Triple System - Lie Triple Systems

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) =, 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, w] makes m into a Lie triple system.