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A Natural Extension of the Universal Enveloping Algebra Functor to Crossed Modules of Leibniz Algebras

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Abstract

The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category \(\mathcal {LM}\) in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.

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Authors and Affiliations

  1. University of Santiago de Compostela, Santiago de Compostela, Spain

    Rafael Fernández-Casado, Xabier García-Martínez & Manuel Ladra

Authors
  1. Rafael Fernández-Casado
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  2. Xabier García-Martínez
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  3. Manuel Ladra
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Corresponding author

Correspondence to Xabier García-Martínez.

Additional information

The authors were supported by Ministerio de Economía y Competitividad (Spain), grants MTM2013-43687-P and MTM2016-79661-P (European FEDER support included). The second and third authors were supported by Xunta de Galicia, grant GRC2013-045 (European FEDER support included). The second author is also supported by an FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain).

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Fernández-Casado, R., García-Martínez, X. & Ladra, M. A Natural Extension of the Universal Enveloping Algebra Functor to Crossed Modules of Leibniz Algebras. Appl Categor Struct 25, 1059–1076 (2017). https://doi.org/10.1007/s10485-016-9472-9

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Mathematics Subject Classifications (2000)