Unit 10. The Lie Algebra of Vector Fields

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1 U 10. The Le Algebra of Vecor Felds ================================================================================================================================================================ Vecor felds ad ordary dffereal equaos; basc resuls of he heory of ordary dffereal equaos (whou roof); he Le algebra of vecor felds ad he geomerc meag of Le bracke, commug vecor felds, Le algebra of a Le grou Defo. A smooh vecor feld X over a dffereable mafold M s a smooh mag of M o s age budle, such ha X() e T M for each e M. Obvously, smooh vecor felds over M form a real vecor sace wh resec o he oeraos (X + Y)() := X() + Y(), (l X)() := l X(), where X, Y are vecor felds, l e R, e M. We ca mully vecor felds by smooh fucos as well, by he rule (f X)() := f() X(). We deoe by X(M) he vecor sace of smooh vecor felds. Assocaed o a local coordae sysem x = (x,...,x ) o M, here s a 1 ( ) bass of T M a each e dom x, formed by he age vecors {d (): 1<<}. 9 0 The mag d : L d () gves a local smooh vecor feld he doma of he char for each. Thus, every smooh vecor feld X ca be wre he form X = S X d, =1 where he X -s are smooh fucos o he doma of x. The fucos X are called he comoes of he vecor feld X. Gve a smooh vecor feld X o a mafold M, we may ose he followg roblem. Fd hose smooh curves g :(a,b) l M o M for whch he seed of g a e (a,b) s X(g()). Such curves are called he egral curves of he vecor feld. Obvously, a resrco of a egral curve oo a suberval s also a egral curve, herefore, s eough o look for he maxmal egral curves whch ca o be exeded o a egral curve defed o a larger erval. 1

2 Tryg o solve he roblem, we fd ha reduces o a ordary dffereal equao of frs order. Ideed, g s a egral curve f ad oly f for each char x = (x,...,x ), he "vecor-valued" fuco 1 f = xqg :(a,b) l R sasfes he dffereal equao -1-1 f () = (X qx qf(),...,x qx qf()). 1 Acually, fdg egral curves of a vecor feld s he same roblem as solvg a ordary dffereal equao, oly he laguage of formulao s dffere. Traslag he basc resuls of he heory of ordary dffereal equaos o he laguage of geomery we ge he followg heorems, we meo whou roof. Theorem ) (Exsece ad uqueess of soluos). Le X be a smooh vecor feld o a dffereable mafold M. The for each o e M here exss a uque maxmal egral curve g : (a,b) l M of he vecor feld X such ha 0 e (a,b) ad g (0) = (a ad b mgh be -8 ad 8 resecvely). ) (sragheg vecor felds). Le X be a arbrary vecor feld o a mafold M ad e M such ha X()$0. The here exss a char x =(x,...,x ) aroud for whch X = d. Ths meas ha he dervave of 1 1 he mag x urs he vecor feld X o a cosa vecor feld o R. ) (Uboudedess of soluos). If a (or b) fe he o comac subse of M coas he mage g ((a,0)) (or g ((0,b)) ). v) (dffereable deedece o he al o). Le us defe he se U C M for e R as follows U = { e M : e dom g }. The U s a oe subse of M ad he mag H :U L M defed by H () = g () s a dffeomorhsm bewee U ad U. If, furhermore, he - exresso H (H ()) s defed, he so s H () ad H (H ()) = H (). The famly {H : e R } s called he oe-arameer famly of dffeomorhsms or he flow geeraed by he vecor feld THE LIE ALGEBRA OF VECTOR FIELDS Sce age vecors o a mafold a a o are defed wh dervaos a he o, vecor felds ca be cosdered o be dffereal

3 oeraors assgg o a smooh fuco aoher smooh fuco by he formula [X (f)]() = [X()](f), where X e X(M), f e F(M), e M. I hs sese, a vecor feld X s a lear mag X:F(M) L F(M), sasfyg he "Lebz rule" X(fg) = X(f)g + fx(g). Defo. Le A ad B be wo lear edomorhsms of a vecor sace V. The he lear mag [A,B] = A q B - B q A s called he commuaor of hem. Prooso. The commuaor of lear mags sasfes he followg dees (A,B,C e Ed((V), l e R) ) [A+B,C] = [A,C]+[B,C] [C,A+B] = [C,A]+[C,B] [la,b] = [A,lB] = l[a,b] (bleary) ) [A,B] = -[B,A] (a-commuao) ) [A,[B,C]]+[B,[C,A]]+[C,[A,B]]= 0 (Jacob dey) Proof. We rove oly ), he res s lef o he reader [A,[B,C]]+[B,[C,A]]+[C,[A,B]] = = [A,(BqC-CqB)]+[B,(CqA-AqC)]+[C,(AqB-BqA)] = = Aq(BqC-CqB)-(BqC-CqB)qA + Bq(CqA-AqC) - -(CqA-AqC)qB + Cq(AqB-BqA)-(AqB-BqA)qC = = AqBqC - AqCqB - BqCqA + CqBqA + +BqCqA - BqAqC - CqAqB + AqCqB + +CqAqB - CqBqA - AqBqC + BqAqC = = Defo. Le us suose ha a lear sace L s edowed wh a blear mag [, ]:LxL L L sasfyg codos ), ), ad ) of he above rooso. The he ar (L,[,]) s called a Le algebra Prooso. Le X ad Y be wo smooh vecor felds o a mafold M Cosderg hem o be lear edomorhsms of he vecor sace of smooh fucos F(M), he commuaor [X,Y] of hem s also a vecor feld. Proof. The commuaor [X,Y] s a lear edomorhsm of F(M) so we oly have o check ha sasfes he Lebz rule. For f,g e [X,Y](fg) = (XqY-YqX)(fg) = X(Y(fg))-Y(X(fg)) = F(M) we have = X ( Y(f)g + fy(g) ) + Y( X(f)g + fx(g) ) 3

