Triple system

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

$(\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.$

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 symmetric R-spaces and bounded symmetric domains.

Lie triple systems

A triple system is said to be a Lie triple system if the trilinear form, 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 L_u,v:V→V, defined by L_u,v(w) = [u, v, w], is a derivation of the triple product.

Jordan triple systems

A triple system is said to be a Jordan triple system if the trilinear form, 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 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

$[L_{u,v},L_{w,x}]:=L_{u,v}\circ L_{w,x}-L_{w,x}\circ L_{u,v}=L_{w,\{u,v,x\}}-L_{\{v,u,w\},x}$

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

References

• Sigurdur Helgason (2001), "Differential geometry, Lie groups, and symmetric spaces", American Mathematical Society, New York (1st edition: Academic Press, New York, 1978).
• Nathan Jacobson (1949), "Lie and Jordan triple systems", American Journal of Mathematics 71, pp. 149–170.
• Kamiya, Noriaki (2001) [1994], "Lie triple system", Encyclopedia of Mathematics, EMS Press.
• Kamiya, Noriaki (2001) [1994], "Jordan triple system", Encyclopedia of Mathematics, EMS Press.
• M. Koecher (1969), An elementary approach to bounded symmetric domains. Lecture Notes, Rice University, Houston, Texas.
• Ottmar Loos (1969), "Symmetric spaces. Volume 1: General Theory. Volume 2: Compact Spaces and Classification", W. A. Benjamin, New York.
• Ottmar Loos (1971), "Jordan triple systems, R-spaces, and bounded symmetric domains", Bulletin of the American Mathematical Society 77, pp. 558–561. (doi: 10.1090/S0002-9904-1971-12753-2)