Nonabelian Extensions and Factor Systems for the Algebras of Loday

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 $\mathscr{P}$ algebra analogues to a series of results from W. R. Scott's $\textit{Group Theory}$, which gives an explicit theory of factor systems for the group case. Here $\mathscr{P}$ 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 $\mathscr{P}$ 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.