Крайние модули над алгеброй Вейля An

Класифікуються деякі класи простих модулів над узагальненими алгебрами Вейля (крайні, поліноміальні, з сильним D-скрутом). Для цих алгебр доводяться аналоги теорем Бернштейна і Стаффорда....

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Datum:1993
1. Verfasser: Бавула, В.В.
Format: Artikel
Sprache:Russian
Veröffentlicht: Інститут математики НАН України 1993
Schriftenreihe:Український математичний журнал
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Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/164584
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Крайние модули над алгеброй Вейля An / В.В. Бавула // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1187–1197. — Бібліогр.: 12 назв. — рос.

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spelling irk-123456789-1645842020-02-11T01:27:45Z Крайние модули над алгеброй Вейля An Бавула, В.В. Статті Класифікуються деякі класи простих модулів над узагальненими алгебрами Вейля (крайні, поліноміальні, з сильним D-скрутом). Для цих алгебр доводяться аналоги теорем Бернштейна і Стаффорда. Certain classes of simple modules over generalized Weyl algebras (extreme modules, polynomial modules, and modules with strong D-torsion) are classified. For these algebras, analogs of the Bernshtein and Stafford theorems are proved. 1993 Article Крайние модули над алгеброй Вейля An / В.В. Бавула // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1187–1197. — Бібліогр.: 12 назв. — рос. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164584 512.552.8 ru Український математичний журнал Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language Russian
topic Статті
Статті
spellingShingle Статті
Статті
Бавула, В.В.
Крайние модули над алгеброй Вейля An
Український математичний журнал
description Класифікуються деякі класи простих модулів над узагальненими алгебрами Вейля (крайні, поліноміальні, з сильним D-скрутом). Для цих алгебр доводяться аналоги теорем Бернштейна і Стаффорда.
format Article
author Бавула, В.В.
author_facet Бавула, В.В.
author_sort Бавула, В.В.
title Крайние модули над алгеброй Вейля An
title_short Крайние модули над алгеброй Вейля An
title_full Крайние модули над алгеброй Вейля An
title_fullStr Крайние модули над алгеброй Вейля An
title_full_unstemmed Крайние модули над алгеброй Вейля An
title_sort крайние модули над алгеброй вейля an
publisher Інститут математики НАН України
publishDate 1993
topic_facet Статті
url http://dspace.nbuv.gov.ua/handle/123456789/164584
citation_txt Крайние модули над алгеброй Вейля An / В.В. Бавула // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1187–1197. — Бібліогр.: 12 назв. — рос.
series Український математичний журнал
work_keys_str_mv AT bavulavv krajniemodulinadalgebrojvejlâan
first_indexed 2025-07-14T17:12:31Z
last_indexed 2025-07-14T17:12:31Z
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fulltext Y,UK 512.552.8 B. B . .6aeyJia, KaHJI. <pH3.-MaT. ttayK (KHes. ytt-T) KPAHHUE MO,[l;Y JIU HA,U AJirEEPOH BEHJIH An Certain classes of simple modules over generalized Wey! algebras (extreme, polynomial, and with strong D-torsion) are classified. For these algebras, the analogs of the Bernstein theorem and the Stafford theorem are proved. KnacmpiKYIOTbC.11 11e.11Ki KJiaCH npocTHX MOAYJiiB Hall y3araJibHeHHMH anre6paMH BeltJI.11 (KpaAHi, rroJiiHOMiaJiblli, 3 CHJlbHHM D -CKPYTOM). ,Un.11 QHX anre6p /lOB0/1.l!TbC.II aHaJIOrH TeopeM oepHWTeltHa i CTaqxpop11a. B HaCTOHlll.etl: CTan,e H3BeCTHbie pe3yJibTaTbl, cnpaBeAJIHBbie AJIH anre6p Betl:JIH (He­ paBeHCTBO H qmJibTpaQHH Eepmirrel-tua, <pyHKQHH rHJib6epra, pa3MepHOCTb H KpaT­ HOCTb KOHequonOpo)KAeHHbIX