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Schwarzian Derivative Related to Modules

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Schwarzian Derivative Related to Modules POISSON GEOMETRY n centerlineBANACH CENTERfPOISSON PUBLICATIONS GEOMETRY g comma VOLUME 5 1 INSTITUTE OF MATHEMATICS nnoindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 5 1 WARSZAWA 2 0 0 n centerlineSCHWARZIANfINSTITUTE DERIVATIVE OF MATHEMATICS RELATED TOg MODULESPOISSON GEOMETRY OF DIFFERENTIALBANACH CENTER .. OPERATORS PUBLICATIONS , VOLUME 5 1 n centerlineON A LOCALLYfPOLISH PROJECTIVE ACADEMY OF MANIFOLD SCIENCESINSTITUTEg OF MATHEMATICS S period .. B OUARROUDJ .. and V periodPOLISH .. Yu ACADEMY period .. OF OVSIENKO SCIENCES n centerlineCentre de PhysiquefWARSZAWA Th acute-e 2 0 0 oriqueg WARSZAWA 2 0 0 CPT hyphen CNRSSCHWARZIAN comma Luminy Case DERIVATIVE 907 RELATED TO MODULES n centerlineF hyphen 1f 3288SCHWARZIAN Marseille Cedex DERIVATIVEOF 9 DIFFERENTIAL comma RELATED France TO MODULES OPERATORSg E hyphen mail : so f-b ouON at cpt A period LOCALLY univ hyphen PROJECTIVE mrs period r-f sub MANIFOLD comma ovsienko at cpt period univ hyphen mrsn centerline period fr fOF DIFFERENTIALS .n Bquad OUARROUDJOPERATORS andg V . Yu . OVSIENKO Abstract period We introduce a 1 hyphenCentre cocycle de onPhysique the group Th ofe´ orique diffeomorphisms Diff open parenthesis M closing parenthesisn centerline of afON smooth A LOCALLY PROJECTIVECPT MANIFOLD - CNRS , Luminyg Case 907 manifold M endowed with a projective connectionF - 1 3288 Marseille period This Cedex cocycle 9 , France represents a nontrivial cohomol hyphen n centerline fS. nquad B OUARROUDJ nquad and V . nquad Yu . nquad OVSIENKO g ogy class of DiffE open - mail parenthesis : so f − b Mou closing@ cpt parenthesis . univ - mrs related:r − f to; ovsienko the Diff open@ cpt parenthesis . univ - mrs M . fr closing parenthesis hyphen modules ofAbstract second order . We linear introduce differential a 1 - cocycle operators on the group of diffeomorphisms Diff (M) of a smooth n centerlineon M periodfmanifold InCentre the oneM de hyphenendowed Physique dimensional with a Th projective case $ nacute comma connectionf thiseg . cocycle$ This orique cocycle coincides representsg with athe nontrivial Schwarzian cohomol derivative - comma while comma ogy class of Diff (M) related to the Diff (M)− modules of second order linear differential operators n centerlinein the multifon hyphenCPTM:−In dimensionalCNRS the one , - Luminy dimensional case comma Case case it ,907 represents thisg cocycle its coincides natural with and the new Schwarzian generalization derivative period , while This work is a continuation, of in open the multi square - dimensional bracket 3 closing case , it square represents bracket its natural where and the new same generalization problems have . This been work treated is a in the one hyphenn centerline dimensionalfcontinuationF − case1 3288 period of [ 3 Marseille ] where the same Cedex problems 9 , have France been treatedg in the one - dimensional case . 1 period .. Introduction 1 . Introduction n centerline1 period 1 periodfE − mail .. The1 .: classical 1 so . $ The Schwarzian f− classicalb$ ou derivative Schwarzian $@$ period derivative cpt Consider . univ . Consider the− mrs group the $Diff group . open Diff r− parenthesisf (S1f) of, g$ S to ovsienko the power$@$ of 1 cpt closing .diffeomor univ parenthesis− - mrs of diffeomor. f r g hyphen phisms of thephisms circle preservingof the circle its preserving orientation its period orientation Identifying . Identifying S to theS power1 with ofR 1P with1; fix R an P affine to the pa power - of 1 comma fixn hspace an affine∗fn paframeter ihyphen l l g Abstractx on S1 such . We that introduce the natural a PSL 1 − (2;cocycleR)− action on is giventhe group by the linear of diffeomorphisms - fractional Diff $(rameter M x on)$ S to ofasmooth the power of 1 such that the natural PSL open parenthesis 2 comma R closing parenthesis hyphen action is given by the linear hyphen fractional transformations : nnoindentLine 1 transformationsmanifold : $ Line M $ 2 x endowed rightax arrow+ b with ax plus a b projective divideda b by cx connection plus d sub comma . This where cocycle Row 1 a b represents Row 2 c d a nontrivial cohomol − x ! where 2 SL(2; R): (1:1) . in SL open parenthesis 2 comma R closingcx + parenthesisd ; periodc open d parenthesis 1 period 1 closing parenthesis nnoindent ogy class of Diff $( M )$ related to the Diff $( M ) − $ modules of second order linear differential operators The classicalThe Schwarzian classical derivativeSchwarzian is derivative then given is by then : given by : Line 1 S open parenthesis f closing parenthesis = parenleftbigg f to the power of prime prime prime open parenthesis nnoindent on $M .