On the structure of graded Leibniz triple systems

Abstract

We study the structure of a Leibniz triple system E graded by an arbitrary abelian group G which is considered of arbitrary dimension and over an arbitrary base field K. We show that E is of the formwith U a linear subspace of the 1-homogeneous component E 1 and any ideal I [j] of E, satisfying {I [j] , E,, where the relation ∼ in 1 = {g ∈ G\{1} : L g = 0}, defined by g ∼ h if and only if g is connected to h.

On the structure of graded $3-$Leibniz algebras

“…We will show that an arbitrary 3−Leibniz algebra T is of the form T = U ⊕ j I j , with U a linear subspace of T 1 , the homogeneous component associated to the unit element 1 in G, and any I j a well described graded ideal of T, satisfying [I j , T, I k ] = [I j , I k , T ] = [T, I j , I k ] = 0, if j = k. Finally, in section 4 we focus on the gr-simplicity and gr-primeness of graded 3−Leibniz algebras by centering our attention in those of maximal length. Our results extend the ones for graded Leibniz algebras [13], graded Leibniz triple systems [11] and for split 3−Leibniz algebras [16]. Finally, Section 5 provides a concrete example which characterizes the inner structure of graded 3−Leibniz algebras.…”

“…We will say that T is a simple 3−Leibniz algebra if [T, T, T ] = 0 and its only ideals are {0}, J and T (see [11]).…”

“…On the other hand, the descriptions of all possible gradings on algebras plays an important role both in the structure theory of finite dimentional and infinite dimentional algebras and its applications (see [5,6,8,10,12,14,16,17,18,25,31]). In particular, the interest in group grading on n−ary algebras has been remarkable in the last years, motivated in part by their application in physics, geometry and topology where they appear as the natural framwork for an algebraic model (see [2,9,11]).…”

On split Lie color triple systems

“…In [14], Calderón introduced techniques of connections of roots in the eld of split Lie color algebras. Recently, the structure of di erent classes of split algebras have been studied by using techniques of connections of roots (see for instance [15][16][17][18]). Our work is based on [13,14] and our aim is to consider the structure of split Lie color triple systems by the techniques of connections of roots.…”

Decompositions of linear spaces induced by n-linear maps

“…Without being exhaustive, these techniques were used, for instance, along with the notions of multiplicative basis and quasi-multiplicative basis not only related with algebras (see Caledrón and Navarro, [3,4]), but also with some n-ary generalizations (see, e.g., the works of Calderón, Barreiro, Kaygorodov and Sánchez in [1,2,7]). Further, connection techniques were also applied in the context of graded Lie algebras (see [5]) and to obtain structural results on graded Leibniz triple systems (see Cao and Chen (2016) [8]).…”