Power-associative algebras that are train algebras

On Some Jordan Baric Algebras

“…Let (A, ω) be a baric algebra satisfying (2). If e is a nonzero idempotent of A, then e = e 2 e 2 = ω(e)e 3 = ω(e)e, so that ω(e) = 1 and e = e 3 .…”

“…• If (A, ω) satisfies (2), then the set of nonzero idempotents of A is the set (3), then the set of nonzero idempotents of A is the set…”

“…As for baric algebras satisfying (4), the following example in [2] shows that they are not Jordan algebras in general. Then (A, ω) is a baric algebra satisfying (4).…”

“…It was shown in [2] how to move from algebras satisfying (2) to algebras satisfying (3) and back. Actually, this is a particular case of the gametization process considered in [8] (for train algebras of rank 3) and in [15], which will play a key role here.…”

“…Then, in Sec. 3, the way to obtain a Jordan algebra of generic degree at most 4 in [5] will be reviewed, and this will be used to prove in a very simple way, not relying on properties of Peirce decompositions, that baric algebras satisfying (2) or (3) are always Jordan algebras, while algebras satisfying (1) are very close to being such. A characterization of the nonzero idempotent elements in the algebras satisfying (1), (2) or (3) will be given in Sec.…”