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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: | http://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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Decompositions of linear spaces induced by n_linear maps_AC_IK_PS_repositorio_UC.pdf | 342.43 kB | Adobe PDF | View/Open |

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