Compatible Peirce decompositions of Jordan triple systems - MSP

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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 ô = 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, «oô = 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 JÔ), 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
90 KEVIN MCCRIMMON a hierarchical family elt e2f . of idempotents in J. If J and J are unital and elf •••, en are supplementary orthogonal idempotents in J, then elf , en are supplementary orthogonal idempotents in J. The same results hold if J, J are Jordan pairs. If J-+J is a projection of Jordan triple systems with nil kernel, and if 1/2 eΦ, then any finite or countable hierarchical family in J. of tripotents in J can be lifted to a hierarchical family Proof Assume {ej is a hierarchical family of tripotents (resp. idempotents in the Jordan algebra case). By the Lifting Lemma 5.1 we can under our hypotheses lift ex to eu which is by itself trivially hierarchical. Assume we have lifted {eu -- ,en} to hierarchical {elf -- ,en}. Then these are in particular compatible, and determine a Peirce decomposition J = φ J (
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Page 3 and 4: 58 KEVIN MCCRIMMON involution a-â
Page 5 and 6: 60 KEVIN MCCRIMMON (k = j or i = j"
Page 7 and 8: 62 KEVIN MCCRIMMON (iii) [P{e)\ P(/
Page 9 and 10: 64 KEVIN MCCRIMMON patibility reduc
Page 11 and 12: 66 KEVIN MCCRIMMON elements z with
Page 13 and 14: 68 KEVIN MCCRIMMON of the underlyin
Page 15 and 16: 70 KEVIN MCCRIMMON For certain purp
Page 17 and 18: 72 KEVIN MCCRIMMON (U3) (U4) {J tjJ
Page 19 and 20: 74 KEVIN MCCRIMMON component in J t
Page 21 and 22: 76 KEVIN MCCRIMMON corresponding to
Page 23 and 24: 78 KEVIN MCCRIMMON ^(2001) = «Muoi
Page 25 and 26: KEVIN MCCRIMMON rigidly imbedded in
Page 27 and 28: 82 KEVIN MCCRIMMON algebra, hence t
Page 29 and 30: 84 KEVIN MCCRIMMON reduce to (*) =
Page 31 and 32: 86 KEVIN MCCRIMMON also vanish and
Page 33: 88 KEVIN MCCRIMMON lifted modulo ni
Page 37 and 38: 92 KEVIN MCCRIMMON For example, tak
Page 39 and 40: 94 KEVIN MCCRIMMON pairs over arbit
Page 41 and 42: 96 KEVIN MCCRIMMON usual to simply
Page 43 and 44: 98 KEVIN MCGRIMMON S and S* preserv
Page 45 and 46: 100 KEVIN MCCRIMMON By (7.8) this i
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