MOAyJiel-t, 6epnwTel-tHOBCKHl-t KJiacc MOAyJiel-t, Teope­ Mbl CTacpcpopii.a. fJIOOaJibHaH pa3MepHOCTb H pa3MepHOCTb KpynJIH), nepeHOCHTCH Ha OAHH Knacc o6o6mem-1h1x anre6p Bel-tJIH (onpeii.eneHHe CM. B n. 2), coii.ep)KamHx anre6pb1 Bel-tJIH A,,, a TaK)Ke npHBOAHTCH HOBbie pe3yJihTaTbl (npocTOTa TeH30pHbIX npOH3Bel).eHHl-t npoCTbfX MOAyJiel-t, KJiaCCH(pHKaQHH HeKOTOpbIX KJiaCCOB npOCTbIX MOAyJiel-t, HanpHMep, IlOJIHHOMHaJibHblX MOAyJiel-t, C CHJlbHblM D-KpyqeuHeM (no­ CJieAHHe HBJIHIOTCH aHaJIOfOM BeCOBblX MOAyJiel-t npOCTbIX anre6p JlH), KJiaCCH<pH­ Kaum1. Kpal-tHHX A,,-MOAyJiel-t, TeopeMa O KpaTHOCTH npocTbIX A,.-MOAyJiel-t). 1. OnpeneJieuue o6o6meuuott aJire6p1,1 BettJI.11 (1 - 3]. TiycTh D- neKo- TOpoe KOJlbQO, <J = (a I' ... , a,.) - ceMel-tCTBO KOMMYTHpylOIJ.(HX aBTOMOpq)H3MOB KOJibQa D, T. e. a,aj =O"/J;, a= (al, ... , a,,) - MHO)KeCTBO neuyneBbIX 3JieMeH­ TOB uenrpa Z(D) KOJihQa D, npHqeM a,(a) = aj AJIH nlo6hrx i * j. 0606U{e1-11-1ou a1ue6pou Beilit.fl A= D(a, a) (coKpamenno OAB) cmene1-1u n c 6a3uCHbtM KOJibQOM D 6yii.eM Ha3bIBaTb KOJibQO, KOTOpoe nonyqaeTCH npHCoeAHHeHHeM K D 2n CHMBOJIOB X( , ... ,X;, Xj"" , ... ,X;;-, YAOBJiernopHIOIJ.(HX CJieAylOIJ.(HM Oilpel).eJIH­ lOlll.HM COOTHOWeHH.ll.M: x:-x+ = a- x+x:- = a.ra -) x±a = cr±1(a)X± Va e D, I I I' I I I~ I ' I J I [X;, x7] = [Xt, Xj] = [Xt, x7] = 0 Vi*f MHO)KeCTBa a H a 6yii.eM Ha3blBaTb COOTBeTCTBeHHO 011peoe1t.fl/OU{UMU 3/teMeH­ maMU H a8moMoptpU3J.taMu o6o6mettHol-t anre6pb1 Bel-tJIH A. EcnH D - neTepooo KOJibQO (COOTBeTCTBeHHO 6e3 AeJIHTeJiel-t HyJIH), TO A - HeTepOBO KOJlbQO (COOT­ seTCTBeHHO 6e3 ii.eJIHTeJiel-t HyJIH) . .[lnH nio6oro uenoqHcnenHoro BeKTopa k = (k1, ... , k,,) e ~" onpeaenHeM vk = vk/1) ... vk_{n), rae AJIH Ka)Kii.oro 1 ~ i ~ n H uenoro m ~ 0 AJIH yao6cTsa 3anHCH nonaraeM v±,,.(i) = (X;)"', v0(i) = 1. B cnyqae n = 1 BMecTo v"'(l) nHWeM V nr HenocpeACTBeHHO H3 onpeaeJIHIOIJ.(HX COOTHOllJeHHl-t cneayer, qTO A = $ Ak ~"-rpaayHposaHHaH anre6pa (A..Ae c Ak+e \;;/ k, e e ~"), rii.e Ak = Dvk. 2. OcHOBHOii 06'hCKT. B aaJibHel-tweM, eCJIH He OfOBOpeHO npOTHBHOe, D = K[H I• ... , H,.] - KOJibQO MHOroqJieHOB OT /l nepeMeHHblX C K03cp<pHQHeH­ TaMH B anre6paHqecKH 3aMKHyTOM none K xapaKTepHCTHKH uynh, cr;(H) = Hj- Su (1 ~ i, j ~ n, o,1 - aenhra-<pyHKUHH KpoHeKepa), MHoroqneHbI a; e K[H;] HMeIOT nOJIO)KHTeJibHYIO cTeneHh. TaKHM o6pa3oM, OAB A-HeTepoBa, 6e3 aen0Tenet1: © B. B. EABY JIA, 1993 ISSN 0041-6053 . YKp. Ma/II. )Kyp11., 1993, Ill. 45, N• 9 1187 1188 B. B. EABY JIA HYJUI, CJieAOBaTeJlbHO, no TeOpeMe roJIAH A HMeeT TeJIO '{aCTHbIX. A112e6poil Beu.11.11 An = An(K) CTenenH 11 HM noneM K na3hrnaeTc_g anre6pa c 211 o6pa3ylOW,HMH X1, . .. , Xn, 81, ... , 8n, YAOBJieTBOp_glOW,HMH COOTHOWeJ-Iffj(M [X;, Xjl = [8;, 8jl = [8,, Xjl = 0 npH i ic j, [8;, X,-] = 1. An 110.11.11emc11 OAB c 6a3uCHbtM KOllbl(OM D = K[H1, ... , H 11] u MHO)KecmooM onpeoe111110U{U.X 3.11.eMeHmoo {a;= H,-11 $; i $; 11} u aomoM.opcf>U3M06 { cr,-}. O'{eBH/lHO, TeH30puoe npoH3Be/leHHe o6o6w,eHHbIX anre6p BettJiil CH0Ba .)IBJlil­ eTC.)I OAB H npoTHBOnono.lKna_g anre6pa A0 "' A. 3. KJiaCcH<l>HK8QHJI (c TO'IHOCTLIO AO HCp33J10:>KHMblX 3JICMCHTOB CBKJIHAO­ BOro KOJILQa) npoCTblX A-MOAYJICH. IlpHBOAHMbie B llaHH0M nyHKTC pe3yJll,TaTbl j(BJIRIOTCR HeCJI0)KHbIM H O'{eBH/lHbIM o6o6w,eHHeM pe3yJibTaTOB pa6oTbl [31 Ha npoTR)KeHim 3Toro nynKTa 5l - OAB c-renenH I c nenyneBbIM onpeAeJIRIOll{HM 3JieMeHTOM a * 0 H 6a3HCHblM AeAeKHHAOBbIM KOJibQOM 'lJ, KOJlbQO '{aCTI-lbIX KOToporo o6o3Ha'{HM '-lepe3 k = S-1 'lJ, S = 'lJ - { 0}. IlpH 3TOM onpe1len.si:i:om1-1tt aBTOMOpqm3M <J YAOBJiernopileT ycnoemo: C ) <J"(P) °i' P ¥0 °i' 11 e ~ . P C 'lJ - npoCTOfi HACaJI. .