Abstract
Let be an arbitrary linear space and
an n-linear map. It is proved that, for each choice of a basis
of
, the n-linear map f induces a (nontrivial) decomposition
as a direct sum of linear subspaces of
, with respect to
. It is shown that this decomposition is f-orthogonal in the sense that
when
, and in such a way that any
is strongly f-invariant, meaning that
A sufficient condition for two different decompositions of
induced by an n-linear map f, with respect to two different bases of
, being isomorphic is deduced. The f-simplicity – an analog of the usual simplicity in the framework of n-linear maps – of any linear subspace
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.
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.