Poisson Vector Fields on Weil Bundles

Transcription

1 dvaces Pure athematcs Publshed Ole November 205 ScRes htt://wwwscrorg/joural/am htt://dxdoorg/04236/am Posso Vector Felds o Wel Budles Norbert ahougou oukala Basle Guy Rchard Bossoto 2 Faculty of Sceces ad Techology are NGOUBI Uversty Brazzavlle Cogo 2 Isttut de Recherche e Sceces Exactes et Naturelles (IRSEN Brazzavlle Cogo Receved 2 October 205; acceted 6 November 205; ublshed 9 November 205 Coyrght 205 by authors ad Scetfc Research Publshg Ic Ths work s lcesed uder the Creatve Commos ttrbuto Iteratoal Lcese (CC BY htt://creatvecommosorg/lceses/by/40/ bstract I ths aer s a smooth mafold of fte dmeso s a local algebra ad s the assocated Wel budle We study Posso vector felds o ad we rove that all globally hamltoa vector felds o are Posso vector felds Keywords Wel lgebra Wel Budle Posso afold Le Dervatve Posso 2-Form Itroducto Wel algebra or local algebra ( the sese of dré Wel [] s a fte dmesoal assocatve commutatve ad utary algebra over whch there exsts a uque maxmum deal m of codmeso I hs case the factor sace m s oe-dmesoal ad s detfed wth the algebra of real umbers Thus = m ad s detfed wth where s the ut of I what follows we deote by a Wel algebra a smooth mafold C the algebra of smooth fuctos o ear ot of x of kd s a homomorhsm of algebras ξ : C such that for ay f C ( f f ( x We deote by ξ m the set of ear ots of x of kd ad x = the set of ear ots o of kd The set s a smooth mafold of dmeso dm dm ad called mafold of ftely ear ots o of kd []-[3] or smly the Wel budle [4] [5] If f : s a smooth fucto the the ma f : = ξ f ( ξ ( d = ξ( d f = ξ( f x x How to cte ths aer: ahougou oukala N ad Bossoto BGR (205 Posso Vector Felds o Wel Budles dvaces Pure athematcs htt://dxdoorg/04236/am

2 N ahougou oukala B G R Bossoto s dfferetable of class C [4] [6] The set C ( commutatve algebra over wth ut ad the ma C C f f s a jectve homomorhsm of algebras The we have: We deote ( such that Thus [4] If of smooth fuctos o ( λ λ f + g = f + g ; f = f ; f g = f g ad Der ( C the set of vector felds o : C C ( ϕψ = ( ϕ ψ+ ϕ ( ψ ϕψ for ay C ( = Der C ( : C C s a vector feld o the there exsts oe ad oly oe -lear dervato : C C called rologato of the vector feld [4] [6] such that ( f = ( f f C for ay Let C be the C -module of Kälher dfferetals of C δ : C C the caocal dervato whch the mage of x C wth f g C C C f f f for ay I [7] et [8] C C wth values o s a the set of -lear mas ad δ geerates the C -module C x = f δ g I: fte e for We deote the -module of Kälher dfferetals of C ( -lear I ths case for ϕ C ( we deote ϕ ϕ the class of ϕ ϕ C ( ( ( C C The ma ( ( C C C C f δ f f = f C C s a dervato ad there exsts a uque -lear dervato such that for ay f C [9] oreover the ma δ : C C δ ( f δ ( f = ( ( whch are 758

3 C C ( x x s a jectve homomorhsm of -modules Thus the ar ( uversal roerty: for every C ( -module E ad every -dervato Φ: C ( E there exsts a uque C ( -lear ma : C Φ ( E such that Φ δ =Φ N ahougou oukala B G R Bossoto ( C δ satsfes the followg I other words there exsts a uque Φ whch makes the followg dagram commutatve ( C δ ( Φ C E Ths fact mles the exstece of a atural somorhsm of C ( Φ -modules Hom C E Der C E ψ ψ δ C I artcular f E C ( = we have C Der C = ( C ( sks C ( C ( For ay Λ = L deotes the C ( module of skew-symmetrc multlear forms of degree from C ( to C ( ad ( ( C C Λ = Λ the exteror C ( -algebra of C ( called algebra of Kähler forms o C ( 0 C ( C Λ = If η C ( Λ = C C Λ ( the η s of the form δ ( f δ ( f f f2 f C C C Λ s geerated by elemets of the form the -module wth ϕ = f ϕ ( = f C Let be the C ( ( η = δ ϕ δ ϕ wth : C C Λ -skew-symmetrc multlear ma such that - Thus 759

