Universal enveloping crossed module of Leibniz crossed modules and representations

Algebra, Analysis and Quantum Probability

Casado, Rafael F.^1 ; García-Martínez, Xabier^1 ; Ladra, Manuel^1

Department of Algebra, Universidade de Santiago de Compostela, Santiago de Compostela

15782, Spain^1

关键词: Crossed module;    Functors;    Universal enveloping algebras;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/697/1/012007/pdf
DOI  :  10.1088/1742-6596/697/1/012007

来源: IOP

【 摘 要 】

The universal enveloping algebra functor UL: Lb → Alg, defined by Loday and Pirashvili [1], is extended to crossed modules. 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. Note that the procedure followed in the proof for the Lie case cannot be adapted, since the actor in the category of Leibniz crossed modules does not always exist.