Triple system

For an account of that concept in combinatorics, see Steiner triple system.

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

(⋅,⋅,⋅):V×V×V->V.

The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949. 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).

## Contents

* 1 Lie triple systems
* 2 Jordan triple systems
* 3 Jordan pairs
* 4 See also
* 5 References

## Lie triple systems

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

[u,v,w]=−[v,u,w]
[u,v,w]+[w,u,v]+[v,w,u]=0
[u,v,[w,x,y]]=u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y.

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

It follows that

𝔤:=𝔥⊕ V

is a ℤ2-graded Lie algebra with 𝔥 of grade 0 and V of grade 1, and bracket

[(L,u),(M,v)]=([L,M]+Lu,v,L(v)−M(u)).

This is called the standard embedding of the Lie triple system V into a ℤ2-graded Lie algebra. Conversely, given any ℤ2-graded Lie algebra, the triple bracket [[u, v], w] makes the space of degree-1 elements into a Lie triple system.

However, these methods of converting a Lie triple system into a ℤ2-graded Lie algebra and vice versa are not inverses: more precisely, they do not define an equivalence of categories. For example, if we start with any abelian ℤ2-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain 𝔥 = span {Lu,v : u, v ∈ V} = {0}.

Given any Lie triple system V, and letting 𝔤=𝔥⊕ V be the corresponding ℤ2-graded Lie algebra, this decomposition of 𝔤 obeys the algebraic definition of a symmetric space, so if G is any connected Lie group with Lie algebra 𝔤 and H is a subgroup with Lie algebra 𝔥, then G/H is a symmetric space. Conversely, the tangent space of any point in any symmetric space is naturally a Lie triple system.

We can also obtain Lie triple systems from associative algebras. Given an associative algebra A and defining the commutator by [a,b]=ab−ba, any subspace of A closed under the operation

[a,b,c]=[[a,b],c]

becomes a Lie triple system with this operation.

## Jordan triple systems

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

{u,v,w}={u,w,v}
{u,v,{w,x,y}}={w,x,{u,v,y}}+{w,{u,v,x},y}−{{v,u,w},x,y}.

The second identity means that if Lu,v:V→V is defined by Lu,v(y) = {u, v, y} then

[Lu,v,Lw,x]:=Lu,v∘Lw,x−Lw,x∘Lu,v=Lw,{u,v,x}−L{v,u,w},x

so that the space of linear maps span {Lu,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra 𝔤0.

A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on 𝔤0. They induce an involution of

V⊕𝔤0⊕V*

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 𝔤0 and −1 on V and V*. A special case of this construction arises when 𝔤0 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).

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

[u,v,w]={u,v,w}−{v,u,w}.

## Jordan pairs

A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V−. The trilinear map is then replaced by a pair of trilinear maps

{⋅,⋅,⋅}+:V−×S2V+->V+
{⋅,⋅,⋅}−:V+×S2V−->V−.

The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being

{u,v,{w,x,y}+}+={w,x,{u,v,y}+}++{w,{u,v,x}+,y}+−{{v,u,w}−,x,y}+

and the other being the analogue with + and − subscripts exchanged. The trilinear maps are often viewed as quadratic maps

Q+:V+->Hom(V−,V+)
Q−:V−->Hom(V+,V−).

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

Lu,v+:V+->V+byLu,v+(y)={u,v,y}+

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

[Lu,v±,Lw,x±]=Lw,{u,v,x}±±−L{v,u,w}∓,x±

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

V+⊗V−->𝔤𝔩(V+)⊕𝔤𝔩(V−)

whose image is a Lie subalgebra 𝔤0, and the Jordan identities become Jacobi identities for a graded Lie bracket on

𝔤:=V+⊕𝔤0⊕V−,

making this space into a ℤ-graded Lie algebra 𝔤 with only grades 1, 0, and -1 being nontrivial, often called a 3-graded Lie algebra. Conversely, given any 3-graded Lie algebra

𝔤=𝔤+1⊕𝔤0⊕𝔤−1,

then the pair (𝔤+1,𝔤−1) is a Jordan pair, with brackets

{X∓,Y±,Z±}±:=[[X∓,Y±],Z±].

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

End(S2V+)≅S2V+*⊗S2V−*≅End(S2V−).

These arise in particular when 𝔤 above is semisimple, when the Killing form provides a duality between 𝔤+1 and 𝔤−1.

For a simple example of a Jordan pair, let V+ be a finite-dimensional vector space and V− the dual of that vector space, with the quadratic maps

Q+:V+->Hom(V−,V+)
Q−:V−->Hom(V+,V−)

given by

Q+(v)(f)=f(v)v
Q−(f)(v)=f(v)f

where v∈V+,f∈V−.

## See also

* Associator
* E7 (mathematics)
* Quadratic Jordan algebra

## References

* Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, 1754, Springer, ISBN 978-3-540-41426-1
* Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, ISBN 978-0-8218-2848-9, https://books.google.com/books?id=a9KFAwAAQBAJ

* Jacobson, Nathan (1949), "Lie and Jordan triple systems", American Journal of Mathematics 71 (1): 149–170, doi:10.2307/2372102
* Hazewinkel, Michiel, ed. (2001), "Lie triple system", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Lie_triple_system .
* Hazewinkel, Michiel, ed. (2001), "Jordan triple system", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Jordan_triple_system .
* Koecher, M. (1969), An elementary approach to bounded symmetric domains, Lecture Notes, Rice University
* Loos, Ottmar (1969), General Theory, Symmetric spaces, 1, W. A. Benjamin, OCLC 681278693, https://archive.org/details/symmetricspaces0000loos_k2a0/mode/2up
* Loos, Ottmar (1969), Compact Spaces and Classification, Symmetric spaces, 2, W. A. Benjamin
* Loos, Ottmar (1971), "Jordan triple systems, R-spaces, and bounded symmetric domains", Bulletin of the American Mathematical Society 77 (4): 558–561, doi:10.1090/s0002-9904-1971-12753-2
* Loos, Ottmar (2006), Jordan pairs, Lecture Notes in Mathematics, 460, Springer, ISBN 978-3-540-37499-2, https://books.google.com/books?id=6Zl8CwAAQBAJ&pg=PR1
* Loos, Ottmar (1977), Bounded symmetric domains and Jordan pairs, Mathematical lectures, University of California, Irvine, http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf
* Meyberg, K. (1972), Lectures on algebras and triple systems, University of Virginia, http://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf
* Rosenfeld, Boris (1997), Geometry of Lie groups, Mathematics and its Applications, 393, Kluwer, p. 92, ISBN 978-0792343905
* Tevelev, E. (2002), "Moore-Penrose inverse, parabolic subgroups, and Jordan pairs", Journal of Lie Theory 12: 461–481, Bibcode: 2001math......1107T, http://www.emis.de/journals/JLT/vol.12_no.2/9.html

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