Concept of a birepresentation for the Moufang loops is elaborated. 2000 MSC: 20N05

A left Bol loop is a loop satisfying x(y(xz)) = (x(yx))z. The commutant of a loop is the set of all elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2k, k odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3, the commutant generat...

In this paper the linear representations of analytic Moufang loops are investigated. We prove that every representation of semisimple analytic Moufang loop is completely reducible and find all nonasso-ciative irreducible representations. We show that such representations are closely associated with the (anti-)self-dual Yang-Mills equations in R 8 .

A quasigroup identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. ((xy)x)z = x(y(xz)). Many well-known varieties of quasigroups are of Bol-Moufang type. We show that there are exactly 26 ...

We first show that every Moufang loop L which contains an abelian associative subloop M of index two and odd order must, in fact, be a group. We then use this to settle the question ”For what value of n = 2m, m odd, must a Moufang loop of order n be associative?” Introduction: This paper is motivated by a question asked by Rajah and Jamal in [19]: If L is a Moufang loop of order 2m with an abel...