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 Lu,v:V→V, defined by Lu,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 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.
Read more about this topic: Triple System
Famous quotes containing the words lie, triple and/or systems:
“How clean the sun when seen in its idea,
Washed in the remotest cleanliness of a heaven
That has expelled us and our images . . .
The death of one god is the death of all.
Let purple Phoebus lie in umber harvest,
Let Phoebus slumber and die in autumn umber....”
—Wallace Stevens (1879–1955)
“And DANTE searched the triple spheres,
Moulding nature at his will,
So shaped, so colored, swift or still,
And, sculptor-like, his large design
Etched on Alp and Apennine.”
—Ralph Waldo Emerson (1803–1882)
“We have done scant justice to the reasonableness of cannibalism. There are in fact so many and such excellent motives possible to it that mankind has never been able to fit all of them into one universal scheme, and has accordingly contrived various diverse and contradictory systems the better to display its virtues.”
—Ruth Benedict (1887–1948)