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On \(\mathsf {Lie}\)-central extensions of Leibniz algebras

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called \(\mathsf {Lie}\)-central extensions. We obtain a six-term exact homology sequence associated to a \(\mathsf {Lie}\)-central extension. This sequence, together with the relative commutators, allows us to characterize several classes of \(\mathsf {Lie}\)-central extensions, such as \(\mathsf {Lie}\)-trivial extensions, \(\mathsf {Lie}\)-stem extensions and \(\mathsf {Lie}\)-stem covers, and to introduce and characterize \(\mathsf {Lie}\)-unicentral, \(\mathsf {Lie}\)-capable, \(\mathsf {Lie}\)-solvable and \(\mathsf {Lie}\)-nilpotent Leibniz algebras.

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Authors and Affiliations

  1. Dpto. Matemática Aplicada, Universidad de Vigo, E. E. Forestal, Campus Universitario A Xunqueira, 36005, Pontevedra, Spain

    J. M. Casas

  2. A. Razmadze Mathematical Institute, Tbilisi State University, Tamarashvili Str. 6, 0177, Tbilisi, Georgia

    E. Khmaladze

Authors
  1. J. M. Casas
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  2. E. Khmaladze
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Correspondence to J. M. Casas.

Additional information

The authors wish to thank Prof. Teimuraz Pirashvili and anonymous referees for their suggestions and help in improving the presentation of this paper. Authors were supported by Ministerio de Economía y Competitividad (Spain) (European FEDER support included), Grant MTM2013-43687-P. E. Khmaladze was supported by Xunta de Galicia, Grant EM2013/016 (European FEDER support included) and by Shota Rustaveli National Science Foundation, Grant FR/189/5-113/14.

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Casas, J.M., Khmaladze, E. On \(\mathsf {Lie}\)-central extensions of Leibniz algebras. RACSAM 111, 39–56 (2017). https://doi.org/10.1007/s13398-016-0274-6

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  • DOI: https://doi.org/10.1007/s13398-016-0274-6

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