Idempotents and Peirce gradings of Jordan algebras
The authors give a necessary condition on Peirce gradings of a Jordan algebra \(J\) for them to come from idempotents in a bigger algebra. The natural extension of \(J\) where the idempotent element should be found is constructed. It is proved that this necessary condition is also sufficient for Jordan algebras without 2-torsion, and it holds automatically in sufficiently regular (for example, nondegenerate) Jordan algebras. Also the authors consider Peirce gradings of associative algebras and show that they always come from idempotents in bigger algebras. The authors prove that not all Peirce gradings in special Jordan algebras are induced by associative Peirce gradings.
scientific article; zbMATH DE number 5378358