[lJIR KpaTKOCTH 3anHCH IlOJIO.lKHM X = Xt H Y = X1. JlOKaJIH3aQHR 'B := 5-l 5I KOJibU.a 5l no S H30Mopqma KOJibU.Y K0CbIX 1mpanoBCKHX MHOrO'{JICH0B klX. x-1; <J], KOTOpoe ilBJiileTCil eBKJIH/l0BbIM (BbinOJIHilCTC.ll JieBbitt H npaBbltt anropHTM 1leJieum1 C OCTaTK0M) 0TII0CHTeJibH0 c:pyHKQHH ",nJIHHbI" /: /(aX"' + ... + 13X") = = 11 - m, a, 13 e k - - He uyJ1eeb1e, m < .. . < n. Ilyc-rb M - 51-MOJ~YJib, TOrAa B CHJIY ycJIOBH.ll Ope tor M:= {me MI sm = 0 AJiil HCKOTOporo S ES} 51BJiileTCjl Il0AMOAYJieM ( 'lJ-Kpy'{eflHjl) , CJieAOBaTCJibHO, Ka.lKAhitt npocTott 5l-Mo,ny11b M 111-16o c 'lJ-Kpy'{CHHeM (M = tor M) n1-16o 6e3 '1J­ Kpy'{e1-11-1_g (tor M = 0). Ha MHO.lKeCTBe Henyn eBbIX npOCTbIX HACaJIOB P:= Specm 'lJ cno6oAHO Aettc-rny­ eT rpynna ~ = <cr> , T. e. op6HTa Ka.lKAOtt TO'{KH H30MOpc:pna ~ (B CHJIY yCJIOBHil C). Ilo3TOMY ac e ecTeCTBeHHble Il0HilTH.H (nop_gA0K, 0TpC30K, nonyocb H /lp.), HC- n0Jib3yeMble llJI51 ~ . 6y.neM (6e3 ornnopoK) npHMeHflb JlJl51 mo6ott op6HTbl. Ha p 3a/laAHM 0HIOllle1me 3 KBHBaJieI-rH-I0CTH: 1) ~ (l ¢:::> p, (l Jie)KaT Ha 0/lH0tt op61-ne H 111o6ott 1-1.nean H3 o-rpe3Ka [P , Q) ne co.nep:>KHT onpe.nenil1ow,ero 3JICMCHTa a. T. e. 06JiaCTb 3KBHBaJieI-ITHOCTH - JIH60 op6HTa, Ka.lK/lbltt 3JieMeHT KOT0pott He co.nep:>KHT a (TaKHe 0p6HTbl I-Ia3bIBalOTCil HeBblp0:>K/leHHbIMH, B np0THBH0M CJJy'{ae - BblpO)K/leHHbIMH), JIH6o MH0:>KeCTB0 BH/la (--00 , P], (P, q], (P, 00), r.ne P, (l 3 a H HH 0/lHJ-l .npyrott 3JieMeHT H3 3THX MH0)KeCTB a ue co.nep)Kfff. Bbip0)K/lelUiblX op6HT (C0OTBeTCTBel-11-IO KOHC'{J-lblX o6nacTett 3KBHBaJieHTHOCTH) K0He'{HOC '{HCJI0. IlpcAJIO:>KCHHC 1. J].1111 11,060w 11eoow (npaoozo) uoea11a J * 0 5l -Mooy11b 5l I J u.Meem Kone1.tny,o 01iuny, c11eooeame11bHO, pa3MepHocmb Kpy111111 a11ze6pbt 5l pa01ta K - dim 5'I. = l . IlyCTb A - neK0T0poe K0JlbU.0 H Q - HCK0T0poe CB0ACTB0 np0CTbIX A-M0- .nynett, m-map1rn1rn-10e 0THOCHTeJibJ-10 H30MOpQ)H3Ma M0/lYJiett. Tor.na '{epe3 A.(Q) 0003J-la'{HM C0B0KynH0CTb KJIUCC0B H30MOpQ)H3Ma npOCTblX A-MO/lYJiett, HMeIOI.QHX CBOfiCTBO Q. Tcopcr.1a 1 (KJrnccu<J>1-1KaQHJ1 npocT1>1x 51-MOJlYJICi'I c '1J-Kpyqe11uer.1). Omo- 6pG.JKeHue Specm ('lJ) /~➔ 5l ('1J-Kpy1.tenue), r ➔ [L(r)] 1101111emc11 611eK1111eu, ioe 1) ec1111 r - He6bip0)1(0eHH(IJI op6uma, mo L(r) = 5'1./ 5'1.P, zoe per; ISSN 0041 -6053. YKp. Mam. )Kyp11 ., 1993, m. 45, N• 9 KPAiilUIB MOJ].Y JUI HAJJ. AJlrE6POit BEitJUI An 1189 2) eo,u r = (-oo, P], rn L(r) = 5t / (5tp + JIX); 3) ec11u f' = (<r"(P),P], mo L(r) = 5t / (Af" + 5tp + JIX); 3mo 6 mottHocmu Bee npocmble 5t-Mooy11u KOHettHoa 011uHbl /~(r)) =Ir I< 00 KQK 'lJ-MoOy11u (ux KOHettHOe 'tUC/10 pa6HO K0/1Uttecm6y KOHettHblX 0611acmea 3K6U- 6a/leHmHOcmu); 4) ec11u f' = (P, +00), mo L(f') = 5t/ (5t<1(P) + 5tY). B 'lJ BBeJJ.eM OTHOilleHHe <: a < 13, eCJIH p < Q AJUI JIIOObIX, npumv~nexall(HX OAHOA op60Te, npoCTbIX HAea.noa p H Q, COAepxamux COOTBeTCTBeHHO (l H 13. 3neMeHT b = V ml3m + ... + f3o, 13 j e! 'lJ, m $ i $ 0, Ha3bIBaeTCH HOpMa/lbHblM, eCJIH 130 < 13m 0 f3o < a. JleMMa 1. a.11.Jl .11,06ow b =v__,,.13_n,+ ... +13oejl 0.IIUHbl m >0, 13;e'lJ, m $ i $ 0, cyu{eCmByiom ueuy11e6ble a, 13 e 'lJ maKue, ttmo f3ba.-1 e 5t - uop­ Ma.llbHblu 311eMeum. flpu 3moM oocmamottHO nollOJKumb a. = TT-s:SISO e1i(l3o), 13 = TT-m-s:Sj:Sl oi(l3o), zi)e S -iJocmamOttHO 60/lblUOe (llelKO 6bl'tUC/IUMOe) HamypallbHOe 'tUCIIO. TeopeMa 2 (KnaccH<J>nKaun.11 npocT1>1x 5t-MOA)'Jieit 6e3 'lJ.Kpyqeun.11). 1. KaH011utteCKoe omo6paJKe11ue S- 1: A (6e3 '1J-Kpytte1tu.R) ➔ B, [ M] ➔ ➔ [S-1M] R6/1Remc.R 6ueKt{uea c 06pam1a,1M Soc: [N] ➔ [SocAN], zoe SocAN - l{OKO/lb MOOYIIJl AN. 2. 