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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). | URI: | https://hdl.handle.net/10316/89499 | DOI: | 10.1080/03081087.2018.1450829 | Rights: | embargoedAccess |
Appears in Collections: | I&D CMUC - Artigos em Revistas Internacionais |
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