A' t -i--*-— w«.„ " n MOCKQBCKM Oïyjjm 'JïEmiHâ, ÛJ^r_- v. üfiÄEHa TP^AOBarO KPACHOrO SH^vGI-L'I rCC.'^-.r^.j—-—. .;-. MBBM M.B.ICMOHOCOBA MexaHHKo-MaQ?eMaTir?ecKSH ÈaKyjiBïeT Kaœeflpa BHCniel asreÔpH Ha ireaBax wkoheciï ïïATOH , .TG TJ . jmom&im rPAjryïïPOMHHHx ajd?ebp m 01.01.06. - MaTeMaTsraecKas ' jrorKKa a^»* ¦es. j Qjiivuuo -à- roujJ.,n -li^o^ JÏHcoepTamïH na coECsaHüe y^enoS CTenem KaHAH^ata $K3HKo-MaTeMaTi2raecKiix EayK. Hay^HES pyKOBoaETejïL ¦qjien-Kopp. AH CGC? npcçeccop AsH.K0CTPIîKllH MOCKM - 1982 f Moscow State University M. V. Lomonosov Faculty of Mechanics and Mathematics Department of Higher Algebra Dušan Pagon Deformations of Graded Lie Algebras PhD thesis (01.01.06 - mathematical logics, algebra and number theory) Moscow, 1982 Abstract In the thesis we study the graded deformations of some natural classes of nilpotent Lie algebras and the class of almost free Lie algebras. We have found local graded deformations (2-cocycles) of this algebras and examined the conditions for existence of their extensions to global graded deformations. The obtained results can be divided into three groups: 1. general results about graded deformations (the definition, closure under the extension operation, existence of obstacles) ; 2. graded deformations of nilpotent Lie algebras, which are the maximal parts of some classical simple Lie algebras, and their specializations; 3. graded deformations of free and almost free Lie algebras were completely described. Diasa le Ochosh leopmï jeœopMaiiHt anreöp . . .¦„".- .'' 8 §1. OnpejrejieHse rpajryzpoEaHKHx Re&QVxaigct e HeKOTOpHe SX CBofOTBa.......'....'.. §2. Bonpoc o cneixHajiEsamîiî napaMe?pa ....... 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Tor^a jyw BTopoË rpynnn rpa^ya-posâHHHx KoroMOJiorufi ajireöpH L KMeew: dU Hl(M .-(°'ecmi ^3-4 3 U npn rv = 3, 4 , KaÄßHM H3 KOI#ÏKJIOB 9fe ,' gy, HBJIHBDIJpXCH OCpasyiQ^KMH rpyiUIH L(v/b cjiyqanx ft=3, 4, ßeÄCTsyeT HenyjieBbiM oÖpasoM toji&ko Ha oftHy napy 6a3HCHHx sjieMeHTOB, a HMeHHo: JIOKaJlbHHe £e$OpMaipiK J y W4eH)T TpHBHaJIbHBie npOßOJECeHHH, a npn cnei^HajiH3an,MM napaneTpa T na ocHOBHoe nojie mh b ksjk-^om cjiyqae nojiyqaeM eÄHHCTBeHHyB, c toxihoctwo ao HsoMopcpESMa, aareöpy