Decompositions of linear spaces induced by n-linear maps

Abstract

Let $\mathbb{V}$ be an arbitrary linear space and $f:\mathbb{V}\times \dots \times \mathbb{V}\to \mathbb{V}$ 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 $f(\mathbb{V},\dots ,V_j,\dots ,V_k,\dots ,\mathbb{V})=0$ when $j\ne k$, and in such a way that any $V_j$ is strongly f-invariant, meaning that $f(\mathbb{V},\dots ,V_j,\dots ,\mathbb{V})\subset V_j.$ 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 analog of the usual simplicity in the framework of n-linear 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 Calderón.

Keywords:

• Linear space
• n-linear map
• orthogonality
• invariant subspace
• decomposition theorem

AMS Subject Classifications:

• 15A03
• 15A21
• 15A69
• 15A86

Acknowledgements

The authors would like to express their gratitude to the referee for his exhaustive and careful review of the paper, as well as for his interesting suggestions which definitely helped to improve the work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by FAPESP [17/15437-6]; by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers [FQM298], [FQM7156] and by the project of the Spanish Ministerio de Educación y Ciencia [MTM2016-76327C31P].