1.Thisis another attempt to define structures on foliated manifolds and Lie algebras associated to those structures(c･f.[K‑3,K‑4])･

Bulletin of the Faculty of Education,Mie University･32服tumlStience(1981),7‑15

SomeLieAJgebrasofVector FieldsandtheirDerivations Case of Strictly Partia"y CLassicalType

by Yukihiro KANIE

山中(JJわ〃川/(〆 几J〟仙川(血･J.八丁ん･n品川小･

1.Thisis another ^{attempt to} define structures ^on foliated manifolds and Lie algebras associated ^to those structures(c･f.[K‑3,K‑4])･

Fix a coordinate systemvl,...,Vpln ^a P‑dimensionalEuclidean ^space U=

択p,and wl,‥.,Wqin ^a q‑dimensionalW=tRq･We considersmoothvector fields onthe(P+q)‑dimensionalspace V=U㊤w=tR掴,and ^{the Lie} algebra9t(V) ofallvectorfieldson

V･Denote忘by∂i(i=1,…,P),and誌

by aq(α=1,.‥,q).Use Latinindicesi,L k... ^for variablesin u and Greekindices α,β,‥. ^for variablesin W,Otherwise stated.

Consider the standard codimension ^q foliation9T on v,defined by parallel

P‑planes:7r㌃(a point),Where ^{7tw is a} canonicalprojection ^{of v onto} W.Let g ^be the Lie algebra of allleaf‑tangent ^VeCtOr fields on V,then by[K‑

2],the ^derivation algebra 9押(g)of gis naturallyisomorphic ^to the Lie algebraJ:of foliation‑preSerVing _vector fields,and̲∫is ^decomposed as

J= g _十J′,

wherel′is naturallyisomorphic ^to the Lie

algebra9t(W).

Let g(u)or g(u))be the Lie subalgebra of g,COnSisting _of vector

fields whose coefficient functions areindependent ^of variablesin ^{W or} U

respectively,thatis,

ど(〃)= 〈方∈g;[∂α,ズ]=♂ (ノ≦α≦ヴ)

g(紺)= 〈ズ∈g;[∂∫,ズ]=β(ノ≦ブ≦♪)

ーしーノ＼しトノ

2.Unimodular Structure. Put p=n,Xi=Vi(i=1,‥,,n),and r=勿<‥.∧血乃.

Lemmal.晰ね ズ∈g肪 ^{ズ= ∑} メrち紺ノ∂f. ^乃g乃 fゐgカ肋緋吉昭■

(･りJJ(///〆りJJJ(肌･(叩†J川んリノ/.

仲⊥gγ=抑ノア一わγふ〃翫 C∞カ乃Cめ乃 ￠勅∈C∞印税 紺ゐg柁 エg ∽gα乃S

′か 上ん･/かブ川/JJ･̀･Hイ肋 ハ･ゆ̀･r､/山.Y:

仲ノズ∈g｢ぴノ α乃d ∑∂J=C 舟γSO椚g ^{CO乃Sぬ乃土c･}

f=J

‑7‑

上ズー=屯r=d(∑｢‑りf _f=J 1血)

= ∑ r‑りf 1(∑ 仏力ノ(転<.れ+∑｢∂αメノ血ノα∧ち)

f=J ノ=J _α=J

=(∑｢∂Jバー+∑ ^∑ r‑りf 1｢乳スノ血Jα∧7f.

Here,aSSume that LxT=￠(勅T ^for some ￠伽)∈C∞｢昭.Then ^we get

∫きヱ∂J=￠｢紺ノ ^{α乃d 乳メ=β}

for alli and α,because the differentialn‑forms f and du)｡<Tiareinde‑

pendent.Hence,X∈g(V)and ^{the function} ￠(W)is constant.Thus,(u implies｢ii).

The converseimplicationis obvious.Q.E.D.

