内部预交叉模的Peiffer乘积和Peiffer交换子
Peiffer product and Peiffer commutator for internal pre-crossed modules
摘要：
本文引入了固定对象B上的内部预交叉模的Peiffer积和Peiffer交换子的概念，并将相应的经典概念推广到任意的半阿贝尔范畴C.证明了在C上较弱的附加条件下，交叉模是指其Peiffer交换子<X，X>平凡的预交叉模X.此外，我们还在C上提供了适当的条件（由一大类代数变元满足，包括群、结合代数、李代数和莱布尼茨代数），在此条件下，Peiffer积实现了B上交叉模范畴中的余积。
英文摘要：
In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator <X,X> is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varietes, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
