Actions in the Category of Precrossed Modules in Lie Algebras |
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| Authors: | J. M. Casas T. Datuashvili M. Ladra E. Ö Uslu |
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| Affiliation: | 1. Departamento de Matemática Aplicada I , Universidad de Vigo , Pontevedra , Spain jmcasas@uvigo.es;3. A. Razmadze Mathematics Institute of Tbilisi State University , Tbilisi , Georgia;4. Departamento de álgebra , Universidad de Santiago de Compostela , Santiago de Compostela , Spain;5. Department of Mathematics , Afyon Kocatepe University, Art and Science Faculty , Afyonkarahisar , Turkeyand Departamento de Matemàtica Aplicada I, Universidade de Vigo, Pontevedra, Spain |
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| Abstract: | For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category of precrossed modules. |
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| Keywords: | Action Actor Center Commutator Derivation Precrossed module Semidirect product Split extension classifier |
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