Aleaf‑tangent _VeCtOr field Xis called strict& Pa71hll& col*7mal& ^uni‑

moduldr(s･p･C･u･),if ^LxT equals■tO ^Cr for some constant c･Moreover, if the constant ^c is zero,Xis called strictb)Partidl&unimoduhlr(s.p.u.).

Then we get two Lie subalgebras of9̀:

gβr= 〈ズ∈g:⊥g7=β〉,

gscT=(X∈g′;LxT=C‑r for someconstant c〉･

Similarly,We get tWO Lie subalgebras of̲r:

Jざ丁= 〈方∈J;⊥gr=β〉,

1scT= 〈X∈J;LxT=C‑r ^{for some} constant c).

Since LxT=O for X∈J:′,these ^{are decomposed as}

Jβ丁= gβT+J′ and･JざCT=筑cr+J･

From Lemma5.2in[K‑1]and ^{Lemmal,We get} eaSily

Proposition2.(u 9tT and gs｡T a柁Subaなeb7W〆g(u).Hence,

[筑｡r,.∫′]=[どぶr,J′]=β.

仲ノ 望二Tα乃d箕｡丁α柁乃α血川砂ゐ0椚βゆ紘わ娩βエわαなβ∂憬餌r｢Uノα乃d

軋r｢Uノ 鱒如痴頑γ｢臓[∬‑j]か櫛形露わ乃S〆乳T｢Uノ α乃d守一｡rルノ1.

/かノ装丁ゐ αCOdグ仇g乃5わ乃JんわαJ〆筑｡丁.

/叫乃gゐco〝如5オ′わ乃S〆Jβ丁α乃dJ∂｡rα作成柁Cf:

LIE ALGEBRAS OF VECTOR FIELDS

J8｡r⊆ 軋T｢Uノ㊤乳｢Ⅳノ,

JβT ⊆ 切イUノ㊥餌｢Ⅳノ･

/り凡γ♪≧之g即=ほ8‥汀βr]=[どぶ｡‥gβ｡r]=[ムT,どぶ｡て]

3.Symplectic Structure･Put P=2n･Xi=Vi,yi=Vi･n(1≦i≦n)･and

u= ∑血f∧めi.

i=J

Lemma 3. Ⅳわね ズ∈9■αS.ズ= ^∑ メ｢ち動彿ノ∂f.乃g乃,娩g舟Jわ抑ま乃g

00㈲粛娩間S α柁l狩衰玩庖彿:

伸上gu=￠｢紺ノu かぶo桝g C∞カ乃C血乃 ^{￠｢紺ノ∈C∞=γ上} 伸ノズ∈汀｢〃ノ α乃d舟γα御 上≦乙ノ≦乃,

∂冊￡=み用ん ∂J.乃=ら片抽 ∂∫￡+ら+"ん乃=仇JC ノiげ…肝(､√冊山〃‥､,什/…∵ 軋 メ上裾人血附加膏血血･

伽〆 エgu=動押

d∑ 〈メ圧動紺ノ血+ん乃｢ち弟抑ノ血行乃)

i=j

[∑=烏Jノあ<みf+Hh乃差ノ動<みf〉+呈侃′ノれ<動]

I2 †‡

∑

i=J メ=J α=j

[.∑一〈｢包ん乃ノ(れ∧̀払+｢ら叫ん乃ノ(れ<̀れ)+ ∑イ∂αん乃ノ血ノα<(払]

