Lie subalgebra of ;
– [(d0, d1, d2), ιi(x ⊗ x′)] = ιi(di(x) ⊗ x′);
–
– [ιi(x ⊗ x′), ιi+1(y ⊗ y′)] = ιi+2((x ∗ y) ⊗ (x′ ★ y′)) (indices modulo 3);
–
THEOREM 2.9 (Elduque (2004)).– Assume char
Different versions of this result using Hurwitz algebras instead of symmetric composition algebras have appeared over the years (see Elduque (2004) and the references therein). The advantage of using symmetric composition algebras is that new constructions of the exceptional simple Lie algebras are obtained, and these constructions highlight interesting symmetries due to the different triality automorphisms.
A few changes are needed for characteristic 3. Also, quite interestingly, over fields of characteristic 3 there are non-trivial symmetric composition superalgebras, and these can be plugged into the previous construction to obtain an extended Freudenthal’s magic square that includes some new simple finite dimensional Lie superalgebras (see Cunha and Elduque (2007)).
2.6. Concluding remarks
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.
2.7. Acknowledgments
This work has been supported by grants MTM2017-83506-C2-1-P (AEI/FEDER, UE) and E22 17R (Gobierno de Aragón, Grupo de referencia “Álgebra y Geometría”, co-funded by Feder 2014–2020 “Construyendo Europa desde Aragón”).
2.8. References
Adams, J.F. (1958). On the nonexistence of elements of Hopf invariant one. Bull. Amer. Math. Soc., 64, 279–282.
Baez, J.C. (2002). The octonions. Bull. Amer. Math. Soc. (N.S.), 39(2), 145–205.
Borel, A., Serre, J.-P. (1953). Groupes de Lie et puissances réduites de Steenrod. Amer. J. Math., 75, 409–448.
Bott, R., Milnor, J. (1958). On the parallelizability of the spheres. Bull. Amer. Math. Soc., 64, 87–89.
Cartan, E. (1925). Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sci. Math., 49, 361–374.
Chernousov, V., Elduque, A., Knus, M.-A., Tignol, J.-P. (2013). Algebraic groups of type D4, triality, and composition algebras. Doc. Math., 18, 413–468.
Conway, J.H., Smith, D.A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A K Peters, Ltd., Natick.
Cunha, I., Elduque, A. (2007). An extended Freudenthal magic square in characteristic 3. J. Algebra, 317(2), 471–509.
Ebbinghaus, H.-D., Hermes, H., Hirzebruch, F., Koecher, M., Mainzer, K., Neukirch, J., Prestel, A., Remmert, R. (1991). Numbers, Ewing, J.H. (ed.). With an introduction by Lamotke, K. Translated by Orde, H.L.S. Springer-Verlag, New York.
Elduque, A. (1997). Symmetric composition algebras. J. Algebra, 196(1), 282–300.
Elduque, A. (2004). The magic square and symmetric compositions. Rev. Mat. Iberoamericana, 20(2), 475–491.
Elduque, A., Myung, H.C. (1990). On Okubo algebras. In From Symmetries to Strings: Forty Years of Rochester Conferences, Das, E. (ed.). World Science Publishing, River Edge, 299–310.
Elduque, A., Myung, H.C. (1991). Flexible composition algebras and Okubo algebras. Comm. Algebra, 19(4), 1197–1227.
Elduque, A., Myung, H.C. (1993). On flexible composition algebras. Comm. Algebra, 21(7), 2481–2505.
Elduque, A., Pérez, J.M. (1996). Composition algebras with associative bilinear form. Comm. Algebra, 24(3), 1091–1116.
Elduque, A., Pérez, J.M. (1997). Infinite-dimensional quadratic forms admitting composition. Proc. Amer. Math. Soc., 125(8), 2207–2216.
Elman, R., Karpenko, N., Merkurjev, A. (2008). The Algebraic and Geometric Theory of Quadratic Forms. American Mathematical Society, Providence.
Faulkner, J.R. (1988). Finding octonion algebras in associative algebras. Proc. Amer. Math. Soc., 104(4), 1027–1030.
Frobenius, F.G. (1878). Über lineare substitutionen und bilineare formen. J. Reine Angew. Math., 84, 1–63.
Gille, P. (2014). Octonion algebras over rings are not determined by their norms. Canad. Math. Bull., 57(2), 303–309.
Hurwitz, A. (1898). Über die komposition der quadratischen formen von beliebig vielen variablen. Nachr. Ges. Wiss. Göttingen, 309–316.
Jacobson, N. (1958). Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo, 7(2), 55–80.
Kaplansky, I. (1953). Infinite-dimensional quadratic forms admitting composition. Proc. Amer. Math. Soc., 4, 956–960.
Kervaire,
