Triple system

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

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

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

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}}. 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 k = span {L_u,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.

Writing m in place of V, it follows that

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

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

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

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 [[u, v], 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:

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

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

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 g₀.

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

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

A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of 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 g₀. 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 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

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

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

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

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

so that conversely, if

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

is a graded Lie algebra, 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
• Quadratic Jordan algebra

References

• Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, 1754, Springer-Verlag, ISBN 978-3-540-41426-1
• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society (1st edition: Academic Press, New York, 1978).
• 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), Symmetric spaces. Volume 1: General Theory, W. A. Benjamin
• Loos, Ottmar (1969), Symmetric spaces. Volume 2: Compact Spaces and Classification, W. A. Benjamin
• Loos, Ottmar (1971), "Jordan triple systems, R-spaces, and bounded symmetric domains", Bulletin of the American Mathematical Society, 77: 558–561, doi:10.1090/s0002-9904-1971-12753-2
• Loos, Ottmar (1975), Jordan pairs, Lecture Notes in Mathematics, 460, Springer-Verlag
• 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, 393, Dordrecht: Kluwer Academic Publishers, 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