SPECIAL IDENTITIES FOR QUASI-JORDAN ALGEBRAS

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities a(bc) = a(cb), (ba)a(2) = (ba(2))a, (b, a(2), c) = 2(b, a, c)a. These identities are satisfied by the product ab = a (sic) b + b proves a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree <= 7 is a consequence of the three identities in degree <= 4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.