Compatible Peirce decompositions of Jordan triple systems - MSP

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COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 77 Proof. One can verify this directly, or use the exchange automorphisms of [7] taking e u e 2 , e z , e±-*e 2 , e u —e i9 — e z and e u e 2 , e z , e,--> -e i9 e 3 , e 2 , -e γ . Π The Peirce decompositions (2.1) and (2.2) simplify in the case of a quadrangle 3.3. Quadrangular Decomposition. // g 7 = {elf e2 , ez , ej is a quadrangle of trίpotents in J then the multiplication operators satisfy ( i ) L(eit ei+ ι) = L(ew , eί+2 ) (indices modulo 4) (ii) Pfe)Pfe +1) = P(ew)P(ew) (iii) L(e l9 e x ) - L(e 2 , e 2 ) + L(β 8 , β 3 ) - I^(β 4 , β 4 ) = 0. Peirce decomposition relative to & is J = == ^ ) =: {^(2200) + ^(0022) + «^(2002) + ^(0220)} ~ ^(1210) + e/(0121)J while all other Peirce spaces vanish. Proof. For (i) we have Lfe, e
78 KEVIN MCCRIMMON ^(2001) = «Muoi) = = ^(2100) «Λuio> = = = = *M02l0) = ^(1200) ~ ^(1002) = «Mioin ~ «MOHI) ~
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Page 9 and 10: 64 KEVIN MCCRIMMON patibility reduc
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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
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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
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Page 45 and 46: 100 KEVIN MCCRIMMON By (7.8) this i
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