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

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

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 $[\cdot ,\cdot ,\cdot ]$, 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 $\mathbb {Z} _{2}$-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 {L_u,v : u, v ∈ V} = {0}.

Given any Lie triple system V, and letting ${\mathfrak {g}}={\mathfrak {h}}\oplus$ V be the corresponding $\mathbb {Z} _{2}$-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

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

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

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

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

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

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

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

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

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

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

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

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

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

See also

• Associator
• E₇ (mathematics)
• Quadratic Jordan algebra

References

• Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, vol. 1754, Springer, ISBN 978-3-540-41426-1
• Helgason, Sigurdur (2001) [1978], Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, ISBN 978-0-8218-2848-9

• Jacobson, Nathan (1949), "Lie and Jordan triple systems", American Journal of Mathematics, 71 (1): 149–170, doi:10.2307/2372102, JSTOR 2372102
• Kamiya, Noriaki (2001) [1994], "Lie triple system", Encyclopedia of Mathematics, EMS Press.
• Kamiya, Noriaki (2001) [1994], "Jordan triple system", Encyclopedia of Mathematics, EMS Press.
• Koecher, M. (1969), An elementary approach to bounded symmetric domains, Lecture Notes, Rice University
• Loos, Ottmar (1969), General Theory, Symmetric spaces, vol. 1, W. A. Benjamin, OCLC 681278693
• Loos, Ottmar (1969), Compact Spaces and Classification, Symmetric spaces, vol. 2, W. A. Benjamin
• Loos, Ottmar (1971), "Jordan triple systems, R-spaces, and bounded symmetric domains", Bulletin of the American Mathematical Society, 77 (4): 558–561, doi:10.1090/s0002-9904-1971-12753-2
• Loos, Ottmar (2006) [1975], Jordan pairs, Lecture Notes in Mathematics, vol. 460, Springer, ISBN 978-3-540-37499-2
• Loos, Ottmar (1977), Bounded symmetric domains and Jordan pairs (PDF), Mathematical lectures, University of California, Irvine, archived from the original (PDF) on 2016-03-03
• Meyberg, K. (1972), Lectures on algebras and triple systems (PDF), University of Virginia
• Rosenfeld, Boris (1997), Geometry of Lie groups, Mathematics and its Applications, vol. 393, Kluwer, p. 92, ISBN 978-0792343905, Zbl 0867.53002
• Tevelev, E. (2002), "Moore-Penrose inverse, parabolic subgroups, and Jordan pairs", Journal of Lie Theory, 12: 461–481, arXiv:math/0101107, Bibcode:2001math......1107T