Structure of Sympathetic Jordan Algebras

This paper is devoted to the structure of complete and perfect Jordan algebras, which are called sympathetic Jordan algebras here. In particular, every perfect Jordan algebra [Formula: see text] contains a greatest special sympathetic ideal [Formula: see text] and a decomposition into direct sum of ideals about the greatest special sympathetic ideal. Besides, each perfect Jordan algebra [Formula: see text] also contains a solvable ideal [Formula: see text] which is greatest among the solvable ideals [Formula: see text] of [Formula: see text] such that [Formula: see text] and a decomposition into direct sum of subalgebras [Formula: see text], where [Formula: see text] is a sympathetic subalgebra of [Formula: see text], which is similar to the Levi decomposition of Lie algebras. Moreover, [Formula: see text] is sympathetic if and only if [Formula: see text]. What is more, a class of ideals of Jordan algebras such that the quotients are sympathetic Jordan algebras are studied and some vital properties about this kind of ideal are highlighted.