Noncommutative matrix Jordan algebras

Abstract

We consider noncommutative degree two Jordan algebras J \mathcal {J} of two by two matrices whose off diagonal entries are from an anticommutative algebra S \mathcal {S} . We give generators and relations for the automorphism group of J \mathcal {J} and determine the derivation algebra Der J \mathcal {J} in terms of mappings on S \mathcal {S} . We also give an explicit construction of all S \mathcal {S} for which Der J \mathcal {J} does not kill the diagonal idempotents and give conditions for nonisomorphic S \mathcal {S} ’s to give isomorphic J \mathcal {J} ’s.