Triple system

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

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

$(\cdot ,\cdot ,\cdot ):V\times V\times V\to V.$

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 $[\cdot ,\cdot ,\cdot ]$, 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 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 $\mathfrak{𝔥}$ = span {L_u,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

It follows that

$\mathfrak{𝔤}:=\mathfrak{𝔥}\oplus$ V

is a ${\mathbb{\mathbb{Z}}}_2$-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 ${\mathbb{\mathbb{Z}}}_2$-graded Lie algebra. Conversely, given any ${\mathbb{\mathbb{Z}}}_2$-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 ${\mathbb{\mathbb{Z}}}_2$-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 ${\mathbb{\mathbb{Z}}}_2$-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain $\mathfrak{𝔥}$ = span {L_u,v : u, v ∈ V} = {0}.

Given any Lie triple system V, and letting $\mathfrak{𝔤}=\mathfrak{𝔥}\oplus$ V be the corresponding ${\mathbb{\mathbb{Z}}}_2$-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 L_u,v:V→V is defined by L_u,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 {L_u,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 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 ${\mathfrak{𝔤}}_0$. 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

$\{\cdot ,\cdot ,\cdot {\}}_{+}:V_{-}\times S^2{V}_{+}\to V_{+}$

$\{\cdot ,\cdot ,\cdot {\}}_{-}:V_{+}\times S^2{V}_{-}\to V_{-}$.

The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being

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

and the other being the analogue with + and − subscripts exchanged. The trilinear maps are often viewed as quadratic maps

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

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

As in the case of Jordan triple systems, one can define, for u in V₋ and v in V₊, a linear map

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

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

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

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

$V_{+}\otimes V_{-}\to \mathfrak{𝔤}\mathfrak{𝔩}(V_{+})\oplus \mathfrak{𝔤}\mathfrak{𝔩}(V_{-})$

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 $\mathbb{\mathbb{Z}}$-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

$\mathrm{E}\mathrm{n}\mathrm{d}(S^2{V}_{+})\cong S^2{V}_{+}^{*}\otimes S^2{V}_{-}^{*}\cong \mathrm{E}\mathrm{n}\mathrm{d}(S^2{V}_{-}).$

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

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

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

given by

$Q_{+}(v)(f)=f(v)v$

$Q_{-}(f)(v)=f(v)f$

where $v\in V_{+},f\in V_{-}$.

See also

• Associator
• E₇ (mathematics)
• Quadratic Jordan algebra

References

• Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, 1754, Springer, ISBN 978-3-540-41426-1 
• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, ISBN 978-0-8218-2848-9, https://books.google.com/books?id=a9KFAwAAQBAJ 

• Jacobson, Nathan (1949), "Lie and Jordan triple systems", American Journal of Mathematics 71 (1): 149–170, doi:10.2307/2372102 
• Hazewinkel, Michiel, ed. (2001), "Lie triple system", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Lie_triple_system .
• Hazewinkel, Michiel, ed. (2001), "Jordan triple system", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Jordan_triple_system .
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• Loos, Ottmar (1969), Compact Spaces and Classification, Symmetric spaces, 2, W. A. Benjamin 
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• Loos, Ottmar (2006), Jordan pairs, Lecture Notes in Mathematics, 460, Springer, ISBN 978-3-540-37499-2, https://books.google.com/books?id=6Zl8CwAAQBAJ&pg=PR1 
• Loos, Ottmar (1977), Bounded symmetric domains and Jordan pairs, Mathematical lectures, University of California, Irvine, http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf 
• Meyberg, K. (1972), Lectures on algebras and triple systems, University of Virginia, http://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf 
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• Tevelev, E. (2002), "Moore-Penrose inverse, parabolic subgroups, and Jordan pairs", Journal of Lie Theory 12: 461–481, Bibcode: 2001math......1107T, http://www.emis.de/journals/JLT/vol.12_no.2/9.html