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Concept
Triple system
Summary
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).
A triple system is said to be a Lie triple system if the trilinear map, 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 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 k = span {Lu,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 -graded Lie algebra, the standard embedding of m, 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.
A triple system is said to be a Jordan triple system if the trilinear map, denoted {.,.,.}, satisfies the following identities:
The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if Lu,v:V→V is defined by Lu,v(y) = {u, v, y} then
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.
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