Hô-H30Mop<|Hy» hcxo^hoä. ^0Ka3aTeJibCTB0. KaK h b npe^romeM naparpafe» hsm $jih HaxoH^eHHH jioKajibHHx Ae$opMaE[iiA npufleTCH Hsy^aTb no oT^eJitnoc- TM KaJSflyK KOMHOHeHTy np0H3B0JïbH0r0 KOI^KKJia / £ H ore K L/L), %o(5h c^ejiaTb BHSHCJieHKH Menée rpcMes^KUMH, BBegeM jpn cTan-flapTHoro Öa3Hca .ajireöpH L cJie^yiDipe o6o3HaqeHHH: TaKHM o6pa30M, L ~'^eU 1 i^hj ^ n ) • I . d^ . 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Ten cskûIM mh hojt^ikm, hïo bch KOMnoHeHTa 1/2,)^ paBKa Hyji». lïOflcraBJiHH npn ^W_/1b (2.65: JViÜi ÄiviCt âpKBOAHT Hac k paBeKCTsy: ^ ^';^w\+àVa ~® • AaJïee, no^cTasjis« aa nojiyqaeM, xïto jtS^kfl) ^+4^+4)4" 0 mm k < Ita -r-4 , j = Zj3, •• -, ITI , nOKaSHBSST, tlTO a/^'^J« 0» HaKOHeq, H3 no^cTaHOBEM b (2.6): a=^m^,^ j^~^m+*ìA > c~&mJ > \~ */%;'• v ^; i-^j% •"} w BbirenaeT paseHCTBO: 3. (&,4. nycTb ^ .^.? 1^1 J -'Dû -pH -4±i*i .^K) A± t skin, (c+j+l-i mn m ^foj = 2 f P eUl-s,s W* 4élékéY\. Tor^a, cornac-«0 C2.II):! ^j .,} j , mm -» î-v j ^ /0 ''' i Js fit v '¦' -¥, „ \ c. l Ua ) *- p ¦ ) — U " --'¦ a. - " - • m -¦ • -». i\ *; f f /\ 3aMetîafl aanee, ^to b cJïyqae FI = 3 fljin ^fo/J-^^ o^£/£ Ha Z/ x 4, k nojiaran: mm b 9T0M cjiyqae flonojnuiTejibHO mmssm: A - « 0 = 0. (223.1) ' /V ì ÄOKajseM Tenepb, q*ro bch KO&moHeHTa Yi% % PaBHHeTCH Hyjno. noflCTaBJiHH npH n £ 4 b (2.3):a=e^K >&=^w p c'e^ ) ^^A-/^ h a - ^ž. ^.-/) ; b~^n.-\n y c= e^ s MH nojiy-qaeM, trro f^'^'^Q (ckaiPtl). (223.2) Ü3 (223.1), (223.2) h no^CTanoBCK b (2.3): a- t4{ y^t; V4 ;. 7 ç \ -m> c~ ^k^ > l-ž,^..^^ , 0 ¦ fczi+l/i+Zj ¦:•-, ïl , MH c y^eTGM (223.3) m&eett: Tene.pt nO£CTaHOBKH <^ '^^ ,v - ei,{ ) c~ w >&-'*¦ 4l ~ n npKB0,4fiT Kac k pasencTBy: ? fe • / e, Ì = /3 (2^.3.5) ..QACTâHOBKH b (2.3): a=S, n ;b~£; j* / C-Ôjj ) fc-&,3,'-ift o k è L f j £ !\ h paBeHCTBa (223.5), (223.6) noKasbraasT, qTO 'F', sO . (223.7) ite (223.7) h no^cTaHOBOK b (2.3): a- £^& ,£> = €./,; f c~£p 0 fh ' =0 npH yKa3aHHHx ¦ 3Ha^eHHflx hkaokcob i^^icji. 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(2.8)-(2.I2)), nosTOMy ecJiK ohh koi^mkjih, to ooRS&remMO HeTpHBHajibHHe. TaKOË cJiyqaH raaeeT Meeio npK 'fl= 4, Korpß effSKCT-BeHHoe OTOÖpaxeHne ynasaHHoro BH^a - Q/t0 yAOBJieïBopïïe? ye~ üobhhm (2.6), (2.7) h, cjieflOBaTejibHO, irpHHaflJiesi'iT Ht» {L.Lj . ïïycTb Tenepb fl£5. ItoflCTaBJUiH b (2.7): ase?