Interior and Exterior Differential Systems for Lie Algebroids

Transcription

1 Aaces Pue Mathematcs o:0436/am05044 Publshe Ole Setembe 0 (htt://wwwscrpog/oual/am) Iteo a Eteo Dffeetal Systems fo Le Algebos Costat M Acuş Secoay School CORELIUS RADU Răeşt Vllage Go Couty Româa E-mal: c_acus@yahoocom c_acus@aesto Recee Mach 6 0; ese August 9 0; accete August 0 0 Abstact A theoem of Maue-Cata tye fo Le algebos s esete Suose that ay ecto subbule of a Le algebo s calle teo ffeetal system (IDS) fo that Le algebo A theoem of obeus tye s obtae Eteg the classcal oto of eteo ffetal system (EDS) to Le algebos a theoem of Cata tye s obtae 000 Mathematcs Subect Classfcato: 00A69 58A5 58B34 Keywos: Vecto Bule Le Algebo Iteo Dffeetal System Eteo Dffeetal Calculus Eteo Dffeetal System Itoucto Usg the eteo ffeetal calculus fo Le algebos (See []) the stuctue euatos of Maue-Cata tye ae establshe Usg the Cata s mog fame metho thee ests the followg Theoem (E Cata) If Ma s a Remaa mafol a X X s otoomal mog fame the thee ests a collecto of -foms uuely efe by the euemets a whee s the cofame (see [3] 5) We ow that a -mesoal stbuto o a mafol s a mag D efe o whch assgees to each ot of a -mesoal lea subsace D of T A ecto fel X belogs to D f we hae X D fo each Whe ths haes we wte X D The stbuto D o a mafol s sa to be ffeetable f fo ay thee ests ffeetable lealy eeet ecto fels X X D a eghbohoo of The stbuto D s sa to be olute f fo all ecto fels X Y D we hae X Y D I the classcal theoy we hae the followg Theoem (obeus) The stbuto D s olute f a oly f fo each thee ests a eghbohoo U a lealy eeet -foms o U whch ash o D a satsfy the coto fo sutable -foms (see [4] 58) Eteg the oto of stbuto we obta the efto of a IDS of a Le algebo A chaacteato of the olutty of a IDS a esult of obeus tye s esete Theoem 47 Ths ae stues the tesecto betwee the geomety of Le algebos a some asects of EDS I the classcal sese a EDS s a a M I cosstg of a smooth mafol M a a homogeeous ffeetally close eal I the algeba of smooth ffeetal foms o M (see [56]) Usg the oto of EDS of a abtay Le algebo I we obtae a ew esult of Cata tye the Theoem 5 I the atcula case of staa Le algebo TM M M ITMIM TM thee ae obtae smla esults those fo stbutos We ow that a submafol S of s sa to be Coyght 0 ScRes

2 46 C M ARCUŞ tegal mafol fo the stbuto D f fo eey ot D coces wth TS The stbuto D s sa to be tegable f fo each ot thee ests a tegal mafol of D cotag As a stbuto D s olute f a oly f t s tegable the the stuy of the tegal mafols of a IDS o EDS s a ew ecto by eseach Pelmaes I geeal f C s a categoy the we eote C the class of obects a fo ay A B C we eote C A B the set of mohsms of A souce a B taget Let L ea lg Mo a B be the categoy of Le algebas moules a ecto bules esectely We ow that f E π M B E π MuMaM E: u π IM a M Ma M R the E π M s a M -moule We ow that a Le algebo s a ecto bule B so that thee ests I B T a also a oeato u wth the followg oetes: LA the eualty hols goo u f u f u I u f fo all u a f LA the 4-tule s a Le -algeba LA3 the Mo -mohsm I s a LeAlg -mohsm of souce a T T taget Let I be a Le algebo Locally fo ay we set t t L t We easly obta that L fo ay L The eal local fuctos L ae calle the stuctue fuctos We assume that s a ecto bule wth tye fbe the eal ecto sace R a stuctue gou a Le subgou of GL R We eote the caocal local cooates o whee a Cose a chage of cooates o The the cooates chage to ' accog to the ule: () If t s abtay the f I t f fo ay f a The coeffcets chage to ule: whee The followg eualtes hol goo: a f f f L () accog to the (3) 3 Iteo Dffeetal Systems Let I (4) (5) be a Le algebo Defto 3 Ay ecto subbule E π M of the ecto bule wll be calle teo ffeetal system (IDS) of the Le algebo I Rema 3 If E π s a IDS of the Le algebo I the we obta a ecto 0 subbule E π 0 of the ual ecto bule so that E π : S0 SEπ The ecto subbule E π wll be calle the ahlato ecto subbule of the IDS E π M Poosto 3 If E π s a IDS of the Le algebo I so that E π S S the t ests lealy eeet so that E π Defto 3 The IDS E π of the Le algebo I wll be calle olute f ST Eπ fo ay ST Eπ Poosto 3 If E π s a IDS of the Le algebo I a S S s a base of the -submoule E π the E π s olute f a oly f S S Eπ fo ay ab a b Coyght 0 ScRes

