Decompositions of linear spaces induced by $n$-linear maps

Abstract

Let $\mathbb V$ be an arbitrary linear space and $f:\mathbb V \times \ldots \times \mathbb V \to \mathbb V$ an $n$-linear map. It is proved that, for each choice of a basis ${\mathcal B}$ of $\mathbb V$, the $n$-linear map $f$ induces a (nontrivial) decomposition $\mathbb V= \oplus V_j$ as a direct sum of linear subspaces of $\mathbb V$, with respect to ${\mathcal B}$. It is shown that this decomposition is $f$-orthogonal in the sense that $f(\mathbb V, \ldots, V_j, \ldots, V_k, \ldots, \mathbb V) =0$ when $j \neq k$, and in such a way that any $V_j$ is strongly $f$-invariant, meaning that $f(\mathbb V, \ldots, V_j, \ldots, \mathbb V) \subset V_j.$ A sufficient condition for two different decompositions of $\mathbb V$ induced by an $n$-linear map $f$, with respect to two different bases of $\mathbb V$, being isomorphic is deduced. The $f$-simplicity -- an analogue of the usual simplicity in the framework of $n$-liner maps -- of any linear subspace $V_j$ of a certain decomposition induced by $f$ is characterized. Finally, an application to the structure theory of arbitrary $n$-ary algebras is provided. This work is a close generalization the results obtained by A. J. Calder\'on (2018).