5t -MoiJy11b M -npocmou 6e3 'lJ-KpytteHuJl ~ M"' (5ta + 'Bb) I '13b, 20 e b e jl - 11ep03110JKUMblU 6 'J3 (u 001103/IO'tHO C mott/lOCntb/0 00 ll0006U.Jl onpei:Je­ /le/lftblti) 3,11e;,.1e11m maKoti, ttmo j3ba-1 - IIOpMOllbHblU Oil.fl fleKomopblX 13 ' a e! ']J (HanpuMep, KOK 6 /leMMe 1). flpuMepbl a11ze6p jl. 1. ']J = K[H] - KOJibQO MHOrQ'IJieHOB HaA noneM K xapaKTep0cT0K0 0, <1(H) = If - 1, a :;; 0. B qacTHOCTH, np0 a = H nonyqaeTCH a.nre6pa BeAnH A 1. 2. TiycTb U = Usl (2) - y110aepca.nhuaH o6epThrna10maH a.nre6pa a.nre6pb1 JlH sl (2) = (X, Y, /-I I [H, X] = X, [If, Y] = -Y, [X, Y] = 2/-/) uaA noneM K , C = = H(H + 1) + YX - 3JieMenT Ka30M0pa. 0'-leant{HO, U"' K[H, C](<1, a= C­ -H(H + 1)), rAe aaT0Mopq.>03M <1 6a30cnoro KOJibQa K[/-1, C] onpet1,enHeTcH cneAyIOmHM o6pa3oM: cr(H) = H - 1, cr(C) = C. Tort1-a AJIH n!o6oro A e K q.>aKTOp­ anre6pa U(A.) := U I U(C -A) H30MOpq.>ua OAB 03 np0Mepa I K[H](<1, a = = '),,, - H(fl + 1)) [4] . 3. PaccMOTpHM K-a.nre6py A(b), KOTOpaH HBJUieTCH t1-eq.>opMaQ0eA Usl (2) 0 nopo)l(t1,aeTCH 6yKBaMH X, Y, fl, YAOBJieTBOpHIOl.l{HMH CJieAyIOl.l{HM COOTHOWeHHHM: [Ii, X] = X, [H, Y] = -Y, [X, Y] = b e K[H] - npoH3BOJibHbIA ne11yneaoA MHOro'IJieH. Tort1-a [2] A(b)"' K[/-1, C](cr, a= C - a), rite cr TaKoe, KaK a np0Mepe 2, a e K[H] YAOBJieTBOpHeT ypaauenmo a - cr(a) = b. 0'-leBHAHO, 1~e1rrp a.11re6pb1 A(b) paaen K[C]. Anai1or0,1110 npeAbIJJ.yIJ.J,eMy np0Mepy AJIH mo6oro A e K q.>aKTOp-anre6pa A(b, A.):= A(b) / A(b)(C - A) H30MOpq.>ua OAB 03 npHMepa I c onpeAeJIHIOl.l{HM 3JieMeI-ITOM paBI-IblM /\. - a. 4. ']J = K[H, (fl - µ(l - A,)- 1)- 1], cr(H) = J..H + µ, rJJ.e A.~ 0 e K He HBJUteTc>i KOpneM 1-13 etJ,HHliQbl, µ e K. ISSN 0041 -6053. Y,.;p . Mal/I . xyp1t., 1993 , 111 . 45, N" 9 1190 B. B. EABY JlA 4. KJJaccH<lmKaQH.11 npocT1,1x A-MOtzyJJeit c CHJJhHLIM D -Kpy'lenHeM, KpHTepHit npocTOTLI, rJJOOBJlhHB.11 p83MCpHOCTL H pa3MCpHOCTh KpyJJJJ.11 8J1re- 6p1,1 A. TiycTb .91.; := 'D;(<J;, a;) - OAB CTene1m 1 c 6a3ucnh1M KOJlbQOM 'D; = = K[H;], 1::; i::; 11. Tor.n.a A = ®;~1 .91.; H na MHO)KeCTBe MaKCHMa.TlbHhIX n.n.ea.noB KOJlbQa D Specm D"' n;~l Specm 'D;"' K11 (rnopeMa rHJlb6eprn O KOpm1x) .11.ettCT· ByeT rpynna ~n = n;~I (<J;) H OnpeJJ.eJleHO OTHOUJeHHe 3KBHBaJielITHOCTH - KaK np.i1Moe npoH3BeJ].euue panee BBeJJ.eHHblX 3KBHBaJleHTHOCTett H3 Specm 'D;. A-MO· .ll.YJlb M Ha3hIBaeTC.II MOJJ.yJ1eM C CU/lbl-lblM D-Kpy'lem,eM (COKpaw:eHHO c.D-Kpyqe­ HHeM) JIH6o BeC08blM, ecJ1u .II.JI.II mo6oro I ::; i::; 11: M-MOJJ.YJlb c 'IJ,-KpyqeuueM. TeopeMa 3 (KJ18CCH4>HKaQHJI npoCTLIX A-MOtzyJJeit C CHJlhHLIM D-Kpy'ICHH• CM). Omo6paJKe1-1ue Specm D !-➔ A (c. D-Kpy'le1-1ue), r = n:1 r, ➔ [L(r) := := ®;'.:1L;(r;)] .RBA.Remc.R 6ueKl{11ei1, zoe L,{r;) maKue , KaK 8 meopo1e 1. B ttacm- 1-1ocmu, 3mo omo6paJKe1-1ue ocy111ecm811Rem 611eKtJUIO Me)Koy KOHel/HbLMu 061zacmRMU 3K8U8a11e1-1m1-1ocmu (KomopblX KOHe•moe l/UC/10) U npocmblMU KOflel/HOJ.1epHblMU A- MOOyll.RMU, dim L(l) = n:1 I r,. 1. lloKaJaTeJlbCTBO uenOCpeJJ.CTBeHHO CJ1e.11.yeT H3 TeopeMhl 1 H Toro O'leBHJJ.HOro cpaKTa, 'ITO JII06ott npOCTOtt A-MOJJ.YJib C CHJibHblM D -Kpy'leHHeM .IIBJl.lleTC.11 cpaKTop-MOJJ.yJ1eM MOJJ.yJ1.11 A I A(H 1 - A- 1, ... , H,. -A,,)"' ®1' .91.; I ~(H, -A,) .II.JI.II neKOTOpbIX A; e K. EcTeCTBeHHO BBOJJ.HTC.11 "0.ll.HOp0.IJ.Hbifi" 6a3HC, npeBpa11.J;a1011.J;Hfi f1.1) B rpaJJ.yH­ pOBaHHbltt A-MOJJ.YJlb. HenocpeJJ.CTBeHHO H3 pe3yJlbTaTOB [3], no11yqeHHbIX npH fl= I, CJ1e.11.yer TaKa.11 TeOpeMa. TeopeMa 4. 1. A11ze6pa A u;weem 11uwb Ko1-1ettHoe 'l11c110 oeycmopo111-1ux uoea/!08, KaJKOblU U3 KOmOpblX 001tOpooe1-1 onmocume/lb/10 zpaoyup08KU. 2. A11zepa A - npocma.11 ~ pa:mocmb ooyx pa311uttHblX KOpHei1 Ka)l(OOW MI-IOZOl//le/la a; He l,{elloe l/UC/10. 3. Ec11u a11ze6pa A npocma.11 u KaJKObli1 M1-1owtt11e1-1 a; ue uMeem KpamHblX Kop1-1eu, mo z1106a11bHaR pa3MepHOcmb gl. dim A = 11. 