Lie algebras of vector fields and generalized foliations

Publicacions Ma emá iques, Vol 37 (1993), 359-367 . Abs ac LIE ALGEBRAS OF VECTOR FIELDS AND GENERALIZED FOLIATIONS JANUSZ GRABOWSKI The main esul is a Pu sell-Shanks ype heo em desc ibing iso- mo phism o he Lie algeb as o ec o ields p ese ing gene alized olia ions . The esul includes as well smoo h as eal-analy ic and holomo phic cases . 1 . In oduc ion . A whole se ies o pape s ollowed he classical esul o Shanks and Pu sell [15] which s a es ha he Lie algeb a X(M) o all smoo h ec- o ields on a smoo h mani old M de e mines he smoo h s uc u e o M, Le . he Lie algeb as X(M1) and X(M 2 ) a e isomo phic i and only i M I and M 2 a e di eomo phic . Some o hese pape s conce n special geome ic si ua ions (hamil onian, con ac , g oup in a ian , e c . ec o ields), as o example he esul s o Omo i [14, Chap e X], Abe [1], o A kin and G abowski [3], and o which speci ic ools we e de eloped in each case . The e is howe e a case when he answe is mo e o less com- ple e in he whole gene ali y . These a e he Lie algeb as o ec o ields which a e modules o e he co esponding ings o unc ions (we shall call hem modula ) . Le us ecall he wo k o Amemiya [2], ou pape [5] whe e de eloped algeb aic app oach made possible o conside analy ic cases as well, and inally he b illian pu ely algeb aic esul o Sk iabin [16] . This inal esul s a es ha , in case when modula Lie algeb as o ec o ields con ain ini e amilies o ec o ields wi h no common ze os (we shall say ha hey a e s ongly non-singula ), isomo phisms be ween hem a e gene a ed by isomo phisms o co esponding algeb as o unc ions, Le . di eomo phisms o unde lying mani olds . The s anda d model o a modula Lie algeb a o ec o ields is he Lie algeb a X(_F) This esea ch was suppo ed in 1993 by KBN, p ojec No . 2 P301 046 03 . 360  J . GRABOWSKI o all ec o ields angen e a gi en (gene alized) olia ion P . How- e e , i we conside no he Lie algeb a o lea p ese ing ec o ields X(_P) bu he Lie algeb a o olia ion p ese ing ec o ields £(7) (Le . ec o ields which gene a e local lows mapping lea es in o lea es), i is no longe modula , since ec o ields o £( .P) ha e ans e sal pa s "cons an " along lea es (see Lemma 1 in Chap e 4), so ha he gen- e al "modula " me hods ail . I makes he desc ip ion o isomo phisms a li le bi ha de and only pa ial solu ions o smoo h classical ( egula ) olia ions we e ound (c . Fukui and Tomi a [4] and Rybicki [19]) . In his no e we p esen a pu ely algeb aic app oach o his ques ion and p e e he Shanks-Pu sell ype esul o he Lie algeb as o ec o ields p ese ing olia ions no only o classical, bu also o a la ge caass o gene alized olia ions . The esul includes as well smoo h as eal-analy ic and holomo phic cases . 2 . S a emen o he main esul . Since we shall be in e es ed mainly in ce ain algeb aic p ope ies o he objec s in ques ion, we shall deal a he same ime wi h ini e dimen- sional mani olds M o di e en classes o smoo hness C : C = C°°, C', l-L, whe e C°° deno es he classical smoo h case, C'-analy ic case, and 7-l deno es holomo phic case o S ein mani olds . Fo de ails we e e o [3] . Fo ins an e, C(M) is he algeb a o caass C unc ions on he mani old M o caass C . No e ha he algeb as C°° (M) and Cw (M) a e eal and he algeb a 7 - L(M) o holomo phic unc ions en he S ein mani old M is complex . I is well-known ha he co esponding Lie algeb a X (M) o all caass C ec o ields can be ega ded as he