Decompositions of linear spaces induced by n-linear maps

• Main Author: Calderón, Antonio Jesús
• Publication Date: 2019
• Other Authors: Kaygorodov, Ivan, Saraiva, Paulo
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: https://hdl.handle.net/10316/89499; https://doi.org/10.1080/03081087.2018.1450829
• topic: Linear space, n-linear map, orthogonality, invariant subspace, decomposition theorem.

Let V be an arbitrary linear space and f : V x ... x V \rightarrow V an n-linear map. It is proved that, for each choice of a basis B of V, the n-linear map f induces a (nontrivial) decomposition V = \oplus V_j as a direct sum of linear subspaces of V, with respect to B. It is shown that this decomposition is f-orthogonal in the sense that f(V, ..., V_j, ..., V_k,..., V) = 0 when j \neq k, and in such a way that any V_j is strongly f-invariant, meaning that f(V, ..., V_j, ..., V) \subset V_j. A sufficient condition for two different decompositions of V induced by an n-linear map f, with respect to two different bases of V, being isomorphic is deduced. The f-simplicity - an analogue 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 (2018).