Vol. 35, No. 3, 1970

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Macdonald’s theorem for quadratic Jordan algebras

Robert Edward Lewand and Kevin Mor McCrimmon

Vol. 35 (1970), No. 3, 681–706

Macdonald’s Theorem says that if an identity in three variables x,y,z which is linear in z holds for all special Jordan algebras, it holds for all Jordan algebras. We show this is equivalent to saying the universal quadratic envelope 𝒰𝒬ℰ(F(2)) of the free Jordan algebra F(2) on two generators x,y is canonically isomorphic to the universal compound linear envelope 𝒰𝒬ℰ(F(2)). We generalize Macdonald’s Theorem from the case of linear Jordan algebras over a field of characteristic 2 to quadratic Jordan algebras over an arbitrary ring of scalars, at the same time improving on the results in the linear case by presenting 𝒰𝒬ℰ(F(2)) in terms of a finite number of generators and relations. Similarly we generalize Macdonald’s Theorem with Inverses concerning identities in x,x1,y,y1,z. Finally, we prove Shirshov’s Theorem that F(2) is special.

Mathematical Subject Classification 2000
Primary: 17C05
Received: 10 December 1969
Published: 1 December 1970
Robert Edward Lewand
Kevin Mor McCrimmon