Compatible Peirce decompositions of Jordan triple systems - MSP

82 KEVIN MCCRIMMON
algebra, hence trivially has unital covering grid g 7 = {1}. It also
has an orthogonal-collinear cover g 7 = {Hiά\i < j}, of all off-diagonal
hermitian matrix units H^ = HH (i < j); here J2{Hij) = D0[ii] +
], Jx{HiS) = ΣΛΦijD[ik] + D[kj], Jmd = Σ*«.i A[**l +
so Hik, Hkl are collinear if they share a common index
and are orthogonal otherwise. However this compatible family is
not a grid in the sense of (3.8) since the collinear tripotents are not
rigidly collinear: J2(Hti) Π J2(Hik) = D0[ii] Φ 0. The hermitian grid
$f = {Hij\i ^ j} IS NOT A GRID; it remains compatible, though no
longer orthogonal-collinear. Π
EXAMPLE 3.14. If J = Φ n
with P(x)y = 2(x, y)x - (x, Sx)Sy for
(x, y) = Σ ίCiί/i the standard inner product on Φ n
and S the reflection
in some subspace of Φ n
, then J contains invertible tripotents
e where (e, e) — 1 and Se = ± e, hence J is an isotope of a Jordan
algebra and has covering grid i? = {e}. D
The remaining basic examples of Jordan triples are really Jordan
pairs. For our present purposes we prefer to consider Jordan pairs
as polarized triple systems, consisting of a Jordan triple system J
together with a decomposition /=/+©/_ such that P(J ε)J e —
{J εJ εJ} = 0 ($ = ±1). Thus the only nontrivial products have the
form P(J ε)J_ ε or {e7 ε/^ e/ e} falling in J e. Then / = (J +, /_) consists
of a pair of spaces acting on each other like Jordan triple systems,
but with trivial action on themselves.
The polarized triples we need to consider have the special form
Jψ J®J obtained by pairing two copies of the same Jordan triple
system J + = J_ = J with P(x ε )y_ ε = P(x)y J is isomorphic to J®Ω
for Ω = Φ+ m 0_, with P{x)y — P(x)y* where the exchange involution
(a?©2/)* =.y©α? is induced from the exchange involution on Ω.
The map x —> x = OJ © x is an isomorphism of J with iϊ(J, *). If
g* = { e J is a compatible or orthogonal-collinear cover or grid for J,
then £? = {e'J is a family of the same sort which covers / since
EXAMPLE 3.15. If J=JφJ results from doubling a Jordan
triple system J having grid g% then the Jordan pair or polarized
Jordan triple system J has grid C S. •
Since all semisimple Jordan pairs with d.c.c. on all inner ideals
are direct sums of simple systems of the above types 3.9-3.15 ([2,
p. 138-139]), as are the semisimple Jordan triple systems finitedimensional
over an algebraically closed field of characteristic Φ2
([3, Th. 10.3, p. 63]), and grids are inherited by direct sums, we