Speciality of Lie–Jordan Algebras

AbstractThe class of so-called Lie–Jordan algebras, which have one binary (Lie) operation [x,y] and one ternary (Jordan) operation {x,y,z}, that satisfy the identitiesx,y=−y,x,x,y,z=z,y,x,x,y,z=x,y,z−y,x,z,x,y,z,t=x,t,y,z+x,y,t,z+x,y,z,t,x,y,z,t,v=x,t,v,y,z−x,y,v,t,z+x,y,z,t,v.is introduced. It is proved that any such algebra is special, that is, isomorphic to a subalgebra of a Lie–Jordan algebra of the type A±, obtained from an associative algebra A via the operations [x,y]=xy−yx, {x,y,z}=xyz+zyx. As an application, we prove the conjecture about associativity of a certain loop constructed by A. Grishkov