乃

∑

i=j ノ=J _α=j

=

∑ 〈代りり+ra用ん乃バ血<動

+ ∑(｢a+乃云ノー｢∂汀"ムノ〉めi∧動+ 革｢観月ノ血/α<(れ

+̀ミ〈伯んノーr∂iん乃ノ)血<あー･∑伐んノ血α<れ

Here,aSSume that LxGJ=￠(u))a,for some￠∈C∞(Wl･From ^the

independence of2‑forms dvi<dLb(i<j)and dvi<dwα,We get

∂αメ=β ｢j≦ブ≦2乃,ヱ≦α≦qノ,

∂ふ乃=らん乃, ∂冊J=往+乃メ ｢j≦乙ノ≦乃ノ,

∂′カ+ら冊ん乃= 仇J￠｢緋ノ ｢J≦乙ノ≦犯ノ･

Hence,X∈g(u)and thefunction￠(u))is constant･Thus,(uimplies｢ii)･

The converseimplicationis obvious. Q･E･D･

Aleaf‑tangent _VeCtOr field Xis called strict&Pa71idl&confbmal&即〝砂Iec‑

tic(s･p･C･S‑),if ^Lxu equals ^to ch)for ^some constant c･Moreover･if

‑9‑

the _constant cis zero,Xis cal1ed strid& paYlhllb7 即mt,h7Ctk(s.p.s).

Then,We _get tWO Lie subalgebras of g:

g紬=〈方∈g;エgu=♂)･

gscu=(X∈g;Lxw=CU ^{for some} constant c).

Similarly,We _{get tWO} Lie subalgebras of

̲ど:

J古山=(方∈∫;エg揖=♂〉,

1sc｡=(X∈J;Lx揖=C̀道 for some constant c).

Since Lxu=O for x∈J′,these ^{are decomposed} as

J紬=gざ餌+J′ 仇ブ｣㌦= 洗｡山+J′.

From Lemma6･5in[K‑1]and ^Lemma3,We get eaSly

Proposition4･仲 g紬 α乃dgβC仙 α柁S〟∂砲e∂闇 〆雷｢〃ノ.助乃Cら [gβ｡山,.∫′]=[g紬,J′]=♂.

伸 ^F‥ _′川〟 ど､‥.〟汀′J"仙崎､ふ〃′拙′舛ん､山/ん̀･〃ぐ′血･わw曾【.rrノ α乃d 9l｡山｢Uノ 柁頭βC如βか｢sge[好一j]舟=勿触拡珊=〆班餓=勉励Ⅶノ.

/諦ノ ≦乙u ゐαCOゐ乃β乃5わ乃JんわαJ〆 監｡u.

仲リT如(奴馴噸朋南山ばq′｣㌔｡(抑d.エ｡仙(那(軌如:

∫∂｡山 ⊆やl｡仙｢Uノ①9けⅣノ, J馳 ⊆飢u仲ノ㊤邪｢Ⅳノ･

/りg古山=[gβ山,g紬]=[どぶ｡山,筑｡u]=[エ｡山∴どぶ｡u].

4.Contact Structure. Put p=2n十1,Xi=ut,yi=Vi.n(1≦i≦n),Z=

〃2乃仙α乃♂ β=成一 ∑神永.

i=J

Lemma 5.1拘滋β.X∈ど必･ズ= ^∑ メ｢ち動Z,抑ノ∂ど.了Ⅵg乃,Jゐg舟肋紺才曙

(､′川d鮎りJJ〟Jl･‖〃JJ川んり止

/り ^エgβ= ￠｢ち動Z,紺ノβカγ50椚g舟形Cよわ乃￠｢ち動Z,紺ノ∈C∞rり;

仲ノ⊥ズβ ⁼ ￠rち動Zノβ カγSO桝g施乃C如犯 ￠｢ち動Zノ∈C∞｢Uノノ

/J項7TJ川‑ふ̀JJ川面′りi〃汀晶JJん｢.r,̲lI.こノ∈ぐ■･='ノ ∫J†(･/J〟M/ルr ^叫IJ

｢j≦グ≦乃ノ,

月=‑∂汀乃包 ん=｢∂fゑノ+γイ筏刷れα乃dん1=ゑ‑ブきJタイ∂汁乃り･

旅呵 々 ゐ0∂ぬま乃βd肪

烏=㍍β=ん1 手痛

LIE ALGEBRAS OF VECTOR FIELDS

PYV〆 ^{Using Cartan,s formula Lx=} dix+ixd,We get

粛gβ=̀晩

エ1･β=

I2

∑

g=J

症朗= ^∑ 作みi‑ん乃血fノ,

[(｢訊烏ノーん乃)血+〈佃汁乃烏ノ+メ)めf]

+r筏糾ノ烏ノ虎 +∑｢∂αゑノ血′.

