Читать книгу Algebra and Applications 1 - Abdenacer Makhlouf - Страница 37

It is impossible to give a thorough account of composition algebras in a few pages, so many things have had to be left out: Pfister forms and the problem of composition of quadratic forms (see Shapiro (2000)), composition algebras over rings (or even over schemes), where Hurwitz algebras are no longer determined by their norms (see Gille (2014)), the closely related subject of absolute valued algebras (see Rodríguez-Palacios (2004)), etc.

The interested reader may consult the following studies: (Conway and Smith 2003; Springer and Veldkamp 2000; Ebbinghaus et al. 1991; Knus et al. 1998; Okubo 1995). Baez (2002) is a beautiful introduction to octonions and some of their many applications.

Let us conclude with the first words of Okubo in his introduction to the monograph (Okubo 1995):

The saying that God is the mathematician, so that, even with meager experimental support, a mathematically beautiful theory will ultimately have a greater chance of being correct, has been attributed to Dirac. Octonions algebra may surely be called a beautiful mathematical entity. Nevertheless, it has never been systematically utilized in physics in any fundamental fashion, although some attempts have been made toward this goal. However, it is still possible that non-associative algebras (other than Lie algebras) may play some essential future role in the ultimate theory, yet to be discovered.