the subspace spanned by all elements of the [orl11 (α1α2.03) forιε V;.えs凶space1 of U 15仁a¥¥edan ide

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A Study on FK‑ternary Algebras

ルlarcb2002

Yoshiaki 'CAN JC tJCHI

C~raduate School ⁰ '1SCl('^T1仁('and Technology I(UNl入札10'T'0UNJVERSITY

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Chapter 1

Generalized Jordan Triple Systems and Their乱1odifications

1.1. FK‑ternary algebra..;;

Let U be a vector spa.ce ovel' 1，:<field P ⁰1" chara.cteristic clift'erent from two and let B:UxUxUー → Ube a trilinea.r ma.ppi昭 Thcnthe pail' (U， B) (or (1) 15 called a. tl'iple system ovel' F. ¥Ve shall often wrItc (αbc) (OI・[a.bc])in stea.d of B(α，b， C). For subspaces l: (i = 1， 2， :~) or U， we denote by (V₁V₂¥/;) the subspace spanned by all elements of the [orl11

(α1α2.03) forιε V;.えs凶space1 of U 15仁a¥¥edan ide(][ if (UU 1) + (U 1U) + (IUU) c 1 Is va.lid. The whole space U a吋 {O}are caUed the tれtr巾 .

if (U U U) =1= {O} a剖I吋 U ha硝sω ω n‑tri 吋a討Ii川d仇leal.An e引I吋O 削〉叩his幻叩S111D of、υ[1 is called a derivation if D(αbc) = (Dαbじ)+ ⁽^σ^ρb c) + ⁽^α⁶Dc)^司 u.ιcE U. We【:lenoteby ::O(U) .the set of all deriva.tiollS of U. ::0 ( U) i討 aLie algebra. under the Ilsual Lie product [DトD_2]:=

D₁₀f)2一角oD₁_・Forα，bE U， let IJS define the endomorphisms L(α，6): Al(α，b)， R(α^守b) andK(α司b)on U by

L(α，b)c := (αbc)司 M(α?b)c:=(αcbL R(仏 b)c:= (cαb)， !¥‑(α: b)じ:=α(cb)ー (bcα). も"^1ehave

)

︐ . ﹄ 1l •

41 官EA •

︑ ︑ [{(α. b)c = .J11(α，b)c ‑M(bα)c = L(o.・，c)b ‑L(b: c)α.

'vVe say that [l .5山sfieslhe conditian (A) if (xuy) = 0 [01' all x J y E U， thenα= 0 (see [3]). We 1'ecall l.he defilution of two kinds of trip¥e sys.tems. A generαlized ]o.，da.n t7・iple system (01' GJTS simply) 01 the 2nd order is il triple system U with a triline孔rproduct [(α. b)c ;= (αbr.') s剖isf)ing.the following conditions for ，.0b， c， dεU (cf[9]):

(GJTSl) [L(αb)， L(c， d)] = L(L(仏 6)cd)‑L(c， L(b守α)d)ぅ

(GJ.TS2) J((}ピ(久b)c，の =L(d、じ)[((a，b) + ¹⁽α(， b)L(c， d}.

A LIf: triple system (or LTS simply) is a triple system T with a trilinea.r product [abc] satおfyin巳thefollowing conditions forα. bームd、￠ εT:

(LTSl) [ααb] ^= O^.

(LTS2) [αbc] + [bcα] + [cab] = 0，

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1.3・ip‑lllO(litica tiOllS

111 ~i 1.1， we delilled .the ψ‑modi自cations0 '1GJTS's of the 2nd order and saw .that they becol11c F KTA '5. 111 1 his sectiol1， we investiga.te relations amongψ‑Il]()( I i /ic aJi ons.

Propositi(υ】111.3.1. λL点e叫ヨ

tυ)f ( 【[/.H)ト.'[日、'h閃附er川1tけh問cG LA .r.(.H) associated w川ith ([υf H) is isollJorphic 10 the G LA 心(B<p)

出討ociatedwith the FKTA (U， Boplε)for ε=(亨ρ)ー

Proof. ¥^le pllt

.

c⁽^B⁾=乞 c.i，.c( ^B^.，:} =乞々

A linear mapping

φ : a @ xー^α^E^Bp-lX~ ⁽α，X E U)

is an LTS吋omorphisll1of .r.‑₁母.r.}onto _.c~l$ .r.]'. Thcrcforcφis exlcnded to the grade‑ preser¥'Ing Lie algebra ison悶 phismof 'c(B) onl.o .r.(Bψ) (see ( 1. 1. 1 0)) .ロ

Hereafter we shall denote simply b)・(U，B<;J the FKTA. (U， H "..ε).

Le .t(【人 B)be a GJTS of the 2nd order wi.th non‑degeneratc .tracc forrllγ. Then we note that (UH) is an FKTA wi.th the automorphisl11 e fd and i isザIllII削 ric. Let T(B) a吋 c.(B)be the LTS and the GLA associated with (U. B) resp虻Lively.We 巾問tebyψ 1

けh陀ead吐djい⁰¹¹^川^lけt.ope肝ra川tor【ofゆ，h^ε Enc吋d(U) wiih r ゼ肘spe(仁け汁:オt.to γ')'. H や^I^S^an^泊^礼^川^，¹ ψ ^寝 = ψ 一1 Iloldい^t^正^お^;^;f什れrrれo川川I川I11(い1.2.iη)ト.Le叶tus Iい}パu川l刊t..

f(lに H)= {ψεGL( U) Iψo L(xJY) = L(ψX， (ψ摘)‑ly)O世for:r，グεU}. The identityψo L(x， y) = L(ψx. (ψつ‑ly)0ψis eqllivalen .t.10

︑EF

‑ ‑

q J

11 (

ゆB(x，y.z) = BI!tx(1/". )一ly，ψz). l'(U. B) becωo山m1附E

and is called .the structure group of (U， B).

Propositio11 1.3.2. Le .t(U， B) be a non‑degenerate GJ.TS of the 2nd order and n' ^t^h^e

set of all 附 akautoll1orphisms of (U， B). Thell 附 ha¥' ( 1.3.2) f(に【B)= I+".

Proof. LetゆEf(U， B). Then (1.3.]) is valid. Using (ψづ一1= (ψ‑))" a.nd L(ιν)"=

L(y， x)， wc I

('1/)* )ー)B(x，払z)= B((ljJ*)一12Tψ仏(ψT)一1z). 1:3

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