Here,aSSume that Lxe=￠(Ⅹ,y,Z･W)e ^for some￠∈C00(V)･Then,We

get

￠=訪軒鬼 ∂誘‑ん乃=‑γf￠, ∂鉦誘+メ=玖 ^乳烏=O

for alliandα,because the differentiall‑formS dvi(1≦i≦2n+1)and ^du)α

｢1≦α≦q)areindependent ^on V.Hence,the function k and ￠ areinde‑

pendent of variablesin W,and further the coefficient functions of X aregiven

aS

メ=‑r∂汀乃烏ノ,ん乃=｢∂i烏ノ+ ツf￠,and ノ;桝1=烏 +∑ッiん Thus,/i)implies/ii),and(Hノimplies/iiiノ.

Now,itis obvious that/iiiノimplies/u. ^Q.E.D.

Aleaf‑tangent Or foliation‑preSerVing _vector field Xis called strict& ^Par‑

tidl&coniact(s.p.c.),if ^X satisfies ^one of the conditions of Lemma5.Then

we get Lie subalgebras 望̀s｡and Js｡Of gand̲r,reSpeCtively:

gs｡= 〈X∈ g;Lxe=￠(xi動Z)e for some￠∈C∞｢tD), Js｡= 〈X∈J;LxO=￠｢Xiy,Z)e ^for some￠∈C∞｢tD)･

Here,nOte that LxO= O for X∈J',and Jseis decomposed as

Jぷβ=どぶβ+ J′.

From Lemma5and by the similar ^arguments for the proof of Proposition

l.8(f)of[K‑4],We ^get eaSily

Proposition6.(i) ^{gs｡is a} subaなeb7tlqf ^g(v).Hence [どぶβ,J′]=β.

仲ノ 三㌔〃 人JJ(J/J†J埴l‑ ふり仙ノゆ/Jん･J〃 仙ぐ ⊥ん･(/なぐわv ^軋イJ'ノ イ〝〃細雨血

〃gCわγノわム太 ^0乃 U｢sββ[好一j]カγ 瑚乃才fわ乃Sノ.

仲ノ乃gゐco〝如SZ如乃〆ござ｡ゐd才柁Cた

‑11一

Jぶ♂ …餌β｢Uノ㊤乳｢Ⅳノ･

仲ノ ≦㌔=[gββ,どぶ♂]=[ムβ,9二♂]･

5.Derivations(Generaltheory).Consider ^{two Lie} algebras L and M.

Assume that Mis aLie subalgebra of L,Or Lis anidealof M.Alinear map D of M ^to Lis called ^a derivation,if D satisfies the equality

β｢[考,義]ノ =[βr凡ノ,羞]+[考,β｢羞ノ]

for _any Xi(i=1,2)of ^{M.Define the} adjoint ^{operator adX for XeL,aS}

｢adズノy=[X y] ′y∈〟ノ, then adX is a derivation of M to L.

Denote by9e声(〟;L)the set of allderivations of M to L･If M

coincides with L,the set9eグ(L)= 9膵(L;L)is an associative algebra,and the set ad(L)of alladjoint operatorsis anidealof 9el(L)as ^{a Lie} algebra.

Proposition 7. Let M be anidealqf ^a LieαなebYtlL such that M=

[朋:｣材].乃g乃,α町 ^{庇わ〝αfわ乃 β} 〆 ^{〟 わ エ ゐ α ゐわ〃α如乃} 〆 ^{〟 わ} ぬ城∴触け 匁

9どイ〟ノ エノ

9印イ〟ノ.

PYOqf Let XeM.Write X as X= ∑[芳,Xi.,]with ^{XieM(1≦i≦}

2r).Then,We get

β｢ズノ=β｢写[ズゎ方汗r]ノ= 写〈[β｢ズiノ,芳.r]+[芳,β｢ズ汀rノ]),

because Mis anidealof L Q.E.D.

