Compatible Peirce decompositions of Jordan triple systems - MSP

64 KEVIN MCCRIMMON
patibility reduces to
1.13. COMPATIBILITY CRITERION FOR IDEMPOTENTS. TWO idempotents
e, f in a Jordan algebra are compatible iff / = / 2 ® / 0 for
orthogonal idempotents f tGJi(e).
Proof. The condition that / = α 2 + α L + a Q is idempotent is
a 2 = at + E 2(e)a\
(1.10') a1 — (α2 + αo) o
α!
α 0 = al + E 0(e)al
and compatibility (1.11) becomes
(1.110 αo 0
^ + αχoα2 = 0 ,
hence a λ — 0 and α 2 = α^ α 0 = αξ are orthogonal idempotents. Conversely,
if /= / 2 + /o then e, / are compatible by (1.8iv). Π
Note that the strong compatibility condition that the operators
P(e), L(e, e) commute with P(f), L(f f) (not merely P(e) 2
and P(/) 2
)
is not an intrinsic condition: it depends on how e, f are imbedded
in J. For example, e = 1[12] and / = 1[13] are collinear and strongly
compatible in £>[12]+D[13] = MU2(D), but not in HID) since P(e)P(f)
1[33] - P(β)l[ll] = 1[22] Φ 0 = P(/)P(e)l[33].
The most important examples of compatible tripotents are either
orthogonal e _L / (each lies in the 0-space of the other) or collinear
eTf (each lies in the 1-space of the other). In the remainder of
this section we investigate what collinearity amounts to in basic
examples of triple systems. Recall that tripotents e, f are collinear
if P(e)f - P(/)e - 0, L(e, e)f = /, L(f f f)e = e.
Let us note that in a Jordan algebra we cannot have collinear
idempotents; collinearity is strictly for tripotents.
PROPOSITION 1.14. Two nonzero idempotents in a Jordan algebra
can never be collinear.
Proof. If eeJjif) and feJ^e) are idempotents then f—{eef)~
e 2 of = e of=eof= {eff} = e, so /= P(/)/= P(f)e - 0 and dually
e = 0. (Alternately, if feJ,(e) then / 2 6 J 2(e) + J 0(e), so the only
idempotent in J^e) is / = 0. Or yet again, the result follows directly
from (1.13).) D