4 = XqY(f)g + Y(f)X(g) + X(f)Y(g) + fxqy(g) -YqX(f)g - X(f)Y(g) - - Y(f)X(g) - fyqx(g) = = [X,Y](f)g + f[x,y](g) Corollary. The commuaor of vecor felds, whch s geerally called he Le bracke of hem, s a bary oerao o X(M), gvg he sace of vecor felds a Le algebra srucure. Prooso. Le us choose a local coordae sysem (x,...,x ) o M ad deoe by d,...,d he assocaed coordae vecor felds. The we have 1 ) [d,d ] = 0; j ) [fx,gy] = fg[x,y] + fx(g)y - gy(f)x for each X,Y e X(M), f,g e F(M); ) f X = S f d, Y = S g d are arbrary vecor felds, =1 =1 he [X,Y] = d g d f S (X(g )-Y(f ))d = S ( S f g )d. j d x j d x =1 =1 j=1 j j Proof. ) The frs ar of he rooso s equvale o Youg s heorem, (kow from mulvarable calculus), whch says ha for ay smooh fuco, defed o a oe subse of R, he mxed aral dervaves d f d f ad are equal. dx dx dx dx j j ) Le h be a arbrary smooh fuco o M, ad aly he oeraor [fx,gy] o. [fx,gy](h) = fx(g Y(h))-gY(fX(h)) = fx(g)y(h)+fgx(y(h))- gy(f)x(h)- gfy(x(h)) & * = fg[x,y] + fx(g)y - gy(f)x (h). 7 8 ) Usg ) ad ) we ge # $ [X,Y] = S f d, S g d = S S [f d,g d ] = j j j j 3=1 j=1 4 =1 j=1 = S S f d (g )d - g d (f )d = S (X(g ) - Y(f ))d j j j j =1 j=1 =1 Suose ha we are gve wo vecor felds X ad Y o a oe subse of R. The corresodg flows H ad G do o commue geeral: H qg $G qh. s s s To measure he lack of commuao of he flows H ad G, we cosder he s dfferece F(s,;) = G qh () - H qg (), for a fxed o. s s 4

5 F s a dffereable fuco of s ad ad s 0 f or s s zero. Ths meas, ha he Taylor exaso of F aroud (0,0;) & d F d F * F(s,;) = F(0,0;)+ s (0,0;) (0,0;) + 7 d s d 8 & s d F d F d F * (0,0;) + s (0,0;) (0,0;) + o(s + ) 7 dsd 8 d s d d F he oly o-zero aral dervave s (0,0;). dsd Clam. Through he aural defcao of he age sace of R a d F wh he vecors of R, he vecor (0,0;) corresods o he age dsd vecor [X,Y](). Proof. Pu X = S f d, Y = S g d, where d deoes he vecor feld =1 =1 d d Le us comue frs he vecor H qg () a s = = 0. dx dsd s & d * & d * We have H qg () (0,) = g (s) (0,) = X(G ()). 7d s s 8 7d s G () 8 Dffereag by, d 1 d 1 & * H qg ()1 = X(G ())1 = S Y(f )d (). dsd s 1s==0 d 1=0 7 8 A smlar comuao shows ha =1 d 1 & * G qh ()1 = S X(g )d (). dsd s 1s==0 7 8 Subracg hese equales we ge =1 d F & * (0,0;) = S (X(g ) - Y(f ))d () = [X,Y]() dsd 7 8 =1 Now reurg o he Taylor exaso of F, we see ha d F F(s,;) = s (0,0;) + o(s + ) = s[x,y]() + o(s + ) dsd I arcular, we oba he followg exresso for [X,Y](). u o [X,Y]() = lm (G qh () - H qg ())/ L0 m Defo. We say ha wo vecor felds are commug f her Le bracke s he zero vecor feld. Theorem. Le {H : e R } ad {G : e R } be he oe-arameer famles

