$k$-Modules over linear spaces by $n$-linear maps admitting a multiplicative basis

Abstract

We study the structure of certain $k$-modules $\mathbb{V}$ over linear spaces $\mathbb{W}$ with restrictions neither on the dimensions of $\mathbb{V}$ and $\mathbb{W}$ nor on the base field $\mathbb{F}$. A basis $\mathfrak{B}=\{v_i{\}}_{i\in I}$ of $\mathbb{V}$ is called multiplicative with respect to the basis ${\mathfrak{B}}^{\prime }=\{w_j{\}}_{j\in J}$ of $\mathbb{W}$ if for any $\sigma \in S_n,$ $i_1,\dots ,i_k\in I$ and $j_{k+1},\dots ,j_n\in J$ we have $[v_{i_1},\dots ,v_{i_k},w_{j_{k+1}},\dots ,w_{j_n}{]}_{\sigma }\in \mathbb{F}{v}_{r_{\sigma }}$ for some $r_{\sigma }\in I$. We show that if $\mathbb{V}$ admits a multiplicative basis then it decomposes as the direct sum $\mathbb{V}={⨁}_{\alpha }{V}_{\alpha }$ of well described $k$-submodules $V_{\alpha }$ each one admitting a multiplicative basis. Also the minimality of $\mathbb{V}$ is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal $k$-submodules, admitting each one a multiplicative basis. Finally we study an application of $k$-modules with a multiplicative basis over an arbitrary $n$-ary algebra with multiplicative basis.

References in corpus (5)

• Generalized derivations of (color) n-ary algebras
• Generalized derivations of multiplicative $n$-ary Hom-$\Omega$ color algebras
• Existence of a multiplicative basis for a finitely spaced module over an aggregate
• Modules over linear spaces admitting a multiplicative basis
• n-Dimensional filiform Leibniz algebras of length (n-1) and their derivations

Cited by in corpus (2)

• Non-associative algebraic structures: classification and structure
• $n$-Ary generalized Lie-type color algebras admitting a quasi-multiplicative basis