Abstract

Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications are numerous and well-known in these common settings. We construct $\mathcal{P}$ algebra analogues to a series of results from W. R. Scott’s Group Theory, which gives an explicit theory of factor systems for the group case. Here $\mathcal{P}$ ranges over Leibniz, Zinbiel, diassociative, and dendriform algebras, which we dub “the algebras of Loday,” as well as over Lie, associative, and commutative algebras. Fixing a pair of $\mathcal{P}$ algebras, we develop a correspondence between factor systems and extensions. This correspondence is strengthened by the fact that equivalence classes of factor systems correspond to those of extensions. Under this correspondence, central extensions give rise to 2-cocycles while split extensions give rise to (nonabelian) 2-coboundaries.

摘要

因子系统是处理代数结构（例如群、李代数和结合代数）的可拓问题的工具。在这些常见的环境中，它们的应用程序众多且广为人知。我们构建$\mathcal{磷}$代数类似于 WR Scott 的Group Theory的一系列结果，它为群案例提供了一个明确的因子系统理论。这里$\mathcal{磷}$范围涵盖 Leibniz、Zinbiel、分离代数和树状代数，我们称之为“洛代代数”，以及 Lie、结合和交换代数。固定一对$\mathcal{磷}$代数，我们开发了因子系统和扩展之间的对应关系。因子系统的等价类对应于扩展的等价类这一事实加强了这种对应关系。在这种对应关系下，中心扩展产生 2-cocycles，而分裂扩展产生（nonabelian）2-coboundaries。