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{{for|an account of that concept in '''combinatorics''' |Steiner triple system |block design}}
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In [[algebra]], a '''triple system''' is a [[vector space]] ''V'' over a field '''F''' together with a [[multilinear map|'''F'''-trilinear map]]
:<math> (\cdot,\cdot,\cdot) \colon V\times V \times V\to V.</math>
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 [[Commutator|anticommutator]]s {''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 space]]s, particularly [[Hermitian symmetric space]]s and their generalizations ([[symmetric R-space]]s 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:
:<math> [u,v,w] = -[v,u,w] </math>
:<math> [u,v,w] + [w,u,v] + [v,w,u] = 0</math>
:<math> [u,v,[w,x,y]] = [[u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y]].</math>
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<sub>''u'',''v''</sub>:''V''→''V'', defined by L<sub>''u'',''v''</sub>(''w'') = [''u'', ''v'', ''w''], is a [[derivation (algebra)|derivation]] of the triple product. The identity also shows that the space '''k''' = span {L<sub>''u'',''v''</sub>: ''u'', ''v'' ∈ ''V''} is closed under commutator bracket, hence a Lie algebra.
 
Writing '''m''' in place of ''V'', it follows that
:<math>\mathfrak g := \mathfrak k \oplus\mathfrak m</math>
can be made into a Lie algebra with bracket
:<math>[(L,u),(M,v)] = ([L,M]+L_{u,v}, L(v) - M(u)).</math>
The decomposition of '''g''' is clearly a [[symmetric space|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:
:<math> \{u,v,w\} = \{u,w,v\} </math>
:<math> \{u,v,\{w,x,y\}\} = \{w,x,\{u,v,y\}\} + \{w, \{u,v,x\},y\} -\{\{v,u,w\},x,y\}. </math>
The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if L<sub>''u'',''v''</sub>:''V''→''V'' is defined by  L<sub>''u'',''v''</sub>(''y'') = {''u'', ''v'', ''y''} then
:<math> [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} </math>
so that the space of linear maps span {L<sub>''u'',''v''</sub>:''u'',''v'' ∈ ''V''} is closed under commutator bracket, and hence is a Lie algebra '''g'''<sub>0</sub>.
 
Any Jordan triple system is a Lie triple system with respect to the product
:<math> [u,v,w] = \{u,v,w\} - \{v,u,w\}. </math>
 
A Jordan triple system is said to be '''positive definite''' (resp. '''nondegenerate''') if the bilinear form on ''V'' defined by the trace of L<sub>''u'',''v''</sub> is positive definite (resp. nondegenerate). In either case, there is an identification of ''V'' with its dual space, and a corresponding involution on '''g'''<sub>0</sub>. They induce an involution of
:<math>V\oplus\mathfrak g_0\oplus V^*</math>
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 '''g'''<sub>0</sub> and &minus;1 on ''V'' and ''V''<sup>*</sup>. A special case of this construction arises when '''g'''<sub>0</sub> preserves a complex structure on ''V''. In this case we obtain dual [[Hermitian symmetric space]]s of compact and noncompact type (the latter being [[bounded symmetric domain]]s).
 
==Jordan pair==
 
A Jordan pair is a generalization of a Jordan triple system involving two vector spaces ''V''<sub>+</sub> and ''V''<sub>&minus;</sub>. The trilinear form is then replaced by a pair of trilinear forms
:<math> \{\cdot,\cdot,\cdot\}_+\colon V_-\times S^2V_+ \to V_+</math>
:<math> \{\cdot,\cdot,\cdot\}_-\colon V_+\times S^2V_- \to V_-</math>
which are often viewed as quadratic maps ''V''<sub>+</sub> → Hom(''V''<sub>&minus;</sub>, ''V''<sub>+</sub>) and ''V''<sub>&minus;</sub> → Hom(''V''<sub>+</sub>, ''V''<sub>&minus;</sub>). The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being
:<math> \{u,v,\{w,x,y\}_+\}_+ = \{w,x,\{u,v,y\}_+\}_+ + \{w, \{u,v,x\}_+,y\}_+ - \{\{v,u,w\}_-,x,y\}_+ \,</math>
and the other being the analogue with + and &minus; subscripts exchanged.
 
