Triple system

{{for|an account of that concept in '''combinatorics''' |Steiner triple system |block design}}

In algebra, a '''triple system''' (or '''ternar''') is a vector space ''V'' over a field '''F''' together with a '''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. 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 <math> [\cdot,\cdot,\cdot] </math>, 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''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 <math>\mathfrak{h}</math> = span {L''u'',''v'' : ''u'', ''v'' ∈ ''V''} is closed under commutator bracket, hence a Lie algebra.

It follows that :<math>\mathfrak{g} := \mathfrak{h} \oplus</math> ''V'' is a <math>\mathbb{Z}_2</math>-graded Lie algebra with <math>\mathfrak{h}</math> of grade 0 and ''V'' of grade 1, and bracket :<math>[(L,u),(M,v)] = ([L,M]+L_{u,v}, L(v) - M(u)).</math> This is called the '''standard embedding''' of the Lie triple system ''V'' into a <math>\mathbb{Z}_2</math>-graded Lie algebra. Conversely, given any <math>\mathbb{Z}_2</math>-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 <math>\mathbb{Z}_2</math>-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 <math>\mathbb{Z}_2</math>-graded Lie algebra, the round trip process produces one where the grade-0 space is zero-dimensional, since we obtain <math>\mathfrak{h}</math> = span {L''u'',''v'' : ''u'', ''v'' ∈ ''V''} = {0}.

Given any Lie triple system ''V'', and letting <math>\mathfrak{g} = \mathfrak{h} \oplus</math> ''V'' be the corresponding <math>\mathbb{Z}_2</math>-graded Lie algebra, this decomposition of <math>\mathfrak{g}</math> obeys the algebraic definition of a symmetric space, so if ''G'' is any connected Lie group with Lie algebra <math>\mathfrak{g}</math> and ''H'' is a subgroup with Lie algebra <math>\mathfrak{h}</math>, 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 <math>[a,b] = ab - ba</math>, any subspace of ''A'' closed under the operation :<math>[a,b,c] = [[a,b],c]</math> 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 <math>\{\cdot,\cdot,\cdot\}</math>, 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 second identity means that if L''u'',''v'':''V''→''V'' is defined by L''u'',''v''(''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''u'',''v'':''u'',''v'' ∈ ''V''} is closed under commutator bracket, and hence is a Lie algebra <math>\mathfrak{g}_0</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''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 <math>\mathfrak{g}_0</math>. 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 <math>\mathfrak{g}_0</math> and −1 on ''V'' and ''V''*. A special case of this construction arises when <math>\mathfrak{g}_0</math> 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 :<math> [u,v,w] = \{u,v,w\} - \{v,u,w\}. </math>

==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 :<math> \{\cdot,\cdot,\cdot\}_+\colon V_-\times S^2V_+ \to V_+</math> :<math> \{\cdot,\cdot,\cdot\}_-\colon V_+\times S^2V_- \to V_-</math>. 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 − subscripts exchanged. The trilinear maps are often viewed as quadratic maps : <math> Q_+ \colon V_+ \to \text{Hom}(V_-, V_+) </math> : <math> Q_- \colon V_- \to \text{Hom}(V_+, V_-) .</math>

As in the case of Jordan triple systems, one can define, for ''u'' in ''V''− and ''v'' in ''V''+, 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−. 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+ and L− are closed under commutator brackets in End(''V''+) and End(''V''−). 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>\mathfrak{g} := V_+\oplus \mathfrak g_0\oplus V_-,</math> making this space into a <math>\mathbb{Z}</math>-graded Lie algebra <math>\mathfrak{g}</math> with only grades 1, 0, and -1 being nontrivial, often called a '''3-graded Lie algebra'''. Conversely, given any 3-graded Lie algebra :<math> \mathfrak g = \mathfrak g_{+1} \oplus \mathfrak g_0\oplus \mathfrak g_{-1},</math> 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''+ = ''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 :<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>.

For a simple example of a Jordan pair, let <math>V_+</math> be a finite-dimensional vector space and <math>V_-</math> the dual of that vector space, with the quadratic maps

: <math> Q_+ \colon V_+ \to \text{Hom}(V_-, V_+) </math> : <math> Q_- \colon V_- \to \text{Hom}(V_+, V_-) </math>

given by

:<math>Q_+(v)(f) = f(v) \,v </math> :<math>Q_-(f)(v) = f(v) \, f </math>

where <math>v \in V_+, f \in V_-</math>.

==References==

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* {{citation|first=Nathan|last= Jacobson|year=1949|jstor=2372102|title= Lie and Jordan triple systems|journal=American Journal of Mathematics|volume= 71|issue= 1|pages=149–170|doi=10.2307/2372102}} * {{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|series=Symmetric spaces |volume=1 |title=General Theory|publisher= W. A. Benjamin |url=https://archive.org/details/symmetricspaces0000loos_k2a0/mode/2up |oclc=681278693}} * {{citation|first=Ottmar|last= Loos|year=1969|series=Symmetric spaces |volume=2 |title=Compact Spaces and Classification|publisher= W. A. Benjamin}} * {{citation|first=Ottmar |last=Loos|year=1971|title= Jordan triple systems, ''R''-spaces, and bounded symmetric domains|journal= Bulletin of the American Mathematical Society|volume= 77|issue=4| pages=558–561|doi=10.1090/s0002-9904-1971-12753-2|doi-access=free}} * {{citation|first=Ottmar|last= Loos |title=Jordan pairs |url=https://books.google.com/books?id=6Zl8CwAAQBAJ&pg=PR1 |date=2006 |publisher=Springer |isbn=978-3-540-37499-2 |series=Lecture Notes in Mathematics |volume=460 |orig-year=1975}} *{{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|url-status=dead|archive-url=https://web.archive.org/web/20160303234008/http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/irvine.pdf|archive-date=2016-03-03}} *{{citation|last=Meyberg|first= K.|title=Lectures on algebras and triple systems|publisher=University of Virginia|year= 1972|url=https://www.math.uci.edu/~brusso/Meyberg(Reduced2).pdf}} * {{citation | last=Rosenfeld | first=Boris | title=Geometry of Lie groups | page=92 | zbl=0867.53002 | series=Mathematics and its Applications | volume=393 | publisher=Kluwer | year=1997 | isbn=978-0792343905 }} * {{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|bibcode= 2001math......1107T |arxiv= math/0101107 }}

Category:Representation theory