$n$-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 $\frak L$ admitting a quasi-multiplicative basis, with restrictions neither on the dimensions nor on the base field $\mathbb F$ and study its structure. If we write $\frak L = \mathbb V \oplus \mathbb W$ with $\mathbb V$ and $0 \neq \mathbb W$ linear subspaces, we say that a basis of homogeneous elements $\mathfrak{B} = \{e_i\}_{i \in I}$ of $\mathbb W$ is quasi-multiplicative if given $0 < k < n,$ for $i_1,\dots,i_k \in I$ and $\sigma \in \mathbb S_n$ satisfies $\langle e_{i_1}, \dots, e_{i_k}, \mathbb V, \dots, \mathbb V \rangle_{\sigma} \subset \mathbb{F}e_{j_{\sigma}}$ for some $j_{\sigma} \in I;$ the product of elements of the basis $\langle e_{i_1}, \dots, e_{i_n} \rangle$ belongs to $\mathbb{F}e_j$ for some $j \in I$ or to $\mathbb V$, and a similar condition is verified for the product $\langle \mathbb V, \dots,\mathbb V \rangle$. We state that if $\frak L$ admits a quasi-multiplicative basis then it decomposes as $\mathfrak{L} ={\mathcal U} \oplus (\sum\limits {\frak J}_{k})$ with any ${\frak J}_k$ a well described color gLt-ideal of $\frak L$ admitting also a quasi-multiplicative basis, and ${\mathcal U}$ a linear subspace of $\mathbb V$. Also the minimality of $\frak L$ 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 $\mu$-quasi-multiplicative basis inherited by the one of $\frak L$.