Triple system

In algebra, a triple system is a vector space "V" over a field F together with a F-trilinear map:$(cdot,cdot,cdot)\; colon\; V\; imes\; V\; imes\; V\; o\; V.$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". 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 form, denoted [.,.,.] , 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 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:$mathfrak\; g\; :=\; mathfrak\; k\; oplusmathfrak\; m$can be made into a Lie algebra with bracket:$[(L,u),(M,v)]\; =\; (\; [L,M]\; +L\_\{u,v\},\; L(v)\; -\; M(u)).$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 form, denoted {.,.,.}, 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 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:$[L\_\{u,v\},L\_\{w,x\}]\; :=\; L\_\{u,v\}circ\; L\_\{w,x\}\; -\; L\_\{w,x\}\; circ\; L\_\{u,v\}\; =\; L\_\{w,\{u,v,x-L\_\{\{v,u,w\},x\}$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:$[u,v,w]\; =\; \{u,v,w,\}\; -\; \{v,u,w\}.$

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:$Voplusmathfrak\; g\_0oplus\; V^*$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 form is then replaced by a pair of trilinear forms:$\{cdot,cdot,cdot\}\_+colon\; V\_-\; imes\; S^2V\_+\; o\; V\_+$:$\{cdot,cdot,cdot\}\_-colon\; V\_+\; imes\; S^2V\_-\; o\; V\_-$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
$L^+\_\{u,v\}:V\_+\; o\; V\_+\; quad\; ext\{by\}\; quad\; L^+\_\{u,v\}(y)\; =\; \{u,v,y\}\_+$and similarly L⁻. The Jordan axioms (apart from symmetry) may then be written:$[L^\{pm\}\_\{u,v\},L^\{pm\}\_\{w,x\}]\; =\; L^\{pm\}\_\{w,\{u,v,x\}\_pm\}-L^\{pm\}\_\{\{v,u,w\}\_\{mp\},x\}$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
$V\_+otimes\; V\_-\; o\; mathfrak\{gl\}(V\_+)oplus\; mathfrak\{gl\}(V\_-)$whose image is a Lie subalgebra $mathfrak\{g\}\_0$, and the Jordan identities become Jacobi identities for a graded Lie bracket on:$V\_+oplus\; mathfrak\; g\_0oplus\; V\_-,$so that conversely, if:$mathfrak\; g\; =\; mathfrak\; g\_\{+1\}\; oplus\; mathfrak\; g\_0oplus\; mathfrak\; g\_\{-1\}$is a graded Lie algebra, 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 forms. Another important case occurs when "V"₊ and "V"₋ are dual to one another, with dual trilinear forms determined by an element of:$mathrm\{End\}(S^2V\_+)\; cong\; S^2V\_+^*\; otimes\; S^2V\_-^*cong\; mathrm\{End\}(S^2V\_-).$These arise in particular when $mathfrak\; g$ above is semisimple, when the Killing form provides a duality between $mathfrak\; g\_\{+1\}$ and $mathfrak\; g\_\{-1\}$.

References

* Wolfgang Bertram (2000), "The geometry of Jordan and Lie structures", Lecture Notes in Mathematics 1754, Springer-Verlag, Berlin, 2000. ISBN: 3-540-41426-6.
* Sigurdur Helgason (2001), "Differential geometry, Lie groups, and symmetric spaces", American Mathematical Society, New York (1st edition: Academic Press, New York, 1978).
* Nathan Jacobson (1949), " [http://www.jstor.org/stable/info/2372102 Lie and Jordan triple systems] ", American Journal of Mathematics 71, pp. 149–170.
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* M. Koecher (1969), An elementary approach to bounded symmetric domains. Lecture Notes, Rice University, Houston, Texas.
* Ottmar Loos (1969), "Symmetric spaces. Volume 1: General Theory. Volume 2: Compact Spaces and Classification", W. A. Benjamin, New York.
* Ottmar Loos (1971), " [http://www.ams.org/bull/1971-77-04/S0002-9904-1971-12753-2/home.html Jordan triple systems, "R"-spaces, and bounded symmetric domains] ", Bulletin of the American Mathematical Society 77, pp. 558–561. (doi: 10.1090/S0002-9904-1971-12753-2)
* Ottmar Loos (1975), "Jordan pairs", Lecture Notes in Mathematics 460, Springer-Verlag, Berlin and New York.
* Tevelev, E (2002), " [http://www.emis.de/journals/JLT/vol.12_no.2/9.html Moore-Penrose inverse, parabolic subgroups, and Jordan pairs] ", Journal of Lie theory 12, pp. 461–481.