-()ö = ^-; C-2^. , ^i-^k-j ' ^^A)--'> ft-k , mseeM ^z(^)eh^\ -û , a noflCTa- HOBKH B (2.6): b = e2^ ;M=f^;^;j « (^3>£3,iJ nOKa3HBaS5T, *ITO ^2, ie^L/i^2>,l'A ~/2 SV^2m~^* ÂOKfiUKeM KH^yK- ipe& no g , UTO ^fe^^J^Û,- #=£3,..., r-rJ » Ks noßcTa-HOBOK B (2.6): öL= Q-îc/k+4 j ^~~£^ * ^"^fe Â-4 cea^ajia shtske-6T» tito f^(egi^ }^krA Ä-^fL'Ö» a 3STeM, nocJie noflCTaHOBOK: a= e,/f , i = eM , <: = ^ ; a= «^ ; £*tfM, c= **«,*-., nojïyxîaeTCH TpeÖyeMHß pe3yjibTaT. TenepB mh HHflyiojiieft no k go-EameM, qtg ßz(^L> ; ?-k+*, k K ~° J &-Z, 3/ — _> fi^ i^K+i. Mr sToro $oeTETOtiHo no^cTaBHTb b (2.65: cl~£9 g- jh~c. , ? floKamëM, HaKOHei;, HH#yKipe$ no fH , qTO iz\6u ^^jJ^-O jiim JÎK50UX */« -m ^^-^f , a 3T0 paBHOCHJÏbHO TOMy, ^TO 6^,/,. -Ö npH H c5. rio^cTaHOBKH b (2.6): £&-£W/wm ; £ ^ £;, ( ? ¦C**m,t > i*h\-,*-A > Ì**l,-,ì>-4> (HI,-,"-*, nOKa3HBaE5T, qTO fzl^tyj^toM/iij ~ 0 . 3aT6M} nOflCTaBJIflH b (2.6): a- e^„„ / t> =£*-<,, t t cst»4 > Äaaee, H3 noACTaHOBOK: û.^ A/^e -jC-t t Jß-^A-^'i ' - \ BHTeKaeT, tjto ^(^ n^^^^.^j - 0 , a ks noscväHQBOK: a~e^7;-.rM > n f 0 'u paBHHeTCH HyJDO, 3. lpt']2_. IlycTb fil Pie-- e,A = Z H' «;¦<-.--& -' s tii^in , Jihtm, iikâ(i°**Î<(!j. cm ., t * BHÖHpan nOÄxo,iyi^HM oÖpasoM saanewisi ff J" s ssh mohsm ^oöetbch . Toro, qTOÖH nocjie BmmTamix hs (y^ j^ KorpamìiiH ö^ BHnoJSïflJiKCb ycJïOBHH : /„ (W eCHß, =0 > ù ~*'l>">n'1 > if^1'1*0' ~'n > Q».t,M__ ni,ï,A,l _ Q }>,.,, 3 _ö (233.1) f % J Z ~" j % / SnatïuT, (^ i -O wgm Hs3. 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IïycTb B&îOHpan HâflJie^ajuHM oöpaaoM KOožcpHn.HeHTH ^f ^ k BbraKïaH as 0T06paH6HHH (^x J2 KOrpaHHICr uCß , MU M0H8M ßOÖKTBCH CJieflVBHpÐC orparoraeHHÄ Ha KOMnoHeHTyf^Jjj: X IÛ \ B cjiywae H. = 3, bh^htoh H3f^»i$onojiHMTejiMìo Korpama KOTopye He bjhmdt Ha KOMnoHeHTy Œj* , mh HKeeM en. e \ „fi r>. h f-> ^*ì ^ « CTaBJIHH b (2.3): ^'4?a ; £~e3,4 ; C~ Z-%/, , noj^aei«, ^vo 0 Jjt>A>4 _. ß li^'h^ ^0> a npK fi £ 5 ks noscT-aKOBOs: a-£>.;?? b'.ei^j c*z-w'> 's^"v n"4 ' BHT8Ka8T' qT0/ -; -ü- Äajiee, noflCTaBJian b (2.3)': a-éU , & = £,-> 5 ^-cU„; £•"" J2 4 »^3^ MH npHKO^KM K pâBeHCTBaM:/ ,t SUJ cj.z.cz,3> a nOACTaHOBKH Ö-=^s ;D=6^;é2^ ^ C = e2^ ,C^ nCKaaKBSDT, tîTO nWft rs 9 h p \ ~{\ (dm* P^?0 -^~^. 