In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of Hodge (2001). In a nal section, we begin a study of the cohomology,of Lie triple systems.
The most significant models are Lie triple frameworks and Jordan triple frameworks. They were presented by Nathan Jacobson in 1949 to examine subspaces of affiliated algebras shut under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. Specifically, any Lie variable based math characterizes a Lie triple framework and any Jordan polynomial math characterizes a Jordan triple framework. They are significant in the speculations of symmetric spaces, especially Hermitian symmetric spaces and their speculations (symmetric R-spaces and their noncompact duals).A triple framework is supposed to be a Lie triple framework if the trilinear map, indicated {displaystyle [cdot ,cdot ]}{displaystyle [cdot ,cdot ]}, fulfills the accompanying personalities:
{displaystyle [u,v,w]=-[v,u,w]}[u,v,w]=-[v,u,w]
{displaystyle [u,v,w]+[w,u,v]+[v,w,u]=0}[u,v,w]+[w,u,v]+[v,w,u]=0
{displaystyle [u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].}[u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].
The initial two personalities unique the slant evenness and Jacobi character for the triple commutator, while the third character implies that the straight guide Lu,v: V → V, characterized by Lu,v(w) = [u, v, w], is an induction of the triple item. The personality likewise shows that the space k = range {Lu,v : u, v ∈ V} is shut under commutator section, consequently a Lie variable based math. Composing m instead of V, it follows that
{displaystyle {mathfrak {g}}:=koplus {mathfrak {m}}}{displaystyle {mathfrak {g}}:=koplus {mathfrak {m}}} can be made into a {displaystyle mathbb {Z} _{2}}mathbb {Z} _{2}-reviewed Lie variable based math, the standard installing of m, with section {displaystyle [(L,u),(M,v)]=([L,M]+L_{u,v},L(v)- M(u)).}[(L,u),(M,v)]=([L,M]+L_{{u,v}},L(v)- M(u)). The disintegration of g is plainly a symmetric deterioration for this Lie section, and consequently if G is an associated Lie bunch with Lie variable based math g and K is a subgroup with Lie polynomial math k, at that point G/K is a symmetric space. Alternately, given a Lie variable based math g with such a symmetric deterioration (i.e., it is the Lie polynomial math of a symmetric space), the triple section [[u, v], w] makes m into a Lie triple framework.