Vol. 16, No. 1, 1966

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Some identities valid in special Jordan algebras but not valid in all Jordan algebras

Charles M. Glennie

Vol. 16 (1966), No. 1, 47–59

A Jordan algebra is defined by tbe identities:

x⋅y = y⋅x,(x⋅y)⋅y2 = (x⋅y2)⋅y.

The algebra AJ obtained from an associative algebra A on replacing the product xy by x y = 12(xy + yx) is easily seen to be a Jordan algebra. Any subalgebra of a Jordan algebra of this type is called special. It is known from work of Albert and Paige that the kernel of the natural homomorphism from the free Jordan algebra on three generators to the free special Jordan algebra on three generators is nonzero and consequently that there exist three-variable relations which hold identically in any homomorphic image of a special Jordan algebra but which are not consequences of the defining identities (1). Such a relation we shall call an S-identity. It is the purpose of this paper to establish that the minimium possible degree for an S-identity is 8 and to give an example of an S-identity of degree 8. In the final section we use an S-identity to give a short proof of the main theorem of Albert and Paige in a slightly strengthened form.

Mathematical Subject Classification
Primary: 17.40
Received: 13 August 1964
Published: 1 January 1966
Charles M. Glennie