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ÏII3-III 7. ?ib*. 14. likpmoB A.K. 0 cboöoähhx KOJimax ile. - Mt. co, , t.45 fis53 WZ, c, 113-122. 15. ÜkpffloB A.M. HöKOTOpHe aj^opiï^ii^eo:::;; zpc-cr.:..^ _-._;. ^„_ lïï. - ChÓ. MaT. syp.5 t.3 flSc2J, 12s e. 2S2-29S. 16. HaroH iL 0 flecfopMaisaHX HEJiLiioTeHîHKx rpasysposäKHEX ajiradr Ih. - becTH. Mock. yH-ïâ. Ma^eM., «exan., i^bi, .D4.. c. 50-54, 17. IlaroH Jl. rpaj^HpoBaHHHe £e$opMamiH ho^tk cboöoäekx adirsi*., Jin. - Béera. Mock. yn-Ta. Ma?eM., Mexan., I9823 ^55 ¦ c. 66-70. ï - -} - > ,JIL r Irr-.- J ^ Ha npasax pyEonucM imrOH JtymaH j^tOEMAlA TEi^itPOMHIfflX ÂŒEEP IM |0i,(£.06 - MaîeMaTffqscEEH moveer« J aireôpa a Teopui weeji| ABTOEEtEPÂT jpcoepïamm me coiCKaïae jrqsHoi CTenens KamroaTa $M3iKo-aaTeMaTffqecKHx mbjb, H3fìATEHbCTBO MOCKOBCKOfO YHWBEPCMTETA-1982 PaóoTa BHïïOjraeHa Ha Katfeßpe BHcmeiî ajireöpu MexaiiHKo-MaTeMaTH^iecKoro $aKyjiLTeTa MocKoBCKoro rocyjiapcTBeHHoro ymiBepcHTeTa hm. M.B.JIoMOHocoBa. HayHHHf pyKoBojoiTejiL - iuieH-KoppeciioHjieHT AH CGGP, npoffîeccop A.M.KoCTpHKHH. (ypmmajihHue onnoneHTH: jtOKTop (îHSîîKo-MaTeHaTiraecKHX Hayic M.M.îIocthiîkob; KaiiHH^aT $H3HKo-MaTe?viaTHxiecKiix nayK A.C.'Jl'&yìAajivun>jia.eB. Bejiyiiiee npe^npanTne - KasaiicKim rocyziapcTBeHHHË yramepcMTeT um. yjihHKOBa (leHHHa) . /? /9 /6'~- 3ainüTa coCTOHTCH "_j»fy'_/j____________,_ 1982 r. b " /;//' nac. na1 3aceji;aHKH cneiiHajffl3KpoBaHHoro GoBeTa )ê2 no MaTewaTHKe npn MD' no ajnpecy: DiaBHoe 3#aHEe Wi , MocKBa, lemmcKne ropH. C jniccepTaEpaeft mosho 03HaK0MHTLCH b HaynHOjì onóJinoTeKe hm. A.iVl.ropLKoro MH . ABTOpe&epaT pa3ocjiaH " **7'/.'-$¦ 1982 r ternit ceKpeTapB cnemi^iH3iipoBaHHoro GoBeTa itò no MaTewaTHKe npn IATI/ KaHjtfmaT 4H3HKo-MaTeMaTHnecKiix nayK B.H. ^yóapHKOB OBiUAÜ XAPAKTEPMCTMK PABOTH AKTyajIBHOCTB TeMH. ilOKRTHe Äe$OpMaiHM B METeMaTHKe H K 6*0, ¦••• Gerstenhaber M. On the deformation of rings and algebras. -Ann. of Math., Ser^ v.79 (1964), N1 , p.59-103. 2. PyßäKOB A.H. ße&opMaiBiM npocTHX auireóp Ih. - MSB. AH CCCP, cep. MaT., T.35 (l97l) , JS5, C.III3-III9. 