The Structure of Alternative Division Rings

Irreducible bimodules over alternative algebras and superalgebras

“…We recall some useful identities that hold in any alternative algebra (see [3], [17], [27]): (6) (z, x, ty) = −(z, t, xy) + (z, x, y)t + (z, t, y)x, (7) (z, x, yt) = −(z, t, yx) + x(z, t, y) + t(z, x, y)..…”

On n-ary Hom–Nambu and Hom–Nambu–Lie algebras

“…In view of Corollary 3.6, we have the following method of constructing Hom-Lie triple systems from Maltsev algebras. Maltsev observed in [37] (see also [11]) that every alternative algebra (A, µ) is Maltsev-admissible, i.e., the commutator algebra A − = (A, [, ] = µ − µ op ) is a Maltsev algebra. The Hom-version of this fact, that Hom-alternative algebras are Hom-Maltsev admissible, is proved in [62], but we do not need that result here.…”

A Class of Simple Moufang Loops

“…We consider the Cayley-Dickson division algebras A over the real field R* and let a basis for A be given by 1, ft, • • • , ft. We recall that 1, ft-, ft, ek, for ii,j, ¿) = (1, 2, 4), (2, 3, 5), (3,4,6), (4, 5, 7), (5, 6, 1), (6, Proof. The center Zm consists of the elements ± 1 and again we will consider only those subloops of M properly containing ZM-Thus, if N is any normal subloop of M under consideration, N contains an element x not in the center of A and x will lie in a quaternion subalgebra Qix).…”

“…It should be noted that these rings are the Cayley algebras that are not division algebras. 3. Moufang loops in simple alternative rings.…”