Freudenthal-Kantor triple system

A triple system considered for constructing all simple Lie algebras (cf. Lie algebra), and introduced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of a generalized Jordan triple system of second order developed by I.L. Kantor.

Recall that a triple system is a vector space over a field together with a -trilinear mapping .

For , a vector space over a field with the trilinear product is called a Freudenthal–Kantor triple system if

(a1)

(a2)

where and .

In particular, a Freudenthal–Kantor triple system is said to be balanced if there exists a bilinear form such that , for all .

This balancing property is closely related to metasymplectic geometry.

Note that if and (identically), then the Freudenthal–Kantor triple system reduces to a Jordan triple system.

As the notion of a Freudenthal–Kantor triple system includes the notions of a generalized Jordan triple system of second order, a structurable algebra, and an Allison–Hein triple system, it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices.

Let be a vector space with a bilinear form . Then is a Freudenthal–Kantor triple system with respect to the triple product . In particular, it is important that the linear span of the set makes a Jordan triple system of with respect to the triple product .

Let be a Freudenthal–Kantor triple system. The vector space becomes a Lie triple system with respect to the triple product defined by

Using this, one can obtain the Lie triple system associated with ; it is denoted be .

Using the concept of the standard embedding Lie algebra associated with a Lie triple system , one can obtain the construction of associated with a Freudenthal–Kantor triple system . In fact, put

equal to the linear span of the endomorphisms

;

;

equal to the linear span of the endomorphisms

equal to the linear span of the endomorphisms

Then one obtains the decomposition

and, more precisely,

These results imply the dimensional formula

This algebra is called the Lie algebra associated with .

The concepts of a triple system and a supertriple system are important in the theory of quarks and Yang–Baxter equations.

Note that a "triple system" in the sense discussed above is totally different from "triple system" in combinatorics (see, e.g., Steiner system).

References

[a1]	H. Freudenthal, "Beziehungen der und zur Oktavenebene I–II" Indag. Math. , 16 (1954) pp. 218–230; 363–386
[a2]	N. Kamiya, "The construction of all simple Lie algebras over from balanced Freudenthal–Kantor triple systems" , Contributions to General Algebra , 7 , Hölder–Pichler–Tempsky, Wien (1991) pp. 205–213
[a3]	N. Kamiya, "On Freudenthal–Kantor triple systems and generalized structurable algebras" , Non-Associative Algebra and Its Applications , Kluwer Acad. Publ. (1994) pp. 198–203
[a4]	N. Kamiya, S. Okubo, "On -Lie supertriple systems associated with -Freudenthal–Kantor supertriple systems" Proc. Edinburgh Math. Soc. , 43 (2000) pp. 243–260
[a5]	I.L. Kantor, "Models of exceptional Lie algebras" Soviet Math. Dokl. , 14 (1973) pp. 254–258
[a6]	S. Okubo, "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press (1995)
[a7]	K. Yamaguti, "On the metasymplectic geometry and triple systems" Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. , 306 (1977) pp. 55–92 (In Japanese)