From this,eaSily ^we obtain

Proposition8.Let 〟1and M2 beidbaね d ^{a Lie} a由ebnlL,SuCh ^that

上=〟1㊦挽 [吼城]=β̀∽d｣怖=[吼〟i] ==J,2ノ.7Ⅵg乃,α町

/〟J･J川/ん川 〃 り/⊥ 入J川ねJ付(l･止(､/川小/バ(‑〟/が

β ⁼ a(i〕戊

汀/J川･上ノ〜｢トJ,ごノ ふ

′=ムリブ川/ん川Jイ.1J,〟〃//ふ函､ぐ〃(バ/ん(･/=/J･ん宥〃J={/‑

β わ 〟f.助乃Cg,

9宣イ｣∴)⊇⊆‡凱叩｢劫1ノ萱9ど声｢肱ノ.

LIE ALGEBRAS ^OF VECTOR FIELDS

Asis well‑known,the first cohomology Hl｢L;L)of ^{a Lie algebra L} withcoefficentsinits adjoint representationisisomorphic ^to the quotient ofits derivation algebra by theidealof allinner derivations:

gl〃ィエノ⊆∴軌膵=リ/ad=リ

(see[C‑1]for definitions).

6.Now we go back to our situation･By Theorems3･3,5･7,5･8and6･8in

[K‑1],We ^get

Theorem9.(u ^Let or=Cl(dimU+1),Ca,Or ^{O.77m all} dbrivations

〆9tJrUノ α柁グ乃乃どれ助ゐ,9β声佃げ｢ぴノノ=ad=‰イLりノ…乳♂｢Uノ･

/fり⊥gJJ=r r成加U≠り ^0γ 肌乃g乃,娩β 滋γょがα≠わ乃α包ゼ∂和〆 飢J=り

ゐ乃α血相砂ゐ0椚0ゆ払わ9t｡イUノ,Jゐαfゐ,9β外用J｢Uノノ= 〈adZl凱イ〃ノ;

Z∈やt｡イこり〉⊆∵軋J【り.

Now,We Can determine the _structure of the derivation algebras of strictly partially classicalLie algebras:

TheoremlO.(i)Let q=CT(P≠1),C6D ^Or e.Alldbrivations 〆 9t｡

α柁f乃乃gれ血fゐ,9̀イgぶJノ

adr範Jノ⊆彗J.ガg乃Cβ,

JJl丹L:且.,ノ ^り･

｢項ノー1イJ一丁r♪i･ノノ.り/･u.7了汀(ム･タブ川/ん川 〝むぐわ川 ^り/‑ ∈｢吋 ふ…′わ〃埴l■

如㈱叫励ヒ ム)g腑,〟適･ね ご‰木曾ふ= 〈adZl洗J;Z∈践｡J〉;監｡い肋乃Cら

ガ1｢筑J;筑Jノ⊆筑｡｡/雷餌⊆⊆lR

｢繭ノJ.̀ゾJ=(･γ｢/)≠Jノ.佃

りJ･β∴l//正直川/晶困り/ エJ̀〃で ^{わ川り1仙J/}

ゐ,9どイJ8Jノ

adLエれ巨⊆｣∴.月占乃Cg, glrエJ;J紬ノ

^0.