4 N ahougou oukala B G R Bossoto for ay x x x C 2 s a uque C s a uque C We deote x x x = x x xˆ x 2 = ad where : C C the uque C ( e -lear ma such that δ = [8] The : C Λ C -skew-symmetrc multlear ma such that ( x x2 x ( x x2 x = ( C ( C ( duces a dervato of degree [9] C : Λ Λ -skew-symmetrc multlear ma such that ( ( x x2 x = x x2 x ( ( : = Λ C Λ C We recall that a Posso structure o a smooth mafold s due to the exstece of a bracket { } such that the ar ( C { } s a real Le algebra such that for ay f C ad( f C C g { f g} s a dervato of commutatve algebra e : { f g h} = { f g} h+ g { f h} for f gh C I ths case we say that C o the ma s a Posso algebra ad s a Posso mafold [0] [] The mafold s a Posso mafold f ad oly f there exsts a skew-symmetrc 2-form such that for ay f ad g C defes a structure of Le algebra over C Posso mafold ad we deote 2 Posso 2-Form o Wel Budles Whe { } s a Posso mafold the ma ω : C C C { f g} ω δ ( f δ ( g = [8] I ths case we say that ω s the Posso 2-form of the ω the Posso mafold of Posso 2-form ω ad : C Der C f ad f such that ad ( f ( g = { f g} for ay g C s a dervato Thus there exsts a dervato ad : C Der C 760

5 N ahougou oukala B G R Bossoto such that Let be a uque C ( = ad f ad f ad : C ( Der C ( -lear ma such that Let us cosder the caocal somorhsm ad let be the ma Proosto [9] If ad δ = ad : C Der C Ψ Ψ δ ad : ( ad C Der C C ( ω s a Posso mafold the the ma ω : C ( C ( C ( such that for ay Y ( C ω ( Y = ad Y s a skew-symmetrc 2-form o C ( such that ω ( x y ω ( xy for ay x ad y C oreover ( = ω s a Posso mafold Theorem 2 [9] The mafold s a Posso mafold f ad oly f there exsts a skew-symmetrc 2-form such that for ay ϕ ad ψ C ( ω : C C C { ϕψ } = ω δ ( ϕ δ ( ψ defes a structure of -Le algebra over C ( { f g } = { f g} oreover for ay f ad g C I ths case we wll say that ω s the Posso 2-form of the -Posso mafold ω the -Posso mafold of Posso 2-form ω [9] ( 3 Posso Vector Feld o Wel Budles Proosto 3 For ay Der C ( ad for ay η C ( Λ ( ( η = ( η Proof If η C the there exsts f f2 f C Λ ( f ( f η = δ δ Thus we have such that ad we deote 76

6 N ahougou oukala B G R Bossoto 3 Le Dervatve ( ( δ ( δ ( ( ( f ( f ( η = δ ( δ ( f f = f f = δ δ ( ( f ( f ( δ ( δ ( = δ δ = f f ( η = The Le dervatve wth resect to D Der ( C L δ δ s the dervato of degree 0 ( ( = + : Λ C Λ C D D D Proosto 4 For ay ( lthe ma L s a uque -lear dervato such that for ay η Λ C Proof For ay η C ( C ( Λ C ( : Λ L ( η = ( η L Λ( we have L ( η = δ ( η + δ ( η = + ( δ η δ η ( δ ( η ( δ ( η ( δ ( η δ ( η = + = + ( η = L vector feld o a Posso mafold wth resect to vashes e L ω = 0 vector feld ω s called Posso vector feld f the Le dervatve of ω : C C o a -Posso mafold of Posso 2-form ω wll be sad Posso vector feld f L 0 ω = ω s a Posso mafold the a vector feld Proosto 5 If s a Posso vector feld f ad oly f s a Posso vector feld Proof deed for ay ( : C C : C C L L [ ] ω = ω 762

