n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis

Abstract

The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of n-Lie algebras and superalgebras, commutative Leibniz n-ary algebras and superalgebras, among others. We focus on the class of generalized Lie-type color algebras L admitting a quasi-multiplicative basis, with restrictions neither on the dimensions nor on the base field F and study its structure. If we write L = V⊕W with V and 0 = W linear subspaces, we say that a basis of homogeneous elements B = {e i } i∈I of W is quasimultiplicative if given 0 < k < n, for i 1 , . . . , i k ∈ I and σ ∈ S n satisfies e i1 , . . . , e i k , V, . . . , V σ ⊂ Fe jσ for some j σ ∈ I; the product of elements of the basis e i1 , . . . , e in belongs to Fe j for some j ∈ I or to V, and a similar condition is verified for the product V, . . . , V . We state that if L admits a quasi-multiplicative basis then it decomposes as L = U ⊕ ( J k ) with any J k a well described color gLt-ideal of L admitting also a quasi-multiplicative basis, and U a linear subspace of V. Also the minimality of L is characterized in terms of the connections and it is shown that the above direct sum is by means of the family of its minimal color gLt-ideals, admitting each one a µ-quasi-multiplicative basis inherited by the one of L.F e i 1 , . . . , e in .Corollary 24. Suppose L is centerless and V is tight, then L decomposes as the direct sum of the color gLt-ideals given in Proposition 17,Proof. By Theorem 21, since U = 0, we just have to show the direct character of the sum. Givenusing the fact J [i] , J [h] , L, . . . , L σ = 0 for [i] = [h] and any σ ∈ S n we obtain x, J [i] , L, . . . , L σ = x, [j] ∈ I/ ∼ j ≁ i J [h] , L, . . . , L σ = 0.

Non-associative algebraic structures: classification and structure

“…The rest of our results in the present topic are dedicated to a generalization of the paper [42] of Calderón. Namely, in a joint work together with Barreiro, Calderón, and Sánchez [22], we obtained an analog of Calderón´s results in the class of color generalized Lie type n-ary algebras. Definition 28.…”

“…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].…”

“…Theorem 56 (Theorem 21, [22]). A color generalized Lie type algebra A = V⊕W admitting a quasi-multiplicative basis of W ̸ = 0 decomposes as…”

Decompositions of linear spaces induced by n-linear maps

“…At this point, a parenthesis is due to underline the considerable amount of recent works where the above mentioned and similar connection techniques are applied as a tool to obtain interesting results in the frameworks of several types of algebras. 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]).…”