Triple system

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

${\displaystyle (\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.}$

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 [[u, v], w] and triple anticommutators {u, {v, w}}, respectively. 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:

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

${\displaystyle [u,v,w]+[w,u,v]+[v,w,u]=0}$

${\displaystyle [u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].}$

The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L_u,v: V → V, defined by L_u,v(w) = [u, v, w], is a derivation of the triple product. The identity also shows that the space of linear operators ${\displaystyle {\mathfrak {h}}}$ = span {L_u,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that

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

is a ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra with ${\displaystyle {\mathfrak {h}}}$ of grade 0 and V of grade 1, and bracket

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

This is called the standard embedding of the Lie triple system V into a ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra. Conversely, given any ${\displaystyle \mathbb {Z} _{2}}$-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 ${\displaystyle \mathbb {Z} _{2}}$-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 ${\displaystyle \mathbb {Z} _{2}}$-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain ${\displaystyle {\mathfrak {h}}}$ = span {L_u,v : u, v ∈ V} = {0}.

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

${\displaystyle [a,b,c]=[[a,b],c]}$

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:

${\displaystyle \{u,v,w\}=\{u,w,v\}}$

${\displaystyle \{u,v,\{w,x,y\}\}=\{w,x,\{u,v,y\}\}+\{w,\{u,v,x\},y\}-\{\{v,u,w\},x,y\}.}$

The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if L_u,v:V→V is defined by L_u,v(y) = {u, v, y} then

${\displaystyle [L_{u,v},L_{w,x}]:=L_{u,v}\circ L_{w,x}-L_{w,x}\circ L_{u,v}=L_{w,\{u,v,x\}}-L_{\{v,u,w\},x}}$

so that the space of linear maps span {L_u,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra ${\displaystyle {\mathfrak {g}}_{0}}$.

A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of L_u,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on ${\displaystyle {\mathfrak {g}}_{0}}$. They induce an involution of

${\displaystyle V\oplus {\mathfrak {g}}_{0}\oplus V^{*}}$

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

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

Given an associative algebra A and defining the anticommutator by ${\displaystyle \{a,b\}=ab+ba}$, any subspace of A closed under the operation

${\displaystyle \{a,b,c\}=\{a,\{b,c\}\}}$

is a Jordan triple system with this operation.

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

${\displaystyle \{\cdot ,\cdot ,\cdot \}_{+}\colon V_{-}\times S^{2}V_{+}\to V_{+}}$

${\displaystyle \{\cdot ,\cdot ,\cdot \}_{-}\colon V_{+}\times S^{2}V_{-}\to V_{-}}$

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

${\displaystyle \{u,v,\{w,x,y\}_{+}\}_{+}=\{w,x,\{u,v,y\}_{+}\}_{+}+\{w,\{u,v,x\}_{+},y\}_{+}-\{\{v,u,w\}_{-},x,y\}_{+}}$

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

${\displaystyle L_{u,v}^{+}:V_{+}\to V_{+}\quad {\text{by}}\quad L_{u,v}^{+}(y)=\{u,v,y\}_{+}}$

and similarly L⁻. The Jordan axioms (apart from symmetry) may then be written

${\displaystyle [L_{u,v}^{\pm },L_{w,x}^{\pm }]=L_{w,\{u,v,x\}_{\pm }}^{\pm }-L_{\{v,u,w\}_{\mp },x}^{\pm }}$

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

${\displaystyle V_{+}\otimes V_{-}\to {\mathfrak {gl}}(V_{+})\oplus {\mathfrak {gl}}(V_{-})}$

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

${\displaystyle {\mathfrak {g}}:=V_{+}\oplus {\mathfrak {g}}_{0}\oplus V_{-},}$

making this space into a ${\displaystyle {\mathfrak {Z}}}$-graded Lie algebra ${\displaystyle {\mathfrak {g}}}$ with only grades 1, 0, and -1 being nontrivial, often called a 3-graded Lie algebra. Conversely, given any 3-graded Lie algebra

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

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

${\displaystyle \{X_{\mp },Y_{\pm },Z_{\pm }\}_{\pm }:=[[X_{\mp },Y_{\pm }],Z_{\pm }].}$

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

${\displaystyle \mathrm {End} (S^{2}V_{+})\cong S^{2}V_{+}^{*}\otimes S^{2}V_{-}^{*}\cong \mathrm {End} (S^{2}V_{-}).}$

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

See also

• Associator
• E₇ (mathematics)
• Quadratic Jordan algebra

References

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