ORIGINAL ARTICLE J Pur Appl Math Vol 2 No 1 April 2018 20 Department of Mathematics, College of Preliminary Year, Umm Al-Qura University, P.O. Box 14035, Makkah Al-Mukarramah 21955, Saudi Arabia. Correspondence: Amir Baklouti, Department of Mathematics, College of Preliminary Year, Umm Al-Qura University, P.O. Box 14035, Makkah Al-Mukarramah 21955, Saudi Arabia. e-mail: ambaklouti@uqu.edu.sa Received: March 30, 2018, Accepted: April 06, 2018, Published: April 09, 2018 This open-access article is distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC) (http:// creativecommons.org/licenses/by-nc/4.0/), which permits reuse, distribution and reproduction of the article, provided that the original work is properly cited and the reuse is restricted to noncommercial purposes. For commercial reuse, contact reprints@pulsus.com Involutve Hom-Lie triple systems Amir Baklout Baklouti A. Involutive Hom-Lie triple systems. J Pur Appl Math. 2018;2(1):20-21. ABSTRACT In this work we we prove that all involutive Hom-Lie triple systems are whether simple or semi-simple. Moreover, we prove that an involutive simple Lie triple system give a rise of InvolutiveHom-Lie triple system. Key Words: Jordan triple system; Lie triple system; Casimir operator; Quadratic lie algebra; TKK construction T he classification of semisimple Lie algebras with involutions can be found in (1). The Hom-Lie algebras were initially introduced by Hartwig, Larson and Silvestrov in (2) motivated initially by examples of deformed Lie algebras coming from twisted discretizations of vector fields. The Killing form K of g is nondegenerate and ˆ Iy  is symmetric with respect to K . In (3), the author studied Hom-Lie triple system using the double extension and gives an inductive description of quadratic Hom-Lie triple system. In this work we recall the definition of involutive Hom-Lie triple systems and some related structure and we prove that all involutive Hom-Lie triple systems are whether simple or semi-simple. Moreover,we prove that an involutive simple Lie triple system give a rise of Involutive Hom-Lie triple system. Definition 0.1 A Hom-Lie triple system is a triple ( ,[ , , ], ) L α −−− consisting of a linear space L , a trilinear map [, , ]: L L L L −−− × × → and a linear map : L L α → such that [, ,] 0 xyz = (skewsymmetry) [, ,] [,,] [,, ] 0 xyz yzx zxy + + = (ternary Jacobi identity) [ ( ), ( ),[ , , ]] u v xyz α α [[ , , ], ( ), ( )] [ ( ),[ , , ], ( )] [ ( ), ( ),[ , , ]], uvx y z x uvy z x y uvz α α α α α α = + + for all , ,,, xyzuvL ε . If Moreover α satisfies ([ , , ]) [ ( ), ( ), ( )] xyz x y z α α α α = (resp. 2 L id α = ) for all , , xyzL ε , we say that ( ,[ , , ], ) L α −−− is a multiplicative (resp. involutive) Hom- Lie triple system. A Hom-Lie triple system ( ,[ , , ], ) L α −−− is said to be regular if α is an automorhism of L . When the twisting map α is equal to the identity map, we recover the usual notion of Lie triple system (4,5). So, Lie triple systems are examples of Hom- Lie triple systems. If we introduce the right multiplication R defined for all , xyL ε by (, )( ): [, ,] Rxy z xyz = , then the conditions above can be written as follow: (, ) ( , ), Rxy Ryx =− (, ) (,) (,) 0, Rxyz Ryzx Rzxy + + = ( ( ), ( ))[ , ,] R u v xyz α α [ (,), ( ), ( )] [ ( ), (,), ( )] [ ( ), ( ), (,)]. Ruvx y z x Ruvy z x y Ruvz α α α α α α = + + We can also introduce the middle (resp. left) multiplication operator (,) : [, , ]( .(,) : [, , ]) Mxzy xyz respL y z x xyz = = for all , , . xyzL ε The equations above can be written in operator form respectively as follows: (, ) (, ) Mxy Lxy =− [1] (, ) (,) (, ) Mxy Myx Rxy − = for all , . xyL ε [2] We can write the equation above as one of the equivalent identities of operators: ( ( ), ( )) (, ) ( ( ), ( )) (,) ( ( (,), ( )) ( ( ), (,))) . R u v Rxy R x y Ruv RRuvx y R x Ruvy α α α α α α α − = + ( ( ), ( )) (,) ( ( ), ( )) (,) ( ( (,), ( )) ( ( ), (,))) . R u v Mxz M x z Ruv MRuvx z M x Ruvz α α α α α α α − = + Definition 0.2 Let ([, , ], ) L α −−− and ( ',[ , , ]', ') L α −−− be two two Hom-Lie triple systems (6). A linear map : ' f L L → is a morphism of Hom-Lie triple systems if ([ , , ]) [ ( ), ( ), ( )]' f xyz fx f y fz = and ' . f f α α =   In particular, if f is invertible, then ' L and ' L are said to be isomorphic. Definition 0.3 Let ( ,[ , , ], L α −−− be a Hom-Lie triple system and I be a subspace of L . We say that I is an ideal of L if [, , ] ILL I ⊂ and () . I I α ⊂ Definition 0.4 A Hom-Lie triple system L is said to be simple (resp. semisimple) if it contains no nontrivial ideal ( . () {0}). resp Rad L = According to a result in [ ? ], if A is aMalcev algebra, then ( ,[ , , ]) A −−− is a Lie triple system with triple product [, ,] 2( ) ( ) ( ). xyz xy z zx y yz x = − − [3] Thus, if A is aMalcev algebra and : A A α → is an algebra morphism, then, ( ,[ , , ] [, , ], A A α α α α = −−− = −−−  is a multiplicative Hom-Lie triple system, where [, , ] −−− is the triple product in [3].