Lie algeb a o de i a ions o C(M) (in analy ic cases we e e o [6]) . De ini ion . A gene alized olia ion .P = { e% EA en a mani old M o caass C is a pa i ion o M in o connec ed submani olds M = UaEA .Fa which a e exac ly he o bi s o composi ions o lows gene a ed by local ec o ields o caass C angen o he lea es o P . In o he wo ds, lea es o a gene alized olia ion consis o maximal in eg al mani olds o an in olu i e gene alized dis ibu ion P C TM o caass C which is in a ian wi h espec o he lows o local ec o ields wi h alues in P (c . [17]) . No e also ha in analy ic cases he assump ion abou in a iance is supe luous (c . [13]) . Gene alized olia ions will be called u he simply olia ions, while he classical olia ions will be called egula olia ions, since he dimension o lea es is cons an . Deno e by X(_P) he Lie algeb a o ec o ields angen o he lea es o .`F (lea p ese ing ec o ields) . This Lie algeb a is modula , Le . i has he na u al s uc u e o an C(M)-module . Conside LIE ALGEBRAS OF VECTOR FIELDS AND FOLIATIONS  36 1 he Lie no malize N( .F) = {x E X(M) : [x, X( .F)] c X( , F)} o X( .F) in X(M) . Rema k 1 . A "s anda d" no malize consis s o olia ion p ese ing ec o ields, bu in he smoo h case i can be la ge . Conside o ins an e he C°°- olia ion F o R consis ing o one 1-dimensional lea (-oo, 0) and non-nega i e poin s being 0-dimensional lea es . I is clea ha N( .F) = X(R), bu no all smoo h ec o ields (e .g . gene a ing ansla ions) a e olia ion p ese ing . Gi en p E M deno e by he .F p he lea con aining p and by .F(p) he angen space Tp-Fp . Fo poin s o M he ob ious equi alen e ela ion " i " means ha q E Fp (p and q belong o he same lea o Y) . De ini ion .  We call a olia ion F ini ely gene a ed i he C(M)- module X(Y) is gene a ed by a ini e amily o ec o ields which span F(p) a e e y pE M and non-singula i he lea es o F a e a leas one-dimensional . I is no ha d o p o e he ollowing (c . [7] o [8]) . Theo em 1 . Regula olia ions o class C a e ini ely gene a ed . No e ha all olia ions gene a ed by hamil onian ec o ields o local Lie algeb as o Ki illo (c . [11]) o , in o he e minology, gene a ed by Jacobi o Poisson s uc u es (c . [9]) a e ini ely gene a ed . Ou main esul is he ollowing . Theo em 2 . Le .I: '2 be a ini ely gene a ed non-singula olia ion on a mani old MZ o class C and le G i be a Lie subalgeb a o N( .FZ) including X(JT2) (e .g . G i = G( .Iz)- he Lie algeb a o class C olia ion p ese ing ec o ields), i = 1, 2 . I lb : £ 1 -` £2 is a Lie algeb a isomo phism hen lb = 0, o a olia ion p ese ing di eomo phism 0 : M l --> M 2 o class C . The abo e esul may be educed o he ollowing . Theo em 3 . Unde he assump ions o Theo em 2 e e y isomo phism lb : G1 -> ,C2 maps X(JF1) in o X(Y2) . I su ices o apply Theo em 5 .5 o [5] o Theo em 3 .2 o [16] o see ha e e y isomo phism P : X(JF1) -> X( .F2) is implemen ed by a olia ion p ese ing di eomo phism . 36 2  J . GRABOWSKI Rema k ha non- egula olia ions ha e no o be ini ely gene a ed as o example he olia ion om Rema k 1 o he C°°- olia ion F = {R {0, 1, 2 , 3, . . .}, {0 }, {1}, { 2 }, { 3 }, . . .} o R, bu his assump ion seems o be a he echnical . Howe e , in analy ic cases we do no o en know whe he he Lie algeb a X( .T) is no i ial . We belie e yes and due o he Theo em A o Ca an and analogous esul o Tognoli [18] o eal-analy ic case i su ices o p o e he ollowing . Conjec u e .  