Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 87
using g, h in place of e, f yields Jm2l) Π J2(k) = Jm21) Π J0(k) = 0 so
/(0121) c JΊ(fe) and dually for J (10i2). Finally (v') follows from (3.3iv),
since by Lemma 3.2 {e, k, g} imbeds in a quadrangle. From these
we immediately obtain the condition (3a), (3b) for {e, f, k}: for (3b)
^(iiD = {^(uii) + ^(noo)} n JM = 0 by (iii), (ii'), J (10ύ) = {J (1012> + e7" (1001)}n
J 0(k) = 0 by (iv'), (ii'), dually J mo) = 0, and J {001) = {J m22) + J mn)
/coooo)} n JJJc) = 0 by (i'), (ϋ'), (V); for (3a) J {22ύ) - J (2200) n J s(Jc) = 0
unless i = 0 by (i'), J (21i> = J (2101) n J s(k) = 0 unless j = 1 by (iv'),
^"(2oy> = ^(2002) Γl Jy(fc) = 0 unless i = 2 by (i'), dually for J U2i), and
^(002) + ^(102) ~Γ ^(012) ==
W (0000) + ^ (0011) ~i~ β' (1001) + ^(1012) 4" ^ (0110) + ^(0121)1 Π
/ 8(fc) = 0 by (v'), (ϋ'), (ii'), (iv'), (ϋ r
), (iv'). Thus the hypotheses
(3a-b) of (4.4) are met, and S {efk} is an automorphism. •
EXAMPLE 4.7. If e = ^Ή, / = 2£ 12 in M p>q(D) then the Peirce
reflection S {β>/} is an automorphism X-> (I p — 2E)X(I q — 2F) for
E= E ne M P(D), F= E n + E 22 e M q(D). If p, q ^ 2 then e, / imbeds
in a quadrangle {β, /, g, h] = {E llf E ί2, E 22, E 2l) as in (4.5). Π
EXAMPLE 4.8. If e = F12, f= F1S in Sn(C) for n ^ 4, then S {e>f}
is not an automorphism if characteristic C Φ 2: i^ and F34 lie in
Ji(&)f Ft* in ^(g 7
), yet {FUF2SF12} - -JP14 lies in J^). Note condition
(4.4(2c)) is violated here: {F3iF23F12} — —Fu is nonzero in
{/ (01)J
(11)β}. The trouble is that CF23 really acts like part of J2—if we take £f' = {e, f, k} = {F12, F1Zj F25} then S {e,
ftk) is an automorphism
X-+ (In - 2£7)X(Jn - 2JS) for E = En + E22 + E^ in Λfn(C). Here
e, / imbeds in the quadrangle [e, f, g, h} = {F12, F13, JP , F 43 42} collinear
with F23 as in (4.6.) •
5* Lifting compatible families* In considering Wedderburn
splittings and the second cohomology group H 2
(J, M), it is important
to be able to lift compatible covering families from J to any
null extension J (i.e., lifting from J = J/M to J modulo a null or
trivial ideal Jl£). In this section we consider the general problem
of lifting a compatible family modulo a nil ideal.
In lifting a family the crucial step is always lifting a single
tripotent.
LIFTING LEMMA 5.1 (1, p. 108). If J^J is a projection of
Jordan algebras whose kernel is nil, then for each idempotent eeJ
and each preimage xeJ (π(x) = e), there is a preimage e = p(x)
which is a polynomial in the given x and is an idempotent in J.
The same holds if J, J are Jordan pairs.
If J and J are Jordan triple systems, then tripotents e can be
+