3- Piper W.S. Algebras of matrices under deformation. - J. of Diff. Geom., v.5 (1971), N3-4, p.437-449. t-/700 2 nojiy^ieHHHe K.C.Jl%yMa.mjrbR&eBìm fi], nosBoJOura eMy yTo^KSTL H3BecT-ny» rHnoT63y A.M.KocTpHKHHa h M. P. UIa$apeBirqa ("2] 0 KJiaccH^HKaiiHH npoCTHX KOHe^iHOMepHHX ajireóp JIh h&r ajireópairqecKH: 3aMKHyTHM no-jteM xapaKTepHCTZKH P > 5. BojiLfflof HHTepec npe#CTaBJiH»T ßerfopMamm HmiLnoTeHTHHx ajireóp Ih. Xopomo H3BecTHo £.3j , ^0 KJiaccH H30Mop$H3Ma muEbnoTeHTHHx Û ajireóp JIh dnMKCMpoBaHHof pasMepHoCTH aajx nojieM ä onpe^ejraoTCH naóopaMH "HenpeptffiHHx" napaMeTpoB - ajieMeHTOB iiojlh ß . Ornano oómasi cHTyanzH isy^ena BectMa cjiaóo, h 3ajra^y 0 pa3yiviH0M oiiHcaHHH HHJiLnoTeHTHHx ajrreóp JIh Tpyjmo Rame c$opMyjia:poBaTL b KoppeKTHofi $opMe. EcTecTBeHHo onmjia.Th, tto ÄerfopMamiH, sœjuwimecn t&kkmr me nHBapiaHTaMH ajireóp, Kan asTOMopcteMH kjem M^éepemmgoBaEVLR, mg-ryT CHrpaTB 3HaHETejTBHyro pojiB b Bonpoce o KJiaccH$HKaiiHH moi^no-TeHTHHX ajireóp Ih. b ä£nccepTaiiKK npejyiaraeTCH H3yqaT& Te ,ne$op-Maura muTBnoTeHTHHx h ÄPyrHx rpajiyjipoBaHHHX ajireóp Ih, KOTopue xopomo corjiacysDTCH e hx rpa^yupoBamioH CTpyKTypofi. UejcL ^paóoTH : roy^emie rpajiyapoBaHKHx fleœopMaiiHH HeKOTopux ecTecTBeHBux KjraccoB HajitnoTeHTHMx ajrreóp JIm, a TaKKe KJiacca no^TH CBoóojiHHX ajireóp Ih. 3to 3Ha-qnT, •qTo jum BmneyKa3aHHHx .ajrreóp Ih L najio peuiHTL cjiejryioiUHe me 3a,naHH: { 1, 1. AKyMajyurbflaeB A.C. OTHocHTejitHHe KoroMojiorHH h flecfopMarnoi ajireóp Ih KapTanoBCKÄS THn.oB.-JIokji.AH CCCP, t . 2b7(l98l)iê5,c .1044-1048. 2. Kostrikin A.I. Variations modulaires sur un theme de Cartan.-Actes Congres intern, math., T.1 (1970), p.285-292. 3. BypóaKH H. TpynriH h ajireópH Ih. Ajireópn Im, CBoöo,nHLie ajrreópH Im, rpyniiH Ih. - M., "Map" 1976. 3 I. BtrascjiHTB BTopyiQ rpyraiy rpa#yHpoBaHHHx KoroMcmorai K (L^L) , to ecTL HafiTH Bee jioKajiBHHe rpajayMpoBaHHHe Äe$opMauHH. li. HccjiejioBaTL Bonpoc o npoÄOJDsaeMocTM: nojry^eHHHX rpaayipoBan-hhx 2-kohhkjiob no rjioóajiBHHx ^eftopMaiiaM ajireópH L • M§I31!§5.J!2151S§-ßce ocHoBHHe peayjiLTara jyaccepTaiira: kbjimjtch hobhmk. Mx MoxcHo paajiejiHTL, Ha Tps rpynrm. 1. Qóme pesyjiLTaTH o rpa^yupoBaiiHiDc AeafopMaimnx. a) Bse^eiio noHHrae rpajiynposaHHoi jreéopMainoï, ôj Äoicasai-ia saMimyTocTB KJiacca rpa,nyspoBaHHHx ^ecpopMartsE OTHocHTejiBHo oneparooä npoAOJiïceHîffl. b) ripuse^eH npMMep HenpoßOJDsaeMot jioKajr&Hof rpa^yiiposaiiHoi Äeq'opMaiiHH. 