As in the case of Jordan triple systems, one can define, for ''u'' in ''V''<sub>&minus;</sub> and ''v'' in ''V''<sub>+</sub>, a linear map
:<math> L^+_{u,v}:V_+\to V_+ \quad\text{by} \quad L^+_{u,v}(y) = \{u,v,y\}_+</math>
and similarly L<sup>&minus;</sup>. The Jordan axioms (apart from symmetry) may then be written
:<math> [L^{\pm}_{u,v},L^{\pm}_{w,x}] = L^{\pm}_{w,\{u,v,x\}_\pm}-L^{\pm}_{\{v,u,w\}_{\mp},x} </math>
which imply that the images of L<sup>+</sup> and L<sup>&minus;</sup> are closed under commutator brackets in End(''V''<sub>+</sub>) and End(''V''<sub>&minus;</sub>). Together they determine a linear map
:<math> V_+\otimes V_- \to \mathfrak{gl}(V_+)\oplus \mathfrak{gl}(V_-)</math>
whose image is a Lie subalgebra <math>\mathfrak{g}_0</math>, and the Jordan identities become Jacobi identities for a graded Lie bracket on
:<math> V_+\oplus \mathfrak g_0\oplus V_-,</math>
so that conversely, if
:<math> \mathfrak g = \mathfrak g_{+1} \oplus \mathfrak g_0\oplus \mathfrak g_{-1}</math>
is a graded Lie algebra, then the pair <math>(\mathfrak g_{+1}, \mathfrak g_{-1})</math> is a Jordan pair, with brackets
:<math> \{X_{\mp},Y_{\pm},Z_{\pm}\}_{\pm} := [[X_{\mp},Y_{\pm}],Z_{\pm}].</math>
 
Jordan triple systems are Jordan pairs with ''V''<sub>+</sub> = ''V''<sub>&minus;</sub> and equal trilinear forms. Another important case occurs when ''V''<sub>+</sub> and ''V''<sub>&minus;</sub> are dual to one another, with dual trilinear forms determined by an element of
:<math> \mathrm{End}(S^2V_+) \cong S^2V_+^* \otimes S^2V_-^*\cong \mathrm{End}(S^2V_-).</math>
These arise in particular when <math> \mathfrak g </math> above is semisimple, when the Killing form provides a duality between <math>\mathfrak g_{+1}</math> and <math> \mathfrak g_{-1}</math>.
 
==See also==
*[[Quadratic Jordan algebra]]
 
==References==
 
* {{citation|first=Wolfgang|last= Bertram|year=2000|title=The geometry of Jordan and Lie structures|series= Lecture Notes in Mathematics|volume=1754|publisher=Springer-Verlag|isbn= 3-540-41426-6}}
* {{citation|first=Sigurdur|last= Helgason|year=2001|title=Differential geometry, Lie groups, and symmetric spaces|publisher= American Mathematical Society}} (1st edition: Academic Press, New York, 1978).
* {{citation|first=Nathan|last= Jacobson|year=1949|url=http://www.jstor.org/stable/info/2372102|title= Lie and Jordan triple systems|journal=American Journal of Mathematics|volume= 71|pages=149–170}}
* {{springer|id=Lie_triple_system|title=Lie triple system|first=Noriaki|last= Kamiya}}.
* {{springer|id=Jordan_triple_system|title=Jordan triple system|first=Noriaki|last= Kamiya}}.
* {{citation|first=M.|last= Koecher|year=1969|title= An elementary approach to bounded symmetric domains|series= Lecture Notes|publisher=Rice University}}
* {{citation|first=Ottmar|last= Loos|year=1969|title=Symmetric spaces. Volume 1: General Theory|publisher=  W. A. Benjamin}}
* {{citation|first=Ottmar|last= Loos|year=1969|title=Symmetric spaces. Volume 2: Compact Spaces and Classification|publisher=  W. A. Benjamin}}
* {{citation|first=Ottmar |last=Loos|year=1971|url=http://www.ams.org/bull/1971-77-04/S0002-9904-1971-12753-2/home.html|title= Jordan triple systems, ''R''-spaces, and bounded symmetric domains|journal=  Bulletin of the American Mathematical Society|volume= 77| pages=558–561}} {{doi|10.1090/S0002-9904-1971-12753-2}}
* {{citation|first=Ottmar|last= Loos|year=1975|title=Jordan pairs|series= Lecture Notes in Mathematics|volume=460|publisher= Springer-Verlag}}
*{{citation|last=Loos|first=Ottmar|title=Bounded symmetric domains and Jordan pairs|series=Mathematical lectures|publisher=University of California, Irvine|year=1977|url=http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf}}
*{{citation|last=Meyberg|first= K.|title=Lectures on algebras and triple systems|publisher=[[University of Virginia]]|year= 1972|url=http://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf}}
* {{citation|last=Tevelev|first= E. |year=2002|url=http://www.emis.de/journals/JLT/vol.12_no.2/9.html|title=Moore-Penrose inverse, parabolic subgroups, and Jordan pairs|journal= Journal of Lie theory|volume=12|pages=461–481}}
 
[[Category:Representation theory]]

Latest revision as of 10:09, 30 July 2014

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