The analy ic shea o ge ms o analy ic ( eal o com- plex) ec o ields angen o a gi en analy ic olia ion is locally ini ely gene a ed . 3 . Algeb aic p epa a ion . Th oughou his sec ion A deno es an associa i e commu a i e uni al algeb a o e a ield k o cha ac e is ic :~ 2and X deno es a subalgeb a o he Lie algeb a De (A) o de i a ions o A wi h he commu a o b acke . No e ha De (A) is an A-module in he ob ious way so ha we ha e he iden i y (3 .1)  [X, Y] = X ( )Y + [X, Y] o all X, Y E De (A), E A . De ini ion .  We call X C De (A) modula i X is an A-submodule o De (A) and s ongly nowhe e anishing i X(A) =A (whe e clea ly X (A) = span{X ( ) : X E Xand EA}) . The las p ope y may be w i en in he homological way as Ho(X, A) = 0 . Ou s anda d model is o cou se A = C(M)- he algeb a o caass C unc ions on he mani old M and X=X ( .F)- he Lie algeb a o caass C ec o ields on M p ese ing lea es o he olia ion F . The Lie algeb a X( .F) is clea ly modula and i is s ongly non-singula i and only i i con ains a ini e numbe o ec o ields wi h no common ze os (c . [5]) . This is he case o F being ini ely gene a ed and non-singula . Rema k ha a modula Lie algeb a o ec o ields is (in a li le bi mo e gene al se ing) called some imes also a di e en ial Lie algeb a (c . By M (A) deno e he se o all ini e codimensional maximal ideals o A and by M (X) he se o all ini e codimensional maximal Lie subalgeb as o X . I is well-known ha in case o A= C(M) we ha e M (A) - M, whe e he co espondence is gi en by ME)p-J(p)={ EC(M) : (p)=0}EM(A) LIE ALGEBRAS OF VECTOR FIELDS AND FOLIATIONS  363 (c . [5, P oposi ion 3 .5]) . Fo ICApu XI :={XEX :X(A)CI},V(I) :={JEM(A) : I C J}, ¡ := nJEV(z) J, and In := span{ i  ...  n : i, . . ., n E I} . Fo LC X pu V(L) := {K E M(X) : L C K} . Due o [5, Theo em 5 .1] elemen s o M(X( .F)) ( .F ini ely gene a ed and non-singula ) a e o he o m X(_F)p := {X E X( .F) : X(p) = 0} o ce ain p E M . Howe e , o ou pu poses we shall need a li le bi s onge esul . I s algeb aic e sion is he ollowing . Theo em 4 . Le X be a modula s ongly non-singula Lie subalgeb a o De (A) . Then o any k = 1, 2, 3, ... he e exis s n(k) such ha o any Lie subalgeb a L o X o codimension < k we ha e IX C L C Xp o an ideal I o A o codimension < n(k) . P oo : Since X(A) = A, he e a e Xl, . . ., X ,  ,, E X and I, . . ., 7  E A such ha _' j Xj( i) = 1 . Pu n(k) = 2m(k + k 2 ) . I L is a Lie subalgeb a o codimension <_ k, hen U := {X E L : [X, X] C L} is a Lie subalgeb a o X o codimension <_ (k + k 2 ) as he ke nel o he adjoin ep esen a ion o L in X/L . Pu W := { E A : X i E U and iXi E U,¡ = 1, . . ., m} . Since dim(A/W) <_ 2m(k + k 2 ) = n(k), J := AW = span{g : gE A, E W} is an ideal o A o codimension < n(k) . Fo X E X and E W he b acke s [ X i , iX] and [X, iXi] belong o L, so calcula ing hei sum wi h he help o (3 .1) we ge Xi( i )X + X ( i ) X i E L . Since X= AX, we conclude ha (3 .2)  J(Xi( i )X + X ( i)Xj) C L  o all X EX, i = 1, . . ., m . In pa icula , (3 .3)  JXi( i)X, C L  o i = 1, . . ., m . On he o he hand, pu ing X := Xi ( i)X in (3 .2), we ge J((X,( i)) 2 X+X¡( i)X( i)Xi) C L and due o (3 .3) (3 .4)  J(X,( á)) 2 X C L  o all X E X, i = 1, . . .,m . 364  J . GRABOWSKI Since E ' 1 Xi( i) = 1, he ideal gene a ed by {(Xi( i))2 : i = 1, . . .,m} equals A and (3 .4) implies inally JX CL . The ideal I := { E A X C L} includes J and is he e o e o codimension <_ n(k) .  