Compatible Peirce decompositions of Jordan triple systems - MSP

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COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 71 β = βiu f— e 22> Jiz = Φβn)' Then K = Φe + J n is an ideal which is not an ideal direct summand, yet it is covered (even spanned) by e and all e t = e + z t for some finite basis {zj for J 12 . These e* are idempotents but are not compatible with e: {eeβi} = 2β + z t ί J 2 (ei) since 2 £ g Λfe). Orthogonal families We may regard (2.2) as an analogue for triple systems of the Peirce decomposition relative to a single idempotent g 7 . If we have mutually orthogonal families g\, •••, S? n (for example, if g^ consists of compatible tripotents from J* in a direct sum J — Jx S ffl «/"„) we have the following triple system analogue of the Peirce decomposition relative to orthogonal idempotents g\, , &n. 2.5. PEIRCE DECOMPOSITION RELATIVE TO ORTHOGONAL COMPATIBLE FAMILIES. If g? = g^ U U g" n is ίfee ^wio^ o/ mutually orthogonal compatible families g^, •••, g^ n of tripotents, then the Jordan triple system J has orthogonal Peirce decomposition for Ju - J*(&i) - Λ(^) n Π Jl&ϊ) ôo — ΓΊ «/o( ^ί) i n Jxί^ ) = ^(ίf,) n J^^ ) n n The Peirce decompositions relative to & and g 7 * are recovered by == Σ JiU JJ&) = Σ Jii + Σ ^0, e7 0 (^) - J oo Σ Λ 0 Jo(î) = Σ Jst j k i The Peirce spaces multiply according to the following rules. A product is zero unless its indices can be linked or linked through 0, (2 6) i W « = {J«JM = o if {k, 1} n [i, j} = 0 or {k, 1}
72 KEVIN MCCRIMMON (U3) (U4) {J tjJ ί0J jk}aJ (k while for distinct linked indices i, j, k, l,0φ (2.8) (P7) (PI) P(J ti)J M C J u + J iΰ, P(J oΰ)Joo C Joo (P2) PiJ^Ju c J 3j + J it + J i0, P( J oί)Λ> c J iS + J ί0, P(Jio)Jti C Ju + Jio + Λo (P3) P(J iS)J i} c / tl + J tj + J i0 + J i0, P(J i0)J i0 c /„ + J 40 (P4) {JtJtiJώ^Jt, (j = 0 allowed) (P5) {JuJuJij} c J i3 (i = 0 allowed), {JiΛJio} c 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: 70 KEVIN MCCRIMMON For certain purp
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 and 34: 88 KEVIN MCCRIMMON lifted modulo ni
Page 35 and 36: 90 KEVIN MCCRIMMON a hierarchical f
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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Compatible Peirce decompositions of Jordan triple systems - MSP

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