仲ノJ･̀イJ一T｢/けJノ ′′′･β.7フJ̀･(ムリ･/川/んり=J毎･わ†=イJ､げ ^ふ 〃〟わ〃t岬l･

如微増滅亡わ.ム｡♂,娩αf料 9eイJ紬ノ=〈adZ】J㍍;Z∈J忘｡J)⊆J5｡J.昂彿喝

〟1｢JsJノムJノ⊆J:｡J/エJ…軋

PYVqf(i)and(iiノareobtainedfrom Propositions2,4and6,andTheorem9･

(iiiノand(ivノare obtained from the above(u,(iiノ,Propositions2,4,6and8,

and the fact that allderivations of 9t(W)are alsoinner. Q.E･D･

ー13‑

7･Product Foliation･The Lie algebras of strictly classicaltype have _not

property(A)(see[K‑1]),SOitisdifficulttoinvestigatethoseLiealgebrason

the whole manifoldsin general･Here,We COnSider only product foliations.Let E=QxP ^{be a} productmanifold･and EFthe foliation.defined by

7T忘(y)

(y∈Q),Where7rQisthe projection ^to themanifold Q.IfPisequippedwith

a classicalstructure ^{6p=Tp,uP Or} ep,then we get ^a strictly partia11y Classicalstructure 6=7rニorp ^{on E,and Lie algebras} gs6(E)and̲rS6(E) Similarly ^as §2‑･§4.

By Propositions2,4and6,eaSily we get the following

Proposition12.⊥g≠ J= ちCちU,Cu ^Oγ β.

仲 乃β⊥わ砲e∂和 装J｢引ゐ乃α知和砂怨0肌0ゆ紘わ弧ケアrPノ･

仲ノ 刀〜ぐ⊥ん,埴･･･わ〟エJ用ノ ふ′〜〟/〜〃叫･品冊仙小姑･/り伽√JJ汀‑､=J…=!/●

9†ゥげノ α乃d 91rQノ･

Now,We get the fo1lowing similarly ^as TheoremlO:

Theorem13･(i)Let o･=CT(P≠1),CGD ^Or e.Allderivatibns〆gs｡(E)

〟JJd∫バー/gノ̀肝ナノ仙､Jl〃(W､̀一.

gl｢洗.rEノノ 洪J｢E〃=0 α乃d 〟1rムJ｢且ノ J∫J｢Eノノ=0.

｢ii)Let o･=T(p≠1)or6J.Assume ^{that Pis} connected.The derivati｡n algebra of̲Cs｡(E)and gsd(E)are naturallyisomorphic to̲ど.ゞ｡q(E)and gscq(E),reSpeCtively.Hence,

ガ1｢ど5J｢Eノノ洗J｢Eノノ=gs｡ノEノ/許Jリり⊆=取αr ^∂

〟1｢エJ｢EノノエJrEノノ=エ｡J｢Eノ/Jふ｢gノ ^{⊇=Rαr ￠}

仙脾りJ,往Jノ1三三 択 げ 〟〃(/りJ巾(/●J′.ふ 〝J=叙J(､/ノiげ川 りJJJI

P7VQf ^Thelast part of(iiノfollows ^from Lemmata4.1and6.1in[K‑1].

Q.E.D.

8･One DimensionaJUnimodular Structure･Let P=1.From ^Lemmal,

We get

gβて= 択∂⊆9lT(択)and 筑｡丁= ^R∂ +取∬∂,

Where ^we omitindiceslof xland ∂..And similarly ^as §6in[K‑4],We

get

Theorem14.(t)9e&(9tT)= ad9二cT ⊆⊆R;Hl(9T,T;9:T)⊆tR.

｢可9g〆｢監｡rノ ⁼ ad監｡丁 ^ノ gl｢監｡Tノ筑｡rノ

0.

仲ノ9拍イ｣ニノ ⁼ _adJ'β｡r; ガ1｢JβT;エrノ ^{… 択.}

LIE ALGEBRAS OF VECTOR FIELDS

/叫9βイJざ｡てノ ⁼ ad̲どざ｡r; ガ1｢Jg｡rノJ占｡rノ

^0･

Rema戒.There are no properly ^outer derivations of gsT _and

̲どsT(c.f.

§6in[K‑4]).

References

[C‑1]C.Cheva11ey,and S.Eilenberg:Cohomol嘩汐

aなeb7W,Trans.A.M.S.,63(1948),

[K‑1]Y.Kanie:Cohomo10gies d Lie砲t7bYW qf

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