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n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis

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Abstract

The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of n-Lie algebras and superalgebras, commutative Leibniz n-ary algebras and superalgebras, among others. We focus on the class of generalized Lie-type color algebras \(\frak L\) admitting a quasi-multiplicative basis, with restrictions neither on the dimensions nor on the base field \(\mathbb F\) and study its structure. We state that if \(\frak L\) admits a quasi-multiplicative basis then it decomposes as \(\mathfrak {L} ={\mathcal U} \oplus (\sum \limits {\frak J}_{k})\) with any \({\frak J}_{k}\) a well described color gLt-ideal of \(\frak L\) admitting also a quasi-multiplicative basis, and \({\mathcal U}\) a linear subspace of \(\mathbb V\). Also the minimality of \(\frak L\) is characterized in terms of the connections and it is shown that the above direct sum is by means of the family of its minimal color gLt-ideals, admitting each one a μ-quasi-multiplicative basis inherited by the one of \(\frak L\).

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Acknowledgments

The first and fourth authors acknowledge financial assistance by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. Second and fourth authors are supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2016-76327C31P. The third author is supported by FAPESP 17/15437-6. The fourth author acknowledges the Fundação para a Ciência e a Tecnologia for the grant with reference SFRH/BPD/101675/2014.

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Authors and Affiliations

  1. CMUC, Universidade de Coimbra, Coimbra, Portugal

    Elisabete Barreiro

  2. Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, Spain

    Antonio Jesús Calderón & José María Sánchez

  3. CMCC, Universidade Federal do ABC, Santo André, Brazil

    Ivan Kaygorodov

Authors
  1. Elisabete Barreiro
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  2. Antonio Jesús Calderón
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  3. Ivan Kaygorodov
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  4. José María Sánchez
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Corresponding author

Correspondence to Ivan Kaygorodov.

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Presented by: Jon F. Carlson

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Barreiro, E., Calderón, A.J., Kaygorodov, I. et al. n-Ary Generalized Lie-Type Color Algebras Admitting a Quasi-multiplicative Basis. Algebr Represent Theor 22, 1371–1386 (2019). https://doi.org/10.1007/s10468-018-9824-2

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