Compatible Peirce decompositions of Jordan triple systems - MSP

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COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 83
GRID THEOREM 3.16. Any semisimple Jordan pair with cLc.c.
on inner ideals, or semisimple Jordan triple system finite-dimensional
over an algebraically closed field of characteristic Φ2, has a
covering grid of tripotents which are pairwise orthogonal or collinear,
•
It will be important when we try to lift tripotents, that the
covers are not merely compatible, but actually orthogonal-collinear.
4* Peirce reflections* A Peirce decomposition J = J 2 0 J 10 J o
relative to a tripotent e determines an important automorphism of
period 2, the Peirce reflection S e(x 2 + x 1 + x Q) = x 2 — Xi + % Q,
(4.1) S e = E 2-E 1 + E o = B(e, 2e) with S e = (-1)' on J t(e) .
These generate a normal subgroup of the group of automorphism,
TS^Γ" 1
— STe, and play an important role in many applications.
We wish to try the same thing for an arbitrary compatible
family of tripotents 8" in place of β. The Peirce reflection S?
relative to this family is defined to be
(4.2) Sv - E 2(ί?) - EJ&) + EJ&), so 8, - (-1)' on
for the Peirce projections E&) of J on J^) in (2.2). These are
normalized by automorphisms,
However, in general the invertible linear operator S# of period 2
is not expressible as a B operator and is not an automorphism of
the triple system. The conditions for it to be an automorphism are
LEMMA 4.3. The Peirce reflection S& relative to a compatible
family & = {e lf , e n } of tripotents is an automorphism of J if
the Peirce decomposition J— J 2 0 JΊ0 J O (for J t ~ Ji($?)) satisfies
the Peirce rules
•: (i) P{J x)Ji^Ju P(/ 2)/ 2cJ 2
( ii ) P(J 1)(/ 2 + Jo) C J 2 + Jo,
(iii) {JJiiJi + Jo)} c J 2 + J o
(iv)
Proof The map S= 10— I on J + (BJ- is an automorphism
if the subspaces J e (ε = ±) satisfy (*) P(J ε )J ε aJ ε , (**) P(J ε )J_ ε c:J_ ε ,
(•***) L(J ε , J ε )J_ ε cJ_ ε . By (4.2), for J + = J 2 + J o and J_ = J x these