On a class of nonflexible algebras

Abstract

An algebra A over a field F of characteristic not two will belong to the class 21 if I. The elements of A satisfy a nontrivial identity of the form ai(zx)y + a2(zy)x + a3y(zx) + cnx(zy) + a6(xz)y + a<¡(yz)x + aiy(xz) + a$x(yz) = 0 for fixed a<£F.II.There is an algebra B over F such that B satisfies (1), B has an identity element, and B is nonflexible; that is, there are elements x and y in B such that x(yx) ^ (xy)x.These conditions are similar to those used by Albert to define almost left alternative algebras [5].Albert's paper led to the study of algebras of (y, 8) type by Kleinfeld and Kokoris [9;11;12].Kokoris has shown that any simple finite dimensional algebra of characteristic prime to 30 of (7, 5) type is either alternative or has an identity element which is an absolutely primitive idempotent(2).Our alteration of Albert's conditions yields a new class of simple power-associative algebras.We note that property II seems more natural in light of Oehmke's results [13] and the remark that most of the well-known nonassociative algebras (Jordan, noncommutative Jordan, Lie, alternative, associative) satisfy the flexible identity x(yx) = (xy)x.In §2 it is shown that if F is algebraically closed then any algebra A over F belonging to 21 is quasi-equivalent in F to an algebra A(n) where A(p) satisfies one of the following identities: (i) (xy)z-x(yz) = (zy)x-z(yx), (ii) x(xy) + (yx)x = 2(xy)x, (iii) x(xy) + (yx)x=(xy)x+x(yx), (iv) x(yz+zy) + (yz+zy)x = (xy+yx)z+z(xy+yx).The remainder of the paper is devoted to the study of algebras which satisfy some one of these four identities.We find that any power-associative Presented to the Society, April 22