Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OP JORDAN TRIPLE SYSTEMS 89
EXAMPLE 5.6. It is somewhat surprising that 1/2 is necessary
for lifting in triple systems, but one can give a "generic" example
of a non-liftable tripotent. One can calculate that there is no
integral polynomial g(t) 6 Z odd [ί] such that g(l) = 1 and g(t) — g(tf is
divisible by (ί - t B )\ Thus if B=Z o ^[i\^tZ[s] (s=ί 2 ), L = (l-s)B=
ί(l - s)Z[s] = (t - f)Z[s], JBΓ = (ί - tJB = β (l - sfB then J = J3/JSΓ—>
JB/L = J has nil (even trivial) kernel LjK, yet e = t is a tripotent
in J with no covering tripotent β = #(*) in /. Thus lifting is not
always possible in triple systems. •
REMARK 5.7. The lift e obtained from an arbitrary x may not
be the "correct" one. For example, if e is the "correct" cover of
e and we choose a preimage x = e — w = e — (w 2 + w t + w 0) for
w teKΓ\ Jiifi) (for trivial K = Ker π: J -> J), then one easily computes
χ 2n+l = e _ ( Wi + Wi + n ^ + w *^ ( n > Q> w , = p( e ) W2y K2
= 0) .
Therefore p(sc) = Σ «n^ 2n+1
= (Σ α«)(β - (wx+w2)) - aQwQ- Σ nan(w2+w%) covers e iff Σ a
n = l I n
general / = β — (^2 + ^ + z0) is tripotent
for ZtβKΓl Ji(e) iff ^o = z2 + rf = 0 (P(/) - / = « - P(e)« - {βe^} =
(« 2 + «! + 30) — 0&! + 22; 2 + zt) by triviality of K), so p(α?) = / is a
tripotent cover of e iff
In this case
Σ α» = 1, «o^o = 0, {1 + 2 Σ ^«n) (w 2 + ^) = 0 .
Thus in general we cannot get rid of the components w x and w 2 , so
no lift feΦ[x] is the correct lift β. •
Once we can lift a single idempotent, we can without further
ado lift a countable family of orthogonal idempotents. It is not
clear that we can always lift compatible families. We will be able
to lift certain families intermediate between orthogonal and compatible.
A linearly-ordered family {ej of tripotents is hierarchical if
β > a implies e β lies in one of the Peirce spaces Ji(e a ). An important
special case is that of an orthogonal family (e β e J 0 (e a )) or a
collinear family (e β e J^O), or more generally an orthogonal-collinear
family where any two e a , e β are either orthogonal or collinear. It
is easy to see by (1.8(iii)) that any hierarchical family is compatible.
COUNTABLE LIFTING PROPOSITION 5.8. If J^J is a projection
of Jordan algebras with nil kernel, then any finite or countable
hierarchical family e u e 2, of idempotents in J can be lifted to