Split regular Hom-Leibniz color 3-algebras

Abstract

We introduce and describe the class of split regular Hom-Leibniz color 3-algebras as the natural extension of the class of split Lie algebras, split Lie superalgebras, split Lie color algebras, split regular Hom-Lie algebras, split regular Hom-Lie superalgebras, split regular Hom-Lie color algebras, split Leibniz algebras, split Leibniz superalgebras, split Leibniz color algebras, split regular Hom-Leibniz algebras, split regular Hom-Leibniz superalgebras, split regular Hom-Leibniz color algebras, split Lie 3-algebras, split Lie 3-superalgebras, split Lie color 3-algebras, split regular Hom-Lie 3-algebras, split regular Hom-Lie 3superalgebras, split regular Hom-Lie color 3-algebras, split Lie triple systems, split Lie triple supersystems, split Lie triple color systems, split regular Hom-Lie triple systems, split regular Hom-Lie triple supersystems, split regular Hom-Lie triple color systems, split Leibniz 3-algebras, split Leibniz 3-superalgebras, split Leibniz color 3-algebras, split regular Hom-Leibniz 3-algebras, split regular Hom-Leibniz 3-superalgebras, and some other algebras. More precisely, we show that any of such split regular Hom-Leibniz color 3-algebras T is of the form T = U + j I j , with U a subspace of the 0-root space T 0 , and I j an ideal of T satisfying for j = k : [T, I j , I k ] + [I j , T, I k ] + [I j , I k , T ] = 0.Moreover, if T is of maximal length, we characterize the simplicity of T in terms of a connectivity property in its set of non-zero roots.

Non-associative algebraic structures: classification and structure

“…The present part is based on the papers written together with Alexandre Pozhidaev, Antonio Jesús Calderón, Elisabete Barreiro, José María Sánchez, Paulo Saraiva, and Yury Popov [22,24,50,168,172].…”

“…In our joint work with Popov [168], we study the structure of split regular Hom-Leibniz 3-algebras of arbitrary dimension and over an arbitrary base field F. Split structures first appeared in the classical theory of (finite-dimensional) Lie algebras, but have been extended to more general settings like, for example, Leibniz algebras [53], Poisson algebras, Leibniz superalgebras, regular Hom-Lie algebras, regular Hom-Lie superalgebras, regular Hom-Lie color algebras, regular Hom-Poisson algebras [14], regular Hom-Leibniz algebras, regular BiHom-Lie algebras [54], and regular BiHom-Lie superalgebras, among many others. As for the study of split ternary structures, see [49] for Lie triple systems, twisted inner derivation triple systems, Lie 3-algebras [49], Leibniz 3-algebras [55], and for Leibniz triple systems.…”

“…Theorem 55 (Theorem 3.16, [168]). Let (T, ϕ) be a split Hom-Leibniz color 3-algebra with multiplication algebra L. Suppose that the root systems Λ T and Λ L are symmetric.…”

Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

“…In recent times, there is a numerous studies on different type of algebras with derivations, see [3], [7], [19]. Similarly, degenerations of algebras is an interesting subject, which were studied in various papers, for the study of degenerations of Leibniz algebras and related structures, see [11], [8], [10]. As Rota-Baxter operator is a kind of generalization of integral operator, therefore, it is natural to consider algebras with Rota-Baxter operator analogous to algebras with differentials.…”