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Triple system

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For an account of that concept in combinatorics, see Steiner triple system and block design.

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 to study subspaces of associative algebras closed under triple commutators , 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).

Lie triple systems

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

[ u , v , w ] = − [ v , u , w ] {\displaystyle =-}
[ u , v , w ] + [ w , u , v ] + [ v , w , u ] = 0 {\displaystyle ++=0}
[ u , v , [ w , x , y ] ] = [ [ u , v , w ] , x , y ] + [ w , [ u , v , x ] , y ] + [ w , x , [ u , v , y ] ] . {\displaystyle ]=,x,y]+,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) = , 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

g := k ⊕ m {\displaystyle {\mathfrak {g}}:=k\oplus {\mathfrak {m}}}

can be made into a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded Lie algebra, the standard embedding of m, with bracket

[ ( L , u ) , ( M , v ) ] = ( [ L , M ] + L u , v , L ( v ) − M ( u ) ) . {\displaystyle =(+L_{u,v},L(v)-M(u)).}

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.

Jordan triple systems

A triple system is said to be a Jordan triple system if the trilinear map, denoted {.,.,.}, 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 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

[ 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}\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 g0.

Any Jordan triple system is a Lie triple system with respect to the product

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

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 g0. 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 g0 and −1 on V and V. A special case of this construction arises when g0 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).

Jordan pair

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_{-}}

which are often viewed as quadratic maps V+ → Hom(V−, V+) and V− → Hom(V+, 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.

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_{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

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

so that conversely, if

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

is a graded Lie algebra, 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 }:=,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}} .

See also

References

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