2. rpa¥nyKpoBaHHH.e..ii;e4opMaiïHz '.H&Ji&noTeHTHHx ajireóp Ih. a) noji-HocTM) Hsy^gHH rpyim;H jioKanLHHX rpajiyHpgBaHHKz ,u;eœop-MausS H2jn>noTeHTHHX HacìeI npocTHX ajrreóp Ih KiaccE^ecKax cepat OTHocHTejiBHo ss ecTecTBeHHoS rpajoysposKS. ó/ ÄoKaaaHo, ^to nojiy^eHHHe rpajryHpoBaHHHe 2-eqecekjih aonyc-KaioT TpaBMajiLHHe npoÄOJixemoi. b) PemeH Bonpoc o cnenaajiroamiM napawieTpa juih rpajrynpoBaHHHX ÄetJ-opManat yKa3aHHHX ajrreóp. 3. rpa^yHpoBaHHHe se^opMaiias jdboooähhx h ho^th cbqooähhx ajrreóp Ih. a) JioicasaHo, hto y cboóo;;hoì h cboóo^hoI HSJibnoTeHTHoi anreóp Im HeT HeTpHBaajr&HHX rpajiyHpoBaHHHX Äe$opMaixsi. ó) Oimcamj Bce rpajoyHpoBaHHHe ßecfapMaura ncmis cboöoähhx ajrreóp Ih. rJpHjio^eHBH. JlMCcepTamn hocht TeopeTireecKirg xapaKTep. PesyjiL-TaTH ÄHGcepTarmH MoryT HatTH npMMeHeioie b TeopHH KoJien, Teopua KoroMojioraf ajrreóp h Mojcyjiefi, h b saj^a^e oimcaHHH KnaccoB k3o- j Mop^3Ma HMJiBnoTeHTHHX ajireóp jfc tacnpoBaHHož paSMepHOCTK. AnpoóaigiH. PesyjiBTara OTccepTamffl soioiaÄHBajmcB Ha KOH$e-peHiwH mmoähx yneHHx MTC , «a ajireópa^ecKzx ceMznapax Mockob-CKoro ymfBepcKTeTa H b HncTHTyTe MaTeMaTKKz * Mexamnüi AH Ka3CCP njrÓjHocara^ CoÄepiamie anccepTamni onyÓJuiKOBaHo b paóoiax, cruicoK kotophx npuBeaeH b KOHne aBTope$epaTa. 0^ÊM.M_9ïEÏKTïEa:EaooTK. AHCcepTaimn comprar 92 CTpamin^ H COCTOHT m BBe^HHü K TPex VJLQB. K ^Cjiy OCHOBHHX OTHOCHTCH ruiaBH II h III. BirójiHorpafeH coaepaaiT I? HainuieHOBaHHÄ. pCHoBHHe pe3yjiLTaTH ÄHCcepTamui onyouHKoBaHH b scjie;nyranHX; padoTax: ! 1. 0 Äe$opManMüx muiBnoTeHTHHX rpajornpoBamaoc ajireóp In. - BecTH. MpcK.iyH-ïa. MaTejji., MexaH., I98I, M, c.50-54. / ! ( \ 2. TpajiyjïpoBàHHHe .necfcopMansni ïïohth cboÖoähhx ajire^ Ih. - BejcTH. . MbcK. yn-Ta. MaTeiL MexaH., 1982, 'Jé5 C.6G-70'}- ¦i\ 3 3axjmveme BHpasaK rjiyooryaa ÖJiaroÄapHocTB Hay^raoMy pyicoBo-OTTejœ, ^JieHy-KoppecijoHÄeHTy AHCCCP, npo^eccopy A.iJ.KocTpmaœy 3a nocTOHHHoe BHHMamie h oojrBuiyc noMomB BpaöoTe Ha^succepTaiiHež. rioan. k neMaTH 5/W/-& ' . *¦ ^ ByM. tan. Ni H3. n. ji. OtÏD Ym.-h3a. a #, » * 3aKa3 THpa* H3A-BO MocKOBCKoro yHHsepcHTeTa. MocKBa, K-9. y*. TepneHa, 5/7. ; THnorpacpHH H3ä-b» MrY, Mociosa, /IeHropbi