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 triple system).
References
[a1] | H. Freudenthal, "Beziehungen der ![]() ![]() |
[a2] | N. Kamiya, "The construction of all simple Lie algebras over ![]() |
[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 ![]() ![]() |
[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) |
Freudenthal-Kantor triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Freudenthal-Kantor_triple_system&oldid=42973