4. Pa:JMepHOCmb Kpy1111J1 a11ze6pb1 A pae1-1a K - dim A= 11. 5. AnaJJor uepaeencTea .6epuUITeilua, 4>yuKQH.11 rHJ11>6epTa, 4>HJ11>TpaQHH. Ha a.nre6pe A paccMorpuM cJ1e.11.yiow:tte <:pHJlbTpau;mi: B: B 111 = L{D;vj I 2I i I +s 1 111 I + ... + s,, Ii,. I ::;;m}, r.n.e { D;} - O6bl'IHa.11 cpttJ1bTpau;tt.11 no crenen.11M KOJlhQa MHorO1.J:J1enoB D, sk - creneHb (onpe.11.e- J1.1110w:ero) MHOrO'IJleHa ak, 1 ::; k ~ II, j = (Jp , J,.) e ~"; O'leBHJJ.HO, B 111 - KOHe'IHOMepuoe npOCT})aHCTBO, m ~ 0; l:: 1:,,, = 1:{ Dv j 11 h I + ... + U,. I ~ m} . rpa.11.y0poBanua.11 anre6pa gr8 (A) nopo)K.11.aerc.11 311 KOMMyr0py10W:HMH nepe- MeHHblMH X;, Y;, h;, KOTOpbie Y.ll.OBJleTBOp.llIOT COOTHOUieHH.IIM hf' = X,.Y;, 1 ::; i::; II. Ey.11.eM rOBOpHTb, 'ITO <PYHKU:lf.11 x: ~+ ➔ 1R HMeeT llOIIWIOMUallbl-lblll pocr, eCJIH JJ.J1.11 neKoroporo narypan1,11oro d cyw:ecrnyer ue1zyJ1eBott npe.11.eJ1 A = lim x<t) / r1. ' t➔oo llJ1.11 (pHJ1brpaQHH B auanorn'IHO CJIY'laIO anre6pb1 BettJ1.11 (5, 6] ua Ka)KJJ.OM KOHe'l- uonopo)KJJ.eHHOM A-MOJJ.yJ1e M cyw:ecrnyer xopouwR <:pHJ1bTpau;tt.11 r = {r;} H ISSN 0041-6053. YKp. lo<0/11. ;,cypH., 1993. 111. 45, N• 9 KP At1HHE MO)lY JUI HA)l AJirEEPOt1 BEi1JI.SI A. 1191 <PYHKQHR n0JIHH0MHaJibH0ro poem (KOTopy10 6yaeM Ha3bIBaTb <PYHKU,HeA rHJib- 6eprn M0AYJIR M) : x(M, r,t) = :t+ ... , esd, de"&.+' s= HOK(s1, . .. ,Sn,2), AJIJI K0T0poA dim r, = X(M, r, J) npH j >> 0, rae TpeMR T0'IKaMH 0003Ha'leHbl 'IJieHbl "MeHbWero nopRAKa" . TipH 3T0M K0HCTaHTbl d = d(M) H e = e(M) 6yaeM Ha3bIBaTb COOTBeTCTBeHH0 pa3AtepHOCmblO H KpanlHOCmblO M0AYJIR M . OHH He 3aBHCRT OT q>HJibTpau;HH r H AJIR HHX cnpaseAJilfBbl see CB0ACTBa, 4T0 H B cnyqae a.nre6pbl BeAJIR. B 'laCTHOCTH, AJIR n106oA T0'IH0A nocnetJ.oBaTeJibH0CTH A-MoayneA O ➔ M ➔ N ➔ L ➔ 0, d(N) = max {d(M), d(L)} H eCJIH d(M) = d(L), TO e(N) = e(N) + e(L). Pa3MepHocnib Te11bcpcmoa - K11pu111108a OAB A pasHa GK -dim A= 2n, noCK0JibKY dim Bi= fl" I (2n)!s 1 ... s,, + ... . KaK H s cnyqae aJire6phI BeAJIR [7], Bhin0JIHeH0 nepaseHCTB0 EepHWTeAHa. Tcope~1a 5. llycmb A - 11pocmaJ1 a11ze6pa u M ,t,. 0 - KONe'lnonopoJKoem1b1il A-MOO)'/lb. Tozoa d(M) ~ 11. ECJIH A - ne npocraR, TO cpeAH TeH30pHbIX C0MH0)KHTeJieA Yl; HMeeTCR 1 ~ t ~ n 1-1e npoCThIX anre6p H s npeAhIAymeA reopeMe cnpaseamrno 1-1epase1-1crso d(M) ~ 11-t. 6. Ecp111UTCHHOBCKHH KJl3CC A-M0/.1.YJJCA. JleMMa 2. llycmb M II N - Kone'l11011opo)l(oennbte Mooy11u Hao 0606UfeN- Nb1,,111 a112e6paM11 Bet/,1J1 A' 11 A" coomoemcmoem,o . Tozoa , o'l eouoHo, meN30pHOe 11pOU38eoem,e MOoy11ei1 M ® N Jl(J /lRemcJI Mooy11eM NGO OAB A'® A" U d(M@ N) =d(M) + d(N), e(M ® N) = e(M)e(N). Knac K0He4H0n0pO)KtJ.emtblX HeHyJieBhIX A-M0/.1.yJieA, HMel0ll(HX MHHHMaJibH0 B03M0)KIIYIO pa3MepH0CTh (11 B c11y4ae A,,), 6yi:.eM Ha3bIBaTb 6epmumei1N08CKUM H o6031-1a4aTb '13 = 'B(A). O4eBH/.I.HO, 6ep11wTetl110BCKHt\ KJJaCC RBJIReTCR n0JIH0tl n0, lK,1Terop11eI1 B KaTeropHH A-M0Aynetl, Ka)KAbIA M0AYJib H3 K0T0poro HMeeT K0He4HYIO ammy. ITyCTb '13', '13'' H '13 - 6epHillTCAl·I0BCKHe KJiaCCbl M0AyJiert /.1.JIR a.nre6p A', A" H A = A' ® A" cooTBeTCTBeImo. Tonla B cIrny npeAbitJ.ymeA neMMbI '13' ® '13'' c '13. B YaCTH0CTH , 113 BKJ110'-teIuD1 A I c 'B(_A 1) cnei:.yeT A 1 ® ... ... ®A I c 'B(A,.), T. e. Te113op11oe npoI13seae1me II npoCTbIX A 1-MoAynert RBJIR­ eTCR A,.-M0/WJieM K0I·IeYH0tl: J~JIHl-lbl H pa3MepH0CTH 11. 