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

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...

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Bibliographic Details
Published in:Applied categorical structures 2017-12, Vol.25 (6), p.1059-1076
Main Authors: Fernández-Casado, Rafael, García-Martínez, Xabier, Ladra, Manuel
Format: Article
Language:English
Subjects:
Algebra
Convex and Discrete Geometry
Geometry
Isomorphism
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Modules
Theory of Computation
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Summary: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 ℒ ℳ in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.
ISSN:0927-2852
1572-9095
DOI:10.1007/s10485-016-9472-9