Clea ly IX CL and L is a Lie algeb a, so in iew o (3 .1) and hence L(I) C I . This in u n implies L(A) C I as shows Lemma 4 .2 in [5] . Co olla y 1 . E e y L E M (X) is o he o m Xj o a unique J E M (A) . The p oo is s aigh o wa d . L(I)X C [L, IX] + I [L, X] CL Co olla y 2 .  Gi en B C M (A) and k = 1, 2 . ... he e is n(k) such ha (nB)n(k)X CL o e e y a mos k codimensional Lie subalgeb a L o X sa is ying V (L) C {Xj : J E B} . P oo . Take L as abo e . Acco ding o Theo em 4 he e is n(k) and an ideal Io A o codimension <_ n(k) such ha IX CL and L(A) C I . Since V(L) C {Xj : JE B} implies V(L(A)) C B, we ha e nB C I . Because o he codimension o I, he descending se ies A D (I+I) D (I+ (Í)2) D ... s abilizes a a mos n(k)- h s ep, so I+(I)n( k ) = I+(I)n(k)+1 and by he Nakayama's Lemma (I)n(k) C I, Le . (n B)n( k )X C IX C L . a Theo em 5 . Gi en ini ely gene a ed non-singula olia ion ., on a mani old M o class C and a posi i e in ege k he in e sec ion naEA La o a amily o a mos k codimensional Lie subalgeb as o X( .97) is ini e codimensional i V(L a ) =V(Lp) o all cx,~3 E A . P oo .. Acco ding o Co olla y 1 and ou ema ks a he beginning o his sec ion, V(L a ) = {X( .F) pi : i = 1, . . ., } o some p1, . ... p E M and all a E A . Due o Co olla y 2, we ha e he inclusion J' (k) X( , T) C naEA L a o J = n2-1 J(pi) being he ideal o unc ions anishing a p1, . . ., p . This ideal is Cea ly ini e codimensional and one can see ha J,(k) is ini e codimensional as well (c . No e 1 .4 in [6] o [10] o C°°case) . The C(M)-module X( .97) is ini ely gene a ed, so Jn(k )X( .F) and hence n l EA L a is ini e codimensional . LIE ALGEBRAS OF VECTOR FIELDS AND FOLIATIONS  36 5 4 . P oo o Theo em 3 . Suppose now ha .F is a ini ely gene a ed non-singula olia ion on a mani old M o class C . Fo any Lie algeb a L o ec o ields on M and any p E M deno e L p := {X E L : X (p) = 0}, Lp := {X E L : X (p) E _F(p)}, L(p) := {X (p) : X E L} . Recall ha N( .F) is he Lie no malize o X(Y) in X(M) and ha "~ " is he equi alen e ela ion gi en by F . Lemma 1 . I p - q, hen N( .7)P =N( .F)e . In o he wo ds, a ec o ield o L is angen o he whole lea i a one poin . P oo - Take Y E N(T)' and X E X (_T), X (p) ~ 0 . Choosing local co- o dina es (xl, . . .x n ) a p E M such ha X = a a l and x +1 = . . . = x, = 0 desc ibes locally a componen o JPp, we can w i e Y = ~ 1 i(x) axi , whe e i (p) = i (0) = 0 o i = +1, . . ., n . Since he ec o ield [X, Y] _ j :i i ax, a~ i belongs o X( .F), we ha e ál (x l , . . ., x, 0, . . ., 0) = 0 o i = + 1, . . ., n . I shows ha i, i = + 1, . . ., n, a e cons an (and hence =0) along ajec o ies o X(F) h ough p, so Y is angen o he lea Yp a any i s poin . Lemma 2 . I L is a Lie ideal o N( .F) wi h L(p) ¢ _ F(p) o ce ain p E M, hen o any q - p we ha e Y(q) C L(q) and V(L q ) = V(Lp), whe e V(L, q) = {K E M (L) : L q C K} . Mo eo e nge .~,p L q is in ini e codimensional in L . P oo . . Take Y E L wi h Y(p) 1 .7(p) . Acco ding o Lemma 1 we ha e Y(q) 0 .F(q) o any q E Fp . Ci en a ini e se {q1, . . ., qs} o poin s o Fp we can ind E C(M) anishing a ql, . . ., qs and such ha Y( )(gi) = ai, i = 1, . . .,s, a e a bi a ily chosen . Fo any X E X(F) he ec o ield [Y, X] belongs o L . Since [Y, X](qi) = ajX(gi), he in e sec ion ngEFP L q is in ini e codimensional and .F(q) C L(q) o any q E .Fp . Take now K E V(L q ) . We claim ha LQ C K . Indeed, LQ is a Lie subalgeb a including L q and Lq ~ K means ha .7(q) ¢ K(q) . K(q) C .97(q) would imply ha K C Lq, bu by he assump ion L9 É L, so by he maximali y o K we would ha e K = Lq and Y(q) C K(q) . Assume he e o e ha he e is Z E K, Z(q) q .F(q) . Since L is a Lie ideal o N ( .F), [Y, X] E L q and hence [Z, [Y, X] ] E K o any X E X ( .F) and any E C(M) anishing a q and such ha Y( )(q) = 0 . Chosing such an sa is ying addi ionally Z( )(q) = 0 and Z(Y( ))(q) = 1 we ha e [Z, [Y, X]](q) = X(q) and hence .77(q) C K(q) . The e o e V(L q ) = 36 6  J . GRABOWSKI {K E M (L) : LQ C K} and since Lq = LP , we ge V(L q ) = V(L p ) o q -p . P oo o Theo em 3 : Suppose L = -P(X( .P1)) is no included in X( .P2) . Then L is a Lie ideal o N(F2) wi h L(p) ¢ .P2(p) o ce ain p E M2 . Acco ding o Lemma 2, V(L q ) =V(L p ) o all qE ( .P2) p and ngF(F2)P Lq is in ini e codimensional . On he o he hand, L is isomo phic o X(PI) and due o Theo em 5 his in e sec ion should be o ini e codimension, since dim(L/L q ) < dim(M2) . Re e ences 1 .  K . ABE, Pu sell-Shanks ype heo em o o bi spaces o G-mani- olds, Publ . Res . Ins . Ma h . Sci . 1 8 (1982), 265-282 . 2 .  I . AMEMIYA, Lie algeb a o ec o ields and complex s uc u e, J . Ma h . Soc . Japan 27 (1975), 545-549 . 3 .  C . J . ATKIN AND J . GRABOWSKI, Homomo phisms o he Lie al- geb as associa ed wi h a symplec ic mani old, Compos . Ma h . 76 (1990),315-349 . 4 .  K . FUKUI AND N . TOMITA, Lie algeb a o olia ion p ese ing ec- o ields, J . Ma h . Kyo o Uni . 22 (1982), 685--699 . 5 .  J . GRABOWSKI, Isomo phisms and ideals o he Lie algeb as o ec- o ields, In en . Ma h . 50 (1978), 13-33 . 6 .  J . GRABOWSKI, De i a ions o he Lie algeb as o analy ic ec o ields, Compos . Ma h . 43 (1981), 239-252 . 7 .  J . GRABOWSKI, The Lie algeb as o ec o ields on mani olds, he- sis, Wa saw Uni e si y, 1981, (Polish) . 8 .  J . GRABOWSKI, Ideals o he Lie algeb as o ec o ields e isi ed, o appea in Geom . and Phys ., P oc . 12 h Win e School, S ni 1992 . 9 .  F . GUEDIRA AND A . LICHNEROWICZ, Géomé ie des algéb es de Lie locales de Ki illo , J . Ma h . Pu es Appl . 63 (1984), 407-484 . 10 . G . KAINZ AND P . MICHOR, Na u al ans o ma ions in di e en ial geome y, Czech . Ma h . J . 37 (1987), 584-607 . 11 . A . A . KIRILLOV, Local Lie algeb as, Russ . Ma h . Su . 31, No 4 (1976), 55-75- 12 . Y . KOSMANN-SCHWARZBACHAND F . MAGRI, Poisson-Nijenhuis s uc u es, Ann . Ins . H . Poinca é Phys . Theo . 53 (1990), 35-81 . 13 . T . NAGANO, Linea di e en ial sys ems wi h singula i ies and an applica ion o ansi i e algeb as, J . Ma h . Soc . Japan 18 (1966), 398-404 . LIE ALGEBRASOF VECTOR FIELDS AND FOLIATIONS  36 7 14 . H . OMORI, "In ini e Dimensional Lie T ans o ma ion G oups," Lec . No es in Ma h . 427, Sp inge Ve lag, 1976 . 15 .  M . E . SHANKS AND L . E . PURSELL, The Lie algeb a o a smoo h mani old, P oc Ame . Ma h . Soc . 5 (1954), 468-472 . 16 .  S . M . SKRIABIN, The egula Lie ings o de i a ions o commu a- i e ings, p ep in WINITI 4403-W87 (1987), (Russian) . 17 .  H . J . SUSSMAN, O bi s o amilies o ec o ields and in eg abili y o dis ibu ions, T ans . Ame . Ma h . Soc . 180 (1973), 151-158 . 18 .  A . TOGNOLI, Some esul s in he heo y o eal analy ic spaces, in "Espaces Analy iques," (Séminai e Bucha es 1969), Ed . Acad . R . S . R ., Bucha es , 1971, pp . 149-157 . 19 .  T . RYBICKI, Lie algeb as o ec o ields and codimension one oli- a ion, Publ . Ma . (UAB) 34 (1990), 311-321 . Ins i u e o Ma hema ics Uni e si y o Wa saw u1 . Banacha 2 PL 02-097 Wa saw POLAND P ime a e sió ebuda el 19 de Feb e de 1993, da e a e si6 ebuda el 30 de Se emb e de 1993