Generalized Derivations of Hom–Lie Triple Systems

3-Derivations and 3-Automorphisms on Lie Algebras

“…The decomposition g = g 0 ⊕ g 1 is called a Z 2 -grading of g. [ad x 1 ,y 1 + z 1 , ad x 2 ,y 2 + z 2 ] S = ([ad x 1 ,y 1 , ad x 2 ,y 2 ] + ad z 1 ,z 2 ) + (ad x 1 ,y 1 (z 2 ) − ad x 2 ,y 2 (z 1 )). (15) for any x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ∈ T. Proposition 5 ([18]). Let (T, {•, •, •}) be a Lie triple system.…”

“…For all x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ∈ T, we only need to verify D satisfies the condition of derivations when it acts on [ad x 1 ,y 1 , ad x 2 ,y 2 ] S , [ad x 1 ,y 1 , z 1 ] S , [z 1 , z 2 ] S these three items. By (15) and the Jacobi identity we have…”

“…There is a close connection between Lie triple systems and Lie algebras, namely, a Lie algebra naturally gives rise to a Lie triple system and conversely a Lie triple system also gives rise to a Lie algebra which is called the standard embedding Lie algebra. Derivations and automorphisms on Lie triple systems were studied in [15,16]. In this article, we establish a relation between derivations (automorphisms) on a Lie triple system and derivations (automorphisms) on the corresponding standard embedding Lie algebra.…”

Central Extensions and Nijenhuis Operators of Hom- δ -Jordan Lie Triple Systems

“…In 2012, Yau showed the concept of Hom-Lie triple systems [17]. Later, generalized derivations of Hom-Lie triple systems were determined [18]. The cohomologies, 1-parameter formal deformations, and central extensions of Hom-Lie triple systems were discussed [19].…”

Double derivations of $n-$Hom-Lie color algebras

“…Their results were generalized by many authors. For example, in the case of color Lie algebras and Hom-Lie color algebras [9,37], Hom-Poisson color algebras [18] Lie triple systems and Hom-Lie triple systems [34,35], color n−ary Ω−algebras and multiplicative n−ary Hom-Ω−algebras [20,8] and many other works. Another generalization of derivations of Lie algebras are Lie triple derivations and generalized Lie triple derivations.…”