7 N ahougou oukala B G R Bossoto Thus L ω = 0 f ad oly f L ω = 0 Proosto 6 Let ( ω be a -Posso mafold The all globally hamltoa vector felds are Posso vector felds Proof Let be a globally hamltoa vector feld the there exsts ϕ C ( such that = ad ( ϕ C φ C e s the teror dervato of the Posso -algebra L ( ( ( = Lad ϕ ( ϕ ( ω δ ( ψ δ ( φ ( ω δ ( ψ δ ad ( φ L ϕ ( ω δ ( ψ L ad δ ( φ ϕ ω δ ψ δ φ ω δ ψ δ φ = ad [6] For ay ψ ad ad ( ϕ ({ ψ φ} ω δ { ϕ ψ} δ ( φ ω δ ( ψ δ { ϕ φ} { ϕ{ ψ φ} } {{ ϕψ} φ} { ψ { ϕφ} } { ϕ{ ψ φ} } { φ{ ϕψ} } { ψ { φϕ} } = = + + = + + = 0 Thus all globally hamltoa vector felds are Posso vector felds Whe ( s a symlectc mafold the ( s a symlectc -mafold [6] [2] For C ϕ we deote ϕ the uque vector feld o C to C ( where such that = ϕ d ( ϕ d : Λ Λ deotes the oerator of cohomology assocated wth the reresetato ( Der C ( Whe ( s a symlectc -mafold the for ay ( L = d ( cosdered as a dervato of Therefore all globally hamltoa vector felds are Posso vector felds ϕ C ad for ay Posso vector feld Y we have Proosto 7 For ay Proof Thus 32 Examle Whe = Y ϕ Y ϕ = Y ϕ Y ϕ Y ϕ ϕ Y ( d ( d ϕ d ( d ϕ ϕ = L = L L = L Y = Y + Y ( ϕ ( ϕ Y( ϕ = d d = d Y = Y Y ϕ = Y ϕ α = dx s a Louvlle form where ( x x s a local system of coordates the cotaget budle T of the ( T = dα = d Λdx = s a symlectc mafold o T [7] Let 763

8 N ahougou oukala B G R Bossoto α be the uque dfferetal -form of degree o T such that Thus d ( α = d( α ( α ( α ( ( = d = d = d Λd x Therefore ( T = d α s a symlectc -mafold For H C ( let H be the globally hamltoa vector feld = d ( H s [3] = x x we have s ad s H = = f + g H = x = d x d = H H + H = x = = H d ( x H d ( + = x = H = f + g j x j xj x j j x = = = f j ( d ( k Λ d ( xk + g j ( d ( k Λd ( xk jk xj x jk j x = jk j x g j d ( k d ( xk d ( k d ( xk jk ( δ δ = g = g j kj k j x H = f + g j j xj j j = = = f = f j ( d ( k Λd ( xk j= xj jk xj = f j d ( k d ( xk d ( k jk x d ( xk j x j jk ( δ δ = f = f j kj k j 764

9 N ahougou oukala B G R Bossoto Thus where f ( g C U H = = = g d x + f d = dh H = H ( ( d x d = x = H f = H g = x tegral curve of H s a soluto the followg system of ordary equato Whe ( x x x Thus 2 2 ( x d H = d t = 2 d ( H = d t x H H H = = x = x s a local system of coordates corresodg at a chart U of where f ( f+ C U s we have Refereces for 2 = dx Λdx + U = ( Λ ( U = = d x d x + = U f + f+ = x = x+ = For ϕ C ( ϕ ϕ = ϕ = x+ x = x x+ ϕ d x ϕ d x ( + = ϕ + = x = x+ { } { ϕψ} ( ψ = ϕ ϕ ψ ϕ ψ ϕψ = = x+ x = x x+ [] Wel (953 Théore des ots roches sur les varétés dfféretables Colloq Géom Dff Strasbourg -7 [2] ormoto (976 Prologato of Coectos to Budles of Iftely Near Pots Joural of Dfferetal Geometry

10 N ahougou oukala B G R Bossoto [3] Okassa E ( Prologemet des chams de vecteurs à des varétés des ots rohes ales de la Faculté des Sceces de Toulouse [4] Bossoto BGR ad Okassa E (2008 Chams de vecteurs et formes dfféretelles sur ue varété des ots roches rchvum athematcum Tomus [5] Kolár P chor PW ad Slovak J (993 Natural Oeratos Dfferetal Geometry Srger-Verlag Berl htt://dxdoorg/0007/ [6] oukala ahougou N ad Bossoto BGR (205 Hamltoa Vector Felds o Wel Budles Joural of athematcs Research htt://dxdoorg/05539/jmrv734 [7] Lauret-Gegoux C Pchereau ad Vahaecke P (203 Posso Structures Grudlehre der mathematsche Wsseschafte 347 wwwsrgercom/seres/38 [8] Okassa E (2007 lgèbres de Jacob et lgèbres de Le-Rehart-Jacob Joural of Pure ad led lgebra htt://dxdoorg/006/jjaa [9] oukala ahougou N ad Bossoto B GR Prologato of Posso 2-Form o Wel Budles [0] Lcherowcz (977 Les varétés de Posso et leurs algèbres de Le assocées Joural of Dfferetal Geometry [] Vasma I (994 Lectures o the Geometry of Posso afolds Progress athematcs 8 Brkhäuser Verlag Basel htt://dxdoorg/0007/ [2] Bossoto BGR ad Okassa E (202 -Posso Structures o Wel Budles Iteratoal Joural of Cotemorary athematcal Sceces [3] Nkou VB Bossoto BGR ad Okassa E (205 New Characterzato of Vector Feld o Wel Budles Theoretcal athematcs ad lcatos