6 of dffeomorhsms geeraed by he vecor felds X ad Y resecvely ad suose ha he vecor felds X ad Y are commug. The he dffeomorhsms H ad G are commug as well he followg sese. s For each o of he mafold here exss a osve e (deedg o ) such ha for ay ar of real umbers s, sasfyg he equaly 1s < e he exressos H (G ()) ad G (H ()) are defed ad s s cocde: H (G ()) = G (H ()). s s Proof. If boh X ad Y vashes a he H () = G () = for ay s ad s ad hus he assero holds rvally. We may hus suose ha oe of he vecors X(), Y(), say X() s o zero. By he heorem o sragheg vecor felds we may suose ha he mafold s a oe subse of R, wh coordaes (x,...,x ), ad he vecor feld X cocdes wh he bass 1 vecor feld d.le Y = S g d be he slg of Y o a lear 1 =1 combao of he bass vecor felds d. By he formula for he Le bracke of vecor felds we have d g d g 0 = [X,Y] = S d = 0 for each. d x d x =1 1 1 Cosequely, he fucos g do o deed o x, hus he vecor feld 1 Y s vara uder raslaos arallel o he vecor e =(1,0,...,0). Ths 1 mles ha f g s a egral curve of he vecor feld Y he so s g + e for ay (he doma of g + e s a oe subse of he doma of 1 1 g). O he oher had, he dffeomorhsm H s jus a raslao by he vecor e. So, for small s ad, we have 1 G (H ()) = G ( + se ) = g () = (g + se )() = g () + se = s 1 + se = G () + se = H (G ()) s THE LIE ALGEBRA OF A LIE GROUP Le F:M L N be a dffeomorhsm bewee wo mafolds. F defes a bjeco bewee X(M) ad X(N), ad hs bjeco s a Le algebra somorhsm. Le G be a Le grou, g e G. Deoe by L he lef raslao by g,.e., g L :G L G, L (h) = gh. L s a dffeomorhsm, s verse s L. A g g g -1 (g ) vecor feld XeX(G) s called lef vara f L (X) = X for all g e G g Sce L (X) = X ad L (Y) = Y mles L [X,Y] = [L (X),L (Y)] = [X,Y], g g g g g lef vara vecor felds form a Le subalgebra of X(G). 6

7 Defo. The Le algebra of lef vara vecor felds of a Le grou s called he Le algebra of he Le grou If XeX(G) s lef vara, he X(g)=L (X(e)), hus, a lef vara g vecor feld s uquely deermed by he vecor X(e) e T G. (e s he u e eleme of he grou G.) Sce every vecor T G exeds o a lef e vara vecor feld hs way, he assgme X9-----LX(e) yelds a lear somorhsm bewee he vecor sace of lef vara vecor felds o G ad T G. As a cosequece, we oba ha he Le algebra of a Le grou s e fe dmesoal, s dmeso s he same as ha of he Le grou. As a examle, le us deerme he Le algebra of Gl(,R). Gl(,R) s a oe subse Ma(,R) = R, so s mafold srucure s gve by oe char, he embeddg. Tage saces a dffere os ca be defed wh he lear sace Ma(,R). For A e Gl(,R), he lef raslao M9-----LAM exeds o a lear rasformao of he whole lear sace Ma(,R). The dervave of a lear rasformao of a lear sace s he lear rasformao self, f we defy he age saces a dffere os wh he lear sace, so a lef vara vecor feld X:Gl(,R) L Ma(,R) has he form X(A) = AM, where M e Ma(,R) s a fxed marx. The egral curves of a lef vara vecor feld o Gl(,R) ca be descrbed wh he hel of he exoeal fuco for marces. If M s a M arbrary square marx, he we defe e as he sum 8 M s =0! :R L Gl(,R) by If we defe he curve g A M g () = A e, A he we oba a egral curve of he vecor feld X(A) = AM. Ideed, M M g () = Ae M = X(Ae ) = X(g ()). A A The flow geeraed by he lef vara vecor feld X cosss of he dffeomorhsms M H (A) = Ae, M ha s, H s a rgh raslao by e. Now le us ake wo lef vara vecor felds X(A)=AM ad Y(A)=AN ad cosder he flows H ad G geeraed by hem. 7

8 Comug G qh (A)-H qg (A) u o o( ), we ge M N N M 1 1 G qh (A)-H qg (A) = A(e e -e e ) = A(I+M+-----(M) )(I+N+-----(N) ) A(I+N+-----(N) )(I+M+-----(M) ) +o( )= A(MN-NM) +o( ). We oba, ha he Le algebra of Gl(,R) s somorhc o he Le algebra of all marces wh Le bracke [M,N] = MN-NM. Furher Exercses Exercse Le d ad d be he wo coordae vecor felds o R 1 deermed by he dey mag. Descrbe he vecor felds X(x,x ) = x d +x d Y(x,x ) = x d -x d, comue her Le bracke, ad deerme he flows geeraed by hem. Exercse 10-. Show ha he Le algebra of SO() s somorhc o he Le algebra of skew-symmerc x marces wh Le bracke [X,Y]=XY-YX. 3 Exercse Show ha R edowed wh he cross-roduc x s a 3-dmesoal Le algebra somorhc o he Le algebra of SO(3). 3 3 Exercse For v e R, le X deoe he vecor feld o R, defed by v X (x) = v x x. v Descrbe he flow geeraed by X, ad rove ha v [X,X ] = - X. v w vxw Exercse Show ha he Le algebra of lef vara vecor felds o a Le grou s somorhc o he Le algebra of rgh vara vecor felds. 8