Triple system

In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map

( ⋅ , ⋅ , ⋅ ) : V × V × V → V . {\displaystyle (\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.}

The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949. 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).

Lie triple systems

A triple system is said to be a Lie triple system if the trilinear map, denoted [ ⋅ , ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ,\cdot ]}, satisfies the following identities:

[ u , v , w ] = − [ v , u , w ] {\displaystyle [u,v,w]=-[v,u,w]}

[ u , v , w ] + [ w , u , v ] + [ v , w , u ] = 0 {\displaystyle [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 ] ] . {\displaystyle [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 h {\displaystyle {\mathfrak {h}}} = span {Lu,v: u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that

g := h ⊕ {\displaystyle {\mathfrak {g}}:={\mathfrak {h}}\oplus } V

is a Z 2 {\displaystyle \mathbb {Z} _{2}}-graded Lie algebra with h {\displaystyle {\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 ) ) . {\displaystyle [(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 Z 2 {\displaystyle \mathbb {Z} _{2}}-graded Lie algebra. Conversely, given any Z 2 {\displaystyle \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 Z 2 {\displaystyle \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 Z 2 {\displaystyle \mathbb {Z} _{2}}-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain h {\displaystyle {\mathfrak {h}}} = span {Lu,v: u, v ∈ V} = {0}.

Given any Lie triple system V, and letting g = h ⊕ {\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus } V be the corresponding Z 2 {\displaystyle \mathbb {Z} _{2}}-graded Lie algebra, this decomposition of g {\displaystyle {\mathfrak {g}}} obeys the algebraic definition of a symmetric space, so if G is any connected Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} and H is a subgroup with Lie algebra h {\displaystyle {\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 ] = a b − b a {\displaystyle [a,b]=ab-ba}, any subspace of A closed under the operation

[ a , b , c ] = [ [ a , b ] , c ] {\displaystyle [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 { ⋅ , ⋅ , ⋅ } {\displaystyle \{\cdot ,\cdot ,\cdot \}}, satisfies the following identities:

{ u , v , w } = { u , w , v } {\displaystyle \{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 } . {\displaystyle \{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 ∘ L w , x − L w , x ∘ L u , v = L w , { u , v , x } − L { v , u , w } , x {\displaystyle [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 g 0 {\displaystyle {\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 g 0 {\displaystyle {\mathfrak {g}}_{0}}. They induce an involution of

V ⊕ g 0 ⊕ V ∗ {\displaystyle 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 g 0 {\displaystyle {\mathfrak {g}}_{0}} and −1 on V and V*. A special case of this construction arises when g 0 {\displaystyle {\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 } . {\displaystyle [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

{ ⋅ , ⋅ , ⋅ } + : V − × S 2 V + → V + {\displaystyle \{\cdot ,\cdot ,\cdot \}_{+}\colon V_{-}\times S^{2}V_{+}\to V_{+}}

{ ⋅ , ⋅ , ⋅ } − : V + × S 2 V − → V − {\displaystyle \{\cdot ,\cdot ,\cdot \}_{-}\colon V_{+}\times S^{2}V_{-}\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 } + {\displaystyle \{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 + : V + → Hom ( V − , V + ) {\displaystyle Q_{+}\colon V_{+}\to {\text{Hom}}(V_{-},V_{+})}

Q − : V − → Hom ( V + , V − ) . {\displaystyle 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 + → V + by L u , v + ( y ) = { u , v , y } + {\displaystyle 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 u , v ± , L w , x ± ] = L w , { u , v , x } ± ± − L { v , u , w } ∓ , x ± {\displaystyle [L_{u,v}^{\pm },L_{w,x}^{\pm }]=L_{w,\{u,v,x\}_{\pm }}^{\pm }-L_{\{v,u,w\}_{\mp },x}^{\pm }}

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 + ⊗ V − → g l ( V + ) ⊕ g l ( V − ) {\displaystyle V_{+}\otimes V_{-}\to {\mathfrak {gl}}(V_{+})\oplus {\mathfrak {gl}}(V_{-})}

whose image is a Lie subalgebra g 0 {\displaystyle {\mathfrak {g}}_{0}}, and the Jordan identities become Jacobi identities for a graded Lie bracket on

g := V + ⊕ g 0 ⊕ V − , {\displaystyle {\mathfrak {g}}:=V_{+}\oplus {\mathfrak {g}}_{0}\oplus V_{-},}

making this space into a Z {\displaystyle \mathbb {Z} }-graded Lie algebra g {\displaystyle {\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

g = g + 1 ⊕ g 0 ⊕ g − 1 , {\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{+1}\oplus {\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{-1},}

then the pair ( g + 1 , g − 1 ) {\displaystyle ({\mathfrak {g}}_{+1},{\mathfrak {g}}_{-1})} is a Jordan pair, with brackets

{ X ∓ , Y ± , Z ± } ± := [ [ X ∓ , Y ± ] , Z ± ] . {\displaystyle \{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

E n d ( S 2 V + ) ≅ S 2 V + ∗ ⊗ S 2 V − ∗ ≅ E n d ( S 2 V − ) . {\displaystyle \mathrm {End} (S^{2}V_{+})\cong S^{2}V_{+}^{*}\otimes S^{2}V_{-}^{*}\cong \mathrm {End} (S^{2}V_{-}).}

These arise in particular when g {\displaystyle {\mathfrak {g}}} above is semisimple, when the Killing form provides a duality between g + 1 {\displaystyle {\mathfrak {g}}_{+1}} and g − 1 {\displaystyle {\mathfrak {g}}_{-1}}.

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

Q + : V + → Hom ( V − , V + ) {\displaystyle Q_{+}\colon V_{+}\to {\text{Hom}}(V_{-},V_{+})}

Q − : V − → Hom ( V + , V − ) {\displaystyle Q_{-}\colon V_{-}\to {\text{Hom}}(V_{+},V_{-})}

given by

Q + ( v ) ( f ) = f ( v ) v {\displaystyle Q_{+}(v)(f)=f(v)\,v}

Q − ( f ) ( v ) = f ( v ) f {\displaystyle Q_{-}(f)(v)=f(v)\,f}

where v ∈ V + , f ∈ V − {\displaystyle v\in V_{+},f\in V_{-}}.

Triple system

Lie triple systems

Jordan triple systems

Jordan pairs

See also