3 C M ARCUŞ 47 4 Eteo Dffeetal Calculus Let I be a Le algebo We eote the set of ffeetal foms of egee If the we obta 0 the eteo ffeetal algeba Defto 4 o ay the alca- to efe by fo ay L L f I f f a L I fo ay a s calle the coaat Le eate wth esect to the secto Theoe m 4 If the a L Defto 4 If L L (4) the the alcato efe by f 0 fo ay f a fo ay s calle the teo ouct as socate to the secto Theoem 4 If the fo ay a we obta the eualty (4) Theoem 43 o ay we obta L L (43) Theoem 44 The alcato efe by fo ay f L f a 0 I 0 ˆ 0 ˆ ˆ fo ay 0 s uue hag the followg oety: L (44) Ths alcato s calle the eteo ffeetato oeato of the eteo ffeetal algeba of the Le algebo I Theoem 45 The eteo ffeetato oeato ge by the eous theoem has the followg oetes: ) o ay a we obta (45) ) o ay we obta L L (46) 3) 0 Theoem 46 (of Maue-Cata tye) If I s a Le algebo a s the eteo ffeetato oeato of the ete o ffeetal -algeba the we obta the stuctu e euatos of Maue-Cata tye t L t t (MC) a t (MC ) whee t s the cofame of the ecto bule These euato s wll be calle the stuctue euatos of Maue-Cata tye assocate to the Le algebo I Poof Let be a btay Sce t esults that t t t L t L t t () Sce L L a t t t t fo ay t esults that L t t L t t () Coyght 0 ScRes

4 48 C M ARCUŞ Usg the eualtes () a () t esults the stuctue euato (MC ) Let be abtay Sce t t esults the stuctue euato (MC) e Rema 4 I the atcula case of the staa Le algebo T I I T T we obta T (MC )' whee s the cofame of the ecto bule T T As T 0 a L 0 fo all we obta 0 L (MC )' These euatos ae the stuctue euatos of Maue-Cata tye assocate to the staa Le algebo T IT I T Theoem 47 (of obeus tye) Let E π be a IDS of the Le algebo I If s a base of submoule E π the the IDS E π s olut- e f a oly f t ests so that Poof Let S S s a base of the -submoule E π Let S S so that S S S S s a base of the -moule Let so that s a base of the -mo- ule o ay ab a we hae the eualtes: a a S S 0 b b b S S a 0 We ema that the set of the -foms a b a b ; ab s a base of the -moule Theefoe we hae A B b c b bc b bc b C whee Abc B a b C abc ae eal local fuctos so that Abc Acb a C C Usg the fomula S Sb Sc I Sb Sc I S S S we obta that c b b c () () A S S (3) bc b c fo ay bc a We amt that E π s a olute IDS of the Le algebo I b c E S As S S π fo ay bc t esults that 0 b fo ay bc a S c Theefoe fo ay bc a we obta A bc 0 a b Bb C b b Bb C b As B b C fo ay t esults the fst mlcato Coesely we amt that t ests so that fo ay (4) Usg the affmatos () () a (4) we obta that A bc 0 fo ay bc a Usg the affmato (3) we obta Sb Sc 0 fo ay bc a Theefoe we hae Sb Sc Eπ fo ay bc Usg the Poosto 3 we obta the seco mlcato e 5 Eteo Dffeetal Systems Le I be a Le algebo Defto 5 Ay eal I of the eteo ffeetal algeba of the Le algebo t Coyght 0 ScRes

5 I C M ARCUŞ close ue ffeetato amely I I s calle ffeetal of the Le algebo I ule E π oeato eal Defto 5 Let I be a ffeetal eal of the Le algebo I fo all If t ests a IDS E π so that a I we hae u u 0 fo ay u u π E the we wll say that I s a eteo ffeetal system (EDS) of the Le algebo I Theoem 5 ( of Cata tye) The ID S E π of the Le algebo I s olutf the eal geeate by the e f a oly -submoule E π s a EDS of the Le algebo I Poof Let E π be a olute IDS of th e Le algebo I be a base of the We ow that Let -submoule E π 0 I E 0 π ; 49 Let be a base of the -submo- As IE π IE π that t ests t esults so that π I E Usg the T heoem 47 thee esults that E π s a olute IDS e 6 Acowlegmets 6 Refeeces Let a be abtay Usg the Theoems 45 a 4 7 we obta As t esults that I E π Theefoe I E π I E π sely let E π be a IDS of the Le alge- Coe bo I so that the -submoule alg I E π ebo I s a EDS of the Le I woul le to tha Matsumae Iteatoal ouato fo the eseach gat at Toa Uesty ug Al-Setembe 008 I woul also le to tha Po- fessos Heo SHIMADA a So Vasle SABAU fom Toa Uesty-Jaa fo useful scussos a the suggestos I memoy of Pof D Gheoghe RADU Decate to Aca Pof D Doc Rau MIRO at hs 84th aesay [] J Gabows a P Ubas Le Algebos a Posso-ehus Stuctues Reots o Mathematcal Physcs Vol 40 o o: 006/S (97)8596- [] C M Male Le Algebos a Le Pseuoalgebas Mathematcs & Physcal Sceces Vol 7 o [3] L I colescu Lectues o the Geomety of Mafols Wol Scetfc Sgaoe 996 o:04/ [4] M e Leo Methos of Dffeetal Geomety Aaltcal Mechacs oth-holla Amsteam 989 [5] R L Byat S S Che R B Gae H L Golschmt a P A Gffths Eteo Dffeetal Systems Sge-Velag ew Yo 99 [6] T A Iey a J M Lasbeg Cata fo Beges: Dffeetal Geomety a Mog ames a Eteo Dffeetal Systems Ameca Mathematcal Socety Poece 003 Coyght 0 ScRes