7. KnaccH<t>HKailH.11 Kpati11Hx A,,-~10i:.yneti. MoAynf, H3 6ept-1illTetl:noscKoro KJiacca, HMe!0ll(HA MHIIHMaJlbH0 l303M0>KHYI0 KpanIOCTb, 6yJ{eM Ha3bll3fffbApaiiNIIM. Kpati1-11-1tl: M0JlYJlb scerw1 np0CT, '.3T0 8 11eK0T0p0M CMblCJle caMblA "MaJiel-lbKHti" npocT0H M0JlYJlb. Qye1mn_110. A,,-MOI\YJJb M-KpalhmA, eCJIH d(M) = 11 1-1 e(M) = 1. O603HalJHM Yepe3 1(,, C0B0KynH0CTb BCex Kpaihmx A,, -M0AYJieA. Tonia B CHJIY 11eMM1>12 1(, ® '1(,,,~~I(_,,+,,, JlJDI Bcex 11,111 e N . TTp1-1MepaM1-1 Kpaih11-1x A,,-MOJly11e1'i RBJHII0TCR W := A,, I A,,(d 1 . ... , d,,) = (K[x 1, .... x,, J. <1/F) = 8F I fo1 - lt)opMaJ11,11a}l np0H3B0/lll3}1, -~/.F) = -\jF - yM11o>Ke111-1e) 1-1 \/ := A,, I A,.(r 1, .... . r,,). TTyCTb Aut8(A,,) - rpy nna anTOM0pltn-13:-10 B a.111"e6p1,1 A,,. coxpall}II0LUHX qm111,­ -rpat.ll-ll0 Eep11wTe1'i1ta (crB,, = B,, VII e N ). 3Ta 1·p y nna ec-rec-r11e11111,IM o6pa30M 0T0>K/leCTBJl}leTC}l c mmrpy nnoii o6parnMbIX oneparopon npoc-rpa11c-rna \ 1 = <I . .r 1, .... .r,,, d1 .. ... d,,>. C0Xpall}ll()W.IIX K0C0Cl·lMMCTJ)l-l'-leCKYI0 (111,liJO)KJ\ellll)'IO) <t)OJJMY [11 , V] = 1/V - VII. ISSN 0041-6053. Y,-,, . -'"""-Ao:_\'f'// . . /'JY3 . I ll. .J5, N" ') 1192 B. B. oABY JIA TeopeMa 6. M-Kpat1111u1 A,,-.At0iJy11b ~ M =- •w (1100Kpy'lew1b1il Mooy,1b) 011R HeKonwpow 't e Auta(A,,). ?.Oe w = A,, I A,,01, ...• o,,). 8. TeH30p111,1e npoH3BCJ].CIIH.II npOCTl,IX MOJzy JICH. B J.J,aHI-IOM nyHKTe Ka)KJ].Oe KOJibUO 'lJ;, l :,; i:,; II, H '1J JIBJIJIIOTCJI KOMMyTaTHBHOti 061iacn,10 C orpam-1'-fel-lHblM yCJIOBHeM MHIIHMaJibHOCTH (T. e. JIIOOOti C06cTBeHltblti cpaKTOp-MO/lYJlb '1J HMeeT KOHe'-fHYIO D.JIHHy) 1-1 KOHe'-f110nOpO)KJ.J,e1111ott K-allre6poti nan. anre6pa1-1Y:eCKH 3aMKHYTbIM noneM K. Ka)KJ.J,blt:\ 3JJeMeHT K(HOporo HenOJ.J,BH)KCl·I OTHOCHTeJlbHO n.ettcrnu.H cr (cr(A.) = A VA e K). JleMMa 3. flycmb Jl. - OAB cme11e11 11 I c 6a311c1tb1M K011btfOM '1J u M - 11pocmm1 Jl.-MOO)'llb 6e3'1J-Kp)''le/-lllfl . TmiJa OllJl .1106ow m ct:- 0 e M cyuiecmoy­ em p e Jl. mai.;oi/, 'Imo Ker (PM)= Km. J(oKa3llllle1lbClll80. 04:eBH/1.HO, M =Jun .. Mp:= Jl./ Jl. n 13p 11.JIJI HeKOTOporo ttepa3llO:>KHMoro B 13 ::i11eMeura p e Jl.. pm= 0. Torna (1) - T04:HUJI noc11e11.0BaTeJ1bHOCTb Jl.-MOLIYJleti, npH4eM \Ip - M0/1.YJlb C V-KpyY:eHH­ eM, c11e11.oaaTe111,110, HomJl.(\lp, M) = 0. Tip11Me1rnx K (1) ¢YHKTOP Hom51(- , M), HMeeM K"" Hom Jl.(MP, M)"" Hom Jl.(Jl. I Jlp, M)"" Ker (pM). fleMMa 11.oKa:mua. TeopeMa 7. flycmb Jl.; - OAB cme11em1 1 c 6a3UCHbrM KO/lbt(OM 'lJ;, 1:,; i:,; I!. TOloa ~I 0 ... 0 ~II C (Jl.1 0 ... 0 Jl.,.)". Eo11 ee 11oopo6110, ll)'Cn!b M;, N; npocmbte Jl.;--'t0oy1111. Tozoa M =@ M; - npocmoii A = @ Jl.;-MOO)'llb u A­ ;1,ooy,111 0 M,- II 0 N; 1130MOJ)Cj}Nbl mozoa U nl0/1bKO mozoa, KOlOG Jl. ,--M00)'/111 M,­ ll N, l/30NOpcj)Nbl 0/lJl K(LJi,;OOlO i. J(oKa3ame1lbCl1180. llm1 11.0Ka3aTeJibCTBa npoCTOTbl A-MOIJ.YJIJI M D.0CTaT04:HO noKa3aTb, 4:TO M nopo)K11.aerc.H JIIOObIM u ct:- 0 e M. TioKa)KeM, 4:TO Au coaep)KHT Hetty 11eBoi-i "011.ttopo11.ub1lf' 3JJeMeHT 01-111.a z = z 1 0 ... 0 z 11 , r 11.e z ,- :t: 0 e M;. Tipeanono.tKHM, 4:TO 3TO TaK. Toraa Au c Az = @ Az; = M, cne11.0BaTeJihHO, M - npocTott Mon.ynh. C11y'lait 1. TI yen, M 1 -Moaynh 6e3 '1J1 -KpyY:eHH.H. Torn.a. KaK uecJIO)KHO BH/1.eTb, II MO:>KHO npeacTaBHTb B BH/1.e cyMMbl Lni ,, u = u-0 w -, u-e M 1, w e ®,M . =N, IJ JJ J _, (2) uanpHMep, m HeHyJieBbIX CJlaraeMblX TaK, '-lT06bI i) Ku; n Kuj = 0, eCJlH i ct:- j; ii) { w) - JIHHettuo He3aBHCHMbie 3JieMeHTbl. B c,rny neMMhJ 3 cymecTByeT p e Jl.1 raKott, 4:TO Ker (pM) = Ku 1• Toraa B 1 CH.11)' ycJIOBiitl i), ii ) 11 1 := pu = I.:' /Jll- 0 w ct:- 0 H 11.0Ka3aTeJibCTBO 3asepwaeTC.H - J J npHMeneuneM HH/1.YKUHH cnaY:ana no m, a 3aTeM no n. Ciy'lafl 2. TiycTb M1-Mo11.y11b c '1J1-Kpy'-leHHeM. Toraa M 1 = L(M1)k cyMMa HeH30MOp(pHbIX Me>KJ.\Y co6oA npocTbIX R-Mo11.y11e8, rae R B 3aBHCHMOCTH OT _CHTyau1m paBHO Jil-100 ']JI JIH6o HeKOTOpoMy KOJihUY L. [8]. 3Ha'-lHT, cyl.l.leCT­ syeT d 1 e R TaKoe, 4:TO du = u1,:@ wk LIJIJI HeKOTOpbtX tt1,: e (M1)1.:, wk e N . .IlOKU3aTeJibCTBO 3asepwaeTC.H npHMel·leHHeM HH/1.YKUHH. 