Nonabelian extensions and factor systems for the algebras of Loday

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