# Triple system

In algebra, a **triple system** (or **ternar**) is a vector space *V* over a field **F** together with a **F**-trilinear map

The most important examples are **Lie triple systems** and **Jordan triple systems**. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[*u*, *v*], *w*] and triple anticommutators {*u*, {*v*, *w*}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).

## Lie triple systems

A triple system is said to be a Lie triple system if the trilinear form, denoted [.,.,.], satisfies the following identities:

The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L_{u,v}:*V*→*V*, defined by L_{u,v}(*w*) = [*u*, *v*, *w*], is a derivation of the triple product. The identity also shows that the space **k** = span {L_{u,v}: *u*, *v* ∈ *V*} is closed under commutator bracket, hence a Lie algebra.

Writing **m** in place of *V*, it follows that

can be made into a Lie algebra with bracket

The decomposition of **g** is clearly a symmetric decomposition for this Lie bracket, and hence if *G* is a connected Lie group with Lie algebra **g** and *K* is a subgroup with Lie algebra **k**, then *G*/*K* is a symmetric space.

Conversely, given a Lie algebra **g** with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket [[*u*, *v*], *w*] makes **m** into a Lie triple system.

## Jordan triple systems

A triple system is said to be a Jordan triple system if the trilinear form, denoted {.,.,.}, satisfies the following identities:

The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if L_{u,v}:*V*→*V* is defined by L_{u,v}(*y*) = {*u*, *v*, *y*} then

so that the space of linear maps span {L_{u,v}:*u*,*v* ∈ *V*} is closed under commutator bracket, and hence is a Lie algebra **g**_{0}.

Any Jordan triple system is a Lie triple system with respect to the product

A Jordan triple system is said to be **positive definite** (resp. **nondegenerate**) if the bilinear form on *V* defined by the trace of L_{u,v} is positive definite (resp. nondegenerate). In either case, there is an identification of *V* with its dual space, and a corresponding involution on **g**_{0}. They induce an involution of

which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on **g**_{0} and −1 on *V* and *V*^{*}. A special case of this construction arises when **g**_{0} preserves a complex structure on *V*. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type (the latter being bounded symmetric domains).

## Jordan pair

A Jordan pair is a generalization of a Jordan triple system involving two vector spaces *V*_{+} and *V*_{−}. The trilinear form is then replaced by a pair of trilinear forms

which are often viewed as quadratic maps *V*_{+} → Hom(*V*_{−}, *V*_{+}) and *V*_{−} → Hom(*V*_{+}, *V*_{−}). The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being

and the other being the analogue with + and − subscripts exchanged.

As in the case of Jordan triple systems, one can define, for *u* in *V*_{−} and *v* in *V*_{+}, a linear map

and similarly L^{−}. The Jordan axioms (apart from symmetry) may then be written

which imply that the images of L^{+} and L^{−} are closed under commutator brackets in End(*V*_{+}) and End(*V*_{−}). Together they determine a linear map

whose image is a Lie subalgebra , and the Jordan identities become Jacobi identities for a graded Lie bracket on

so that conversely, if

is a graded Lie algebra, then the pair is a Jordan pair, with brackets

Jordan triple systems are Jordan pairs with *V*_{+} = *V*_{−} and equal trilinear forms. Another important case occurs when *V*_{+} and *V*_{−} are dual to one another, with dual trilinear forms determined by an element of

These arise in particular when above is semisimple, when the Killing form provides a duality between and .

## See also

## References

- Bertram, Wolfgang (2000),
*The geometry of Jordan and Lie structures*, Lecture Notes in Mathematics,**1754**, Springer-Verlag, ISBN 3-540-41426-6 - Helgason, Sigurdur (2001),
*Differential geometry, Lie groups, and symmetric spaces*, American Mathematical Society (1st edition: Academic Press, New York, 1978). - Jacobson, Nathan (1949), "Lie and Jordan triple systems",
*American Journal of Mathematics*,**71**: 149–170, doi:10.2307/2372102, JSTOR 2372102 - Kamiya, Noriaki (2001), "Lie triple system", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4. - Kamiya, Noriaki (2001), "Jordan triple system", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4. - Koecher, M. (1969),
*An elementary approach to bounded symmetric domains*, Lecture Notes, Rice University - Loos, Ottmar (1969),
*Symmetric spaces. Volume 1: General Theory*, W. A. Benjamin - Loos, Ottmar (1969),
*Symmetric spaces. Volume 2: Compact Spaces and Classification*, W. A. Benjamin - Loos, Ottmar (1971), "Jordan triple systems,
*R*-spaces, and bounded symmetric domains",*Bulletin of the American Mathematical Society*,**77**: 558–561, doi:10.1090/s0002-9904-1971-12753-2 doi:10.1090/S0002-9904-1971-12753-2 - Loos, Ottmar (1975),
*Jordan pairs*, Lecture Notes in Mathematics,**460**, Springer-Verlag - Loos, Ottmar (1977),
*Bounded symmetric domains and Jordan pairs*(PDF), Mathematical lectures, University of California, Irvine - Meyberg, K. (1972),
*Lectures on algebras and triple systems*(PDF), University of Virginia - Rosenfeld, Boris (1997),
*Geometry of Lie groups*, Mathematics and its Applications,**393**, Dordrecht: Kluwer Academic Publishers, p. 92, ISBN 0792343905, Zbl 0867.53002 - Tevelev, E. (2002), "Moore-Penrose inverse, parabolic subgroups, and Jordan pairs",
*Journal of Lie theory*,**12**: 461–481