04:eBI-1/1.HO, Jl.--MOIJ.Ylib @M . JIBJIJteTC51 noJJynpoCTblM H Ka:>Kaoe npocToe CJtara- 1 J ISSN 004/-6053. YKp. Mam . ;,cyp11., 1993, m. 45 , N' 9 KPAUHJfE MOJlY JUI HA.U AME6POU BEtl.!151 A,. I 193 eMoe H30Mopq.mo M,. C11e1wnaTe11b110, A-M0JWJIH ®M,- H ®N; H30Mopq_rnb1 T0rtla H T0JlbK0 Tor Aa, K0rtla M; =- N; V I ~ i ~ 11 • TeopeMa tl0Ka3a1-1a. A O A CieiJcmoue. I. A 1 ® . .. ®A I C A,,. 2. sl (2)" ® .. . ® sl (2)A c (sl (2) x ... x sl (2)t. 9. KpaTHOCTb npOCTOl'O A,.-MO/O'JHI pa3MCpHOCTH n. Teoper.1a 8. l(11J1 i.;a>KiJow 11cm1.ypa.11b1ww m e ~ cyu~ecmayem npocmoii A,,­ MOO)'Ab M pa3Mep1-1ocm11 d(M) = 11 11 Kpan111ocmu e(ffl = m. aoKa3ameAbC11l60. B CHJlY JieMMbl 2 Ii Te0peMhl 7 A0CTaT0'-IH0 J.I_0Ka3aTb Te0- peM)' B c1iyt.Iae A 1. Jler.ar.aa 4. nycmb p = I - ax. a "# 0 e K[H] , deg a = v ( coonwemcmae1mo p = 13 - X, ziJe M1ww 1I11e1-1 13 e K[H] cme11e111, µ ~ 1 11e uMeem t{e.llbtX Kop11eaJ, M = A 1 / A 1p. To2iJa <J o6oux c1y 1WJ1X M -npocnwt1 A 1-MOOY,1b pa3Mepl-locmu d(M) = 1 u ,.:.panl/-locmu e(M) = 2v + I (coonwemcmaeww e(M) = 2µ). aoKa3Qlll&lbCIJl60. PaCCMOTpHM nepm,Ilt CJiyt.Ialt (BT0p0lt paCCMaTp1rnaeTC.){ a~Ia­ JlOrH'-11-10). O603Hat.IHM t.IepT0il (K0T0py10 6yAeM 110 B03M0)KH0CTH onycKaTb) ecTe- CTBeHHhllt. 3ITHMOp<pH3M M0AyJ1ett A 1 ➔ M, v ➔ v. TorAa M = K[H] $ {KHiXj lO~ i~v-1, 1 ~ j}. Ecm1 v = 0, TO M = K[H] 11 AJ1_)i JII06orn 1.(e11oro k ~ 0 pa3MepH0CTb npoci-pa1-1CTBa Mk:= Bk· T = < 1, H, . . . , Hk> pmma k + I , r·Ae {Bk} - cp1111bTJ)aI.\Irn .6ep1-1wTelt.11a 1-1a A 1• C11eAOBaTe.n1,no, Kpan1ocTb 11 cTeneHb M pasua I. Ec11H v ~ I. TO, o6o3Ha­ t.IaR F 11 :=<HiXjl 2i+j~n> n G,,:=<HiYJI 2i+j$n), B,, =F,, +G,,. nMeeM F- ·-<Hi Ji 'Xkl 1< "< O< "< -I 2 · "<2 > d" F---: - ( 1)2 211 .- , J \ - I_ /l, -) - V , J + I - fl II lffi 2,, - V + fl+ ... npH fl>> 0. Ot.IeBHAH0, JiiY2<11-il ➔ JiiIT{ ok(y) I 2.(-11 + i) < k $ 0} - MI-I0f'Ot.I11ett c-rene1m d,,,; = i + (v + 1)2(11 - i). rAe y = o-1(a)a. lfa paBencTB d 11 , , - d,.,;+ 1 = 2v + 1, dn ,i - d11-l.i-l = 1 CJleAyIOT BKJIIO'-leHIUI K[H](v+I)211--{2v+ J)2 C G 211 C K[H](v+I )211• rAe np0CTpaHCTB0 K[H] r C0CT0HT H3 MH0r0'-iJlel-IOB CTene1-m ue 60J11,we ,. OTCIOAa dim G 211 = (V + 1)211 + ... H dim (F 211 n G 211) = 211 + ... npn II>> 0. CJleAOBa­ TeJlbH0, e(M) = 2v + 1 11 d(M) = 1. B 0OOHX CJ!y'-la.){X np0CT0Ta M0AYJl_)i M CJleAyeT 113 npeAJJO)KeHH.ll 8 H CJleACTBlf.){ 1 113 [4]. npHMeHeHI·IblX B CJiyt.Iae a= H. TeopeMa A0Ka3aHa. 10. IloJIHHOMHaJI1,u1,1e MOAYJIH. M0AYJib HaA OAB 6yAeM Ha3blBaTb no­ .11u1-10Mua.11b/-lbtM, ec1111 OH .){8J[.){eTC.){ CBOO0AHblM M0AyJ1eM pattra l HaA 6a3HCHblM K0Jlb~0M OAB. TiyCTb 51. - OAB CTeneI-IH 1 C 6a3HCHbIM K0Jlbl..(0M 'lJ, .)18Jl.){l01J.(HMC.ll K0MM)'­ TaTHBH0R 0611acTI,IO r 11aBHbIX HAeaJl0B C yc11os11eM C), uanpHMep 'lJ = K[H], <J(H) = H - l, J1H6o 'lJ = K[H, H- 1], <J(H) = qH, q # 0 e K - ue Kopeub H3 1, 1106o np0H3B0JibHa.H J10KaJIH3al..(H.H 3THX K0Jlel..( . TipHBeJJ.eM CB0lt.CTBa n0JlHH0MHaJlbHbIX 5l-MOJJ.yJ1elt. Ill. Ka)Kabt(i ll0/1./lflOJ,IUQ.l/b/-lblii 51.-J,IOO)'llb J18/IJ1emCJ1 !-lepQ3,10)KUJ.tblJ,,I MOOyAeM KOHe•moil OAUHbt u U30Mop<jJeu HeKomopo.,,,y MoiJy1110 auiJa w1 := 51./ 51.(X - f. Y - - a I o-1(/)), lOe f "# 0 e 'lJ maKoa, 'Imo fl <J(a), aep!-lo u 06pan11-1oe. ISSN 0041-6053. YKp . -'«1111. JKYP"·, 1993, Ill . 