Lie triple system

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

A vector space with triple product is said to be a Lie triple system if

(a1)

(a2)

(a3)

for all .

Setting , then (a3) means that the left endomorphism is a derivation of (cf. also Derivation in a ring). Thus one denotes by .

Let be a Lie triple system and let be the vector space of the direct sum of and . Then is a Z2-graded Lie algebra with respect to the product

where , .

This algebra is called the standard embedding Lie algebra associated with the Lie triple system . This implies that is a homogeneous symmetric space (cf. also Homogeneous space; Symmetric space), that is, it is important in the correspondence with geometric phenomena and algebraic systems. The relationship between Riemannian globally symmetric spaces and Lie triple systems is given in [a4], and the relationship between totally geodesic submanifolds and Lie triple systems is given in [a1]. A general consideration of supertriple systems is given in [a2] and [a5].

Note that this kind of triple system is completely different from the combinatorial one of, e.g., a Steiner triple system.

References

[a1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[a2]	N. Kamiya, S. Okubo, "On -Lie supertriple systems associated with -Freudenthal–Kantor supertriple systems" Proc. Edinburgh Math. Soc. , 43 (2000) pp. 243–260
[a3]	W.G. Lister, "A structure theory of Lie triple systems" Trans. Amer. Math. Soc. , 72 (1952) pp. 217–242
[a4]	O. Loos, "Symmetric spaces" , Benjamin (1969)
[a5]	S. Okubo, N. Kamiya, "Jordan–Lie super algebra and Jordan–Lie triple system" J. Algebra , 198 : 2 (1997) pp. 388–411