VI - Cubic Jordan algebras

Summary

The fundamentals of cubic norm structures and Jordan algebras are laid out in the first two sections of the chapter. We then derive elementary principles of building up “big” cubic norm structures out of “smaller” ones before we proceed to study cubic Jordan matrix algebras, the most important “hands-on” examples of cubic Jordan algebras. Next we turn to elementary idempotents, which will be used to present a special version of the Jacobson Coordinatization Theorem. Proceeding to Freudenthal algebras, we show that they exist only in ranks 1, 3, 6, 9, 15, and 27, with those of rank 27 being (finally!) called Albert algebras. We define the notion of a split Freudenthal algebra and prove, in analogy with composition algebras, that all Freudenthal algebras are split by some faithfully flat extension, though not always by an étale cover. After having investigated isotopies and norm similarities, with an important characterization of isotopes in Jordan matrix algebras over LG rings as its central result, we study reduced Freudenthal algebras over fields by exhibiting various classifying quadratic form invariants, particularly those pertaining to the invariants mod 2 of Albert algebras.