$ Intheone − dimensional case , this cocycle coincides with the Schwarzian derivative , while , x closing parenthesis divided by f to the power off 000 prime(x) open3 f 00( parenthesisx) x closing parenthesis minus 3 divided by 2 in the multi − dimensional caseS(f) ,= ( it represents− ( )2)( itsdx natural)2; (1:2) and new generalization . This work is a parenleftbigg f to the power of prime prime openf parenthesis0(x) 2 f 0 x(x closing) parenthesis divided by f to the power of prime opencontinuation parenthesis x of closing [ 3 parenthesis ] where parenrightbiggthe same problems 2 parenrightbigg have been open treatedparenthesis in dx the closing one parenthesis− dimensional to the case . wheref 2 Diff(S1): power of 2 comma open parenthesis 1 period 2 closing parenthesis Line 2 where f in Diff open parenthesis S to the power ofn centerline 1 closing parenthesisf1 . nquad period1 . 2Introduction . The Schwarziang derivative as a 1 - cocycle . It is well known that the 1 period 2 periodSchwarzian .. The Schwarzian derivative as a 1 hyphen cocycle period It is well known that the Schwarzian n hspacederivative∗fn canfderivative i l l beg1 intrinsically . can1 . benquad intrinsically definedThe as the classical defined unique as 1 the hyphen Schwarzianunique cocycle 1 - cocycle derivative on Diffon open Diff parenthesis ( .S1 Consider) with values S to the the in power group of Diff 1 closing$ ( parenthesis S ^f 1theg withspace)values $ of quadratic of in diffeomor differentials− on S1; equivariant with respect to the M o¨ b ius group the space of quadratic differentials on S to the power of 1 comma equivariant with respect to the M dieresis-o b ius groupnnoindent phisms of the circle preserving its orientation . Identifying $ S ^f 1 g$ with $ Rhline P ^f 1 g , $ fix an affine pa − 2000 Mathematics Subject Classification : Primary 1 7 B 56 comma 1 7 B 66 comma 1 3 N 1 0 semicolon Secondary 81nnoindent T 70 periodrameter2000 Mathematics $x$ on Subject $S^ Classificationf 1 g$: suchPrimary that 1 7 B the 56 , 1 naturalPSL 7 B 66 , 1 3 N 1 0 $ ; Secondary ( 2 , R ) The− paper$ action is81 in T final 70 is . form The given and paper byno is version in the final linear ofform it will and− benof versionpublishedr a c t i o of n it a elsewhere l will be published period elsewhere . open square bracket 1 5 closing square bracket n [ n begin f a l i g n e d g transformations : nn [15] x nrightarrow n f r a c f ax + b gf cx + d g f , g where n l e f t (n begin f array gf cc g a & b nn c & d nendf array gn right ) n in SL(2,R).(1.1 ) nendf a l i g n e d g n ] nnoindent The classical Schwarzian derivative is then given by : n [ n begin f a l i g n e d g S ( f ) = ( n f r a c f f ^f nprime nprime nprime g ( x ) gf f ^f nprime g ( x ) g − n f r a c f 3 gf 2 g ( n f r a c f f ^f nprime nprime g ( x ) gf f ^f nprime g ( x ) g ) 2 ) ( dx ) ^f 2 g , ( 1 . 2 ) nn where f n in Di f f ( S ^f 1 g ). nendf a l i g n e d g n ] n hspace ∗fn f i l l g1 . 2 . nquad The Schwarzian derivative as a 1 − cocycle . It is well known that the Schwarzian nnoindent derivative can be intrinsically defined as the unique 1 − cocycle on Diff $ ( S ^f 1 g ) $ with values in the space of quadratic differentials on $ S ^f 1 g , $ equivariant with respect to the M $ nddotfog $ b i u s group n begin f a l i g n ∗g n r u l e f3emgf0.4 pt g nendf a l i g n ∗g 2000 Mathematics Subject Classification : Primary 1 7 B 56 , 1 7 B 66 , 1 3 N 1 0 ; Secondary 81 T 70 . The paper is in final form and no version of it will be published elsewhere . n [ [ 1 5 ] n ] 1 6 .. S period BOUARROUDJ AND V period YU period OVSIENKO nnoindentPSL open parenthesis1 6 nquad 2 commaS . BOUARROUDJ R closing parenthesis AND V subset . YU Diff . OVSIENKO open parenthesis S to the power of 1 closing parenthesis comma cf period .... open square bracket 2 comma 6 closing square bracket period That means comma the map open parenthesisnnoindent 1PSL period $ 2 closing ( 2 parenthesis , R satisfies ) thensubset following$ two D i f f $ ( S ^f 1 g ) , $ c f . n h f i l l [ 2 , 6 ] . That means , the map ( 1 . 2 ) satisfies the following two Line 1 conditions : Line 2 S open parenthesis f circ g closing parenthesis = g to the power of * S open parenthesis f closingn [ n begin parenthesisf a l i1 g 6n e plus d gS . S BOUARROUDJconditions open parenthesis AND : V g .nn closing YU . OVSIENKO parenthesis comma open parenthesis 1 period 3 closing parenthesis 1 Swhere ( f to f thePSL powern (2circ; R of) ⊂ *Diff isg the (S ) natural); cf = . Diff [ 2g ,open ^ 6f ] . ∗ parenthesis That g S(f)+S(g),( means S , to the the map power ( 1 . of 2 1 )closing satisfies parenthesis the following hyphen action on1 the .
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