J.D. Phillips
On Moufang A-loops

Comment.Math.Univ.Carolinae 41,2 (2000) 371-375.

Abstract:In a series of papers from the 1940's and 1950's, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck's colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are Moufang ([5]). In [2] we relaunched this now over 50 year old program by examining conditions under which general --- not necessarily commutative --- diassociative A-loops are, in fact, Moufang. Here, we finish part of the program by characterizing Moufang A-loops. We also investigate simple diassociative A-loops as well as a class of centrally nilpotent diassociative A-loops. These results, {in toto}, reveal the distinguished positions two familiar classes of diassociative A-loops --- namely groups and commutative Moufang loops--play in the general theory.

Keywords: diassociative, A-loop, Moufang
AMS Subject Classification: Primary 20N05; Secondary 20A