Yiqian Shi ◽  
Yun Gao
Keyword(s):  
Lie Algebra ◽  
Triple System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lie Triple System ◽  
Necessary And Sufficient

Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2(S) of the Steinberg Lie algebra st2(S) and give a necessary and sufficient condition for I2(S)=0.

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