Triple system

Lie triple systems

A triple system is said to be a Lie triple system if the trilinear map, denoted [\cdot,\cdot,\cdot] , 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 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 of linear operators \mathfrak{h} = span {Lu,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that :\mathfrak{g} := \mathfrak{h} \oplus V is a \mathbb{Z}2-graded Lie algebra with \mathfrak{h} of grade 0 and V of grade 1, and bracket :[(L,u),(M,v)] = ([L,M]+L{u,v}, L(v) - M(u)). This is called the standard embedding of the Lie triple system V into a \mathbb{Z}_2-graded Lie algebra. Conversely, given any \mathbb{Z}_2-graded Lie algebra, the triple bracket [[u, v], w] makes the space of degree-1 elements into a Lie triple system.

However, these methods of converting a Lie triple system into a \mathbb{Z}_2-graded Lie algebra and vice versa are not inverses: more precisely, they do not define an equivalence of categories. For example, if we start with any abelian \mathbb{Z}_2-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain \mathfrak{h} = span {Lu,v : u, v ∈ V} = {0}.

Given any Lie triple system V, and letting \mathfrak{g} = \mathfrak{h} \oplus V be the corresponding \mathbb{Z}_2-graded Lie algebra, this decomposition of \mathfrak{g} obeys the algebraic definition of a symmetric space, so if G is any connected Lie group with Lie algebra \mathfrak{g} and H is a subgroup with Lie algebra \mathfrak{h}, then G/H is a symmetric space. Conversely, the tangent space of any point in any symmetric space is naturally a Lie triple system.

We can also obtain Lie triple systems from associative algebras. Given an associative algebra A and defining the commutator by [a,b] = ab - ba, any subspace of A closed under the operation :[a,b,c] = [[a,b],c] becomes a Lie triple system with this operation.

Jordan triple systems

A triple system V is said to be a Jordan triple system if the trilinear map, denoted {\cdot,\cdot,\cdot}, 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 second identity means that if Lu,v:V→V is defined by Lu,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 span {Lu,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra \mathfrak{g}_0.

A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on \mathfrak{g}_0. They induce an involution of :V\oplus\mathfrak g_0\oplus V^* which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on \mathfrak{g}_0 and −1 on V and V*. A special case of this construction arises when \mathfrak{g}_0 preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type (the latter being bounded symmetric domains).

Any Jordan triple system is a Lie triple system with respect to the operation : [u,v,w] = {u,v,w} - {v,u,w}.

Jordan pairs

A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V−. The trilinear map is then replaced by a pair of trilinear maps : {\cdot,\cdot,\cdot}+\colon V-\times S^2V_+ \to V_+ : {\cdot,\cdot,\cdot}-\colon V+\times S^2V_- \to V_-. The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being : {u,v,{w,x,y}+}+ = {w,x,{u,v,y}+}+ + {w, {u,v,x}+,y}+ - {{v,u,w}-,x,y}+ and the other being the analogue with + and − subscripts exchanged. The trilinear maps are often viewed as quadratic maps : Q_+ \colon V_+ \to \text{Hom}(V_-, V_+) : Q_- \colon V_- \to \text{Hom}(V_+, V_-) .

As in the case of Jordan triple systems, one can define, for u in V− and v in V+, a linear map : L^+{u,v}:V+\to V_+ \quad\text{by} \quad L^+{u,v}(y) = {u,v,y}+ and similarly L−. The Jordan axioms (apart from symmetry) may then be written : [L^{\pm}{u,v},L^{\pm}{w,x}] = L^{\pm}{w,{u,v,x}\pm}-L^{\pm}{{v,u,w}{\mp},x} which imply that the images of L+ and L− are closed under commutator brackets in End(V+) and End(V−). Together they determine a linear map : V_+\otimes V_- \to \mathfrak{gl}(V_+)\oplus \mathfrak{gl}(V_-) whose image is a Lie subalgebra \mathfrak{g}0, and the Jordan identities become Jacobi identities for a graded Lie bracket on :\mathfrak{g} := V+\oplus \mathfrak g_0\oplus V_-, making this space into a \mathbb{Z}-graded Lie algebra \mathfrak{g} with only grades 1, 0, and -1 being nontrivial, often called a 3-graded Lie algebra. Conversely, given any 3-graded Lie algebra : \mathfrak g = \mathfrak g_{+1} \oplus \mathfrak g_0\oplus \mathfrak g_{-1}, then the pair (\mathfrak g_{+1}, \mathfrak g_{-1}) is a Jordan pair, with brackets : {X_{\mp},Y_{\pm},Z_{\pm}}{\pm} := [[X{\mp},Y_{\pm}],Z_{\pm}].

Jordan triple systems are Jordan pairs with V+ = V− and equal trilinear maps. Another important case occurs when V+ and V− are dual to one another, with dual trilinear maps determined by an element of : \mathrm{End}(S^2V_+) \cong S^2V_+^* \otimes S^2V_-^*\cong \mathrm{End}(S^2V_-). These arise in particular when \mathfrak g above is semisimple, when the Killing form provides a duality between \mathfrak g_{+1} and \mathfrak g_{-1}.

For a simple example of a Jordan pair, let V_+ be a finite-dimensional vector space and V_- the dual of that vector space, with the quadratic maps

: Q_+ \colon V_+ \to \text{Hom}(V_-, V_+) : Q_- \colon V_- \to \text{Hom}(V_+, V_-)

given by

:Q_+(v)(f) = f(v) ,v :Q_-(f)(v) = f(v) , f

where v \in V_+, f \in V_-.

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