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Title: Decompositions of linear spaces induced by n-linear maps
Authors: Calderón, Antonio Jesús 
Kaygorodov, Ivan
Saraiva, Paulo
Keywords: Linear space, n-linear map, orthogonality, invariant subspace, decomposition theorem.
Issue Date: 2019
Publisher: Taylor & Francis
Project: UID/MAT/00324/2019 
Serial title, monograph or event: Linear and Multilinear Algebra
Volume: 67
Issue: 6
Abstract: 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).
DOI: 10.1080/03081087.2018.1450829
Rights: embargoedAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

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