45, N• 9 1194 B. B. EABY JIA 112. K(l)l(Oblll co6cnwewtbll"i 1100MOO_Y/lb ( ec11u OH cyu{ecmoyem) 1w11uHOMttallbHOW J.tOOYIIJl wt Jl8/IJlemcJl II0/11/HOMI/G/lbHblM II cy11feCm8eHHb/M. Eo11ee noopo6HO, 'Dh- llOOMOO)'llb o Wf<=> cr(h)f = ha OIIJl 1ieKomopow a -t: 0 e 1J maKow, '<mo a I cr(a) . KaJKOblti co6cmoeHflbU"i <.paKmop-:,.woy,1b 1101111HoMua11bflOW MOO)'/lJI .f/8.11 . .Remc.R J.WOy.lleM C 'lJ-Kpy'<efl lleJ.f, II , C/leooaame.llbf/0, UMeem KOHe'ill)'/0 0.llllH)' KGK 1J-MOO)'llb. 113 . . fll060ii l-/el-l)'.lle80U W MOMOJ)c/JU3M 1/0./l.ll/lOMUQ.llbltblX MOO)'.lleL/ .fl8.llJlen1C.R 0110JKe11ueM. Ec;1u 1w11u110,1wa11.bHbte Mooy11u M u N He 1130M.opc/J1tbt u Hom5t(M, N) -t: 0, mo o6.fl.3(Jme11bNO Hom 5t(N, M) = 0. 114. Mooy.1111 Wt II W g u30M.opcfmb1 o mo,11 u mo11bKO o moM c11y'<ae, Kozoa cyu{ecmayem 06pamuMbU1 3/JeMeJtm A e 1J nwKoii, 'Imo cr(A)g = Af Ec11u M1t0JKecmoo o6pamttMbL\. 3/leJ.te1-1moo 1J • KOllblfa 'lJ coenaoaem c :,.,mo­ )KecmooM Henooou)l{1tbtX mo'leK aomo:,.wpc/Ju3Ma CJ, m. e. 1J •='IP (HanpuMep, 1J=K[H], cr(H)=H - 1), mo 114'. Wf"' Wg <=> f = g. Ha MHO)KecTBe nOJlHIIOMHa.TlbHblX MOJJ.YJieA Pol BBeJ].eM OTHOllieHIUl nopjJJJ.Ka < H 3KBHBa.ne1-rrHOCTH ~M < N (M ~ N), eCJm HomJ!_(M, N) -t: 0 (COOTBeTCTBeHHO cymeCTByeT B Pol KOHe'-{HcUI u,enO'-{Ka MOJJ.YJieA: C lfa'-{aJIOM M H KOHIJ,OM N TaKa}[, '-{TO cymec-rny10T HeHyJJeBhie roMOMOpq)H3Mbl Me)KJJ.Y COCef_lllHMH MOJJ.yJJjJMH, nopjiJJ.OK ue HMeeT 3Ha'-{emrn). 115'. KaJl(OllJI 0611acmb 3Kouea,1e11mnocm11 cocmoum ,111U.tb u3 KOHe'<HOlO I/UC/la 3/le✓\·teflm.08 II 8 Hell cyu1ecmoyem fl(l!Ute/tbUlllll 3/le,11e1m1, KOm.Opbnl 8 CUi i)' C80L/Cf/U3(l IJ3 COOep)KltnlCJl 8 11I060M. MO(J_)'.lle 113 3m.Oii 06.11acm.11. aoKa:Jame,u,emeo. fl!. flyCTb M = 1J - nOJIHHOMHaJJblibJtl: MOJJ.YJib. TOrt:(a XI = f H YI = g /J.Jl}l 1-1eKOTOpbix f, g e 'lJ, KpoMe Toro, a= YXI = <J"" 1(f)g. CJJeJJ.OBareJJbHO, M"' wf - MOJJ.YJlb KOHe'-{HOA: JJ.JIHHbl B CHJJY npeJJ.JIO)Kel-IHjl 1, a 1-1epa3JI0)KHMOCTb CJleJJ.yeT 113 pa.BeHCTBa 'DM= 1J. O6panme 0'-{eBl1JJ.HO. II2. a(h)f=Xh = ah 11 Yh = ~h JJ.JJ.H 11eK0Top1,1x a.~ e 'lJ. Tor11.a a (a)h = = XYh = X(~h) = a (~)ah. OcTaJibHOe MeB1-1JJ.HO. II3. lloKa3aTeJibCTBO HenocpeJJ.CTBeHIIO CJ1e11.yeT 113 Toro, '-{TO nOJJHI-fOMHaJlbHbilt MOLIYJlb HMeeT KOHe'-{HyIO J].JIHHY H He HMeeT 'lJ-Kpy'-{eHirn . II4. .l:130MOpcpH3M <p: WI-) wg nOJll-fOCTblO onpeJJ.eJJ.HeTCji CBOHM 31Ia'-{eHHeM <p(l) = A, KOTopoe, MeBHJJ.HO, j{BJ1.Herc>1 o6parnMhIM B 'lJ. Tor11a lf = <p(X) = = U = cr(A)g. II4' c11e11.yeT 113 II4 H paneHCTBa 'lJ* = 'IP. ITS'. Ilycn, M - neK0Top1,1tl MmmMaJ1b11b1A 3;1eMeHT 113 o6JJaCTH 3Kn11na­ J1e11THOCTH £ (HanpHMep, nepece'-{e1-rne BCex 1-1e11yJteBl,IX n0AMOAYJleA HeKOT0poro nOJIHHOMI-IaJlbH0ro M0/J.YJUI 113 £) 11 F - C0BOKynH0CTb N e £: Hom 5t(M, N) 7' 0. O'-{e1.m11.110, MO/WJJb M -npocrotl. ECJm F-t: E, TO cy111eci-13yer MO/WJtb Le£ \ F TaKolt, '-{TO Hom Jl(L . N) -t: 0 JJ.Jrn 1-1eKornporo N e F. Tori,1,a Hom 51(M, L) -t: 0, noc­ Ko111,Ky M0;KHO C'-{HTaTb, '-{T() M H L .l!BJl.l!IOTC.ll nOAM0/WJl.llMH B N, a M K TOMY )Ke npOCTOrl H cy111eCTBeHilblfi, f10;1y'-{elll-10e npOTHBOpe'-{He J::IBepwaeT JJ.OKa3aTeJlbCTBO, CoBOKymtOCTb 11epa3JJO)KHMblX 3JieMelIT0B 'Jvf. K0JlbUa 1J lla3bl8aeTC}[ Ob/Oe­ ;1e111t0ii , eCJJH mo6otl nepa3JJO)KHMblit 3JJeMellT 113 'lJ Kpan11,1i1 HeKOTOp0MY :-:lJie­ MeHTY 113 'Jvf. 11 ec;m np11 3TOM p = 11,a'(q) LlJl.ll 11eKOTOp1,1x p, q e Af, i e ~. ISSN 0041 -6053 . .Y1q1. MWll . Jl<Yf'"· · 1993. Ill . ./5, N" 9 KPJ\ttHHE M OL{YJll'I HA{( /\JirEGPOtl BEtlJt5l. An 1195 A e ']J* , ro A= l. 3aq_mKCHpyeM HCK0Topoe BbI/J,CJ1e1moe MHO.tKeCTuo 'Jv[ (Harrp1-1- Mep, ec;ui '1J = K [H], TO 'Jv[ - COBOK Y ITHOCTh 1-1epa3JI0:>KHMbrx 110111m0Mon c K03q ) ­ cj)HL(HenroM npa c rapwef-i cTenem1 p am11>1M l ). T or Aa Ha 'Jv( ,~eticniyeT r·pynna ~ = <er>. J].;rn ; 11o6oro a ;tc. 0 e I(_ (= no.,1e 'laCTUhIX 'D) 4epe3 O(a) o6o3H3'lHM COBOKynHOCTh op6Iff 113 'Jv{j ~ - KOTOpbIM np1-mal].JIC)KaT Hep a3j 1Q)KHMl>IC COMilO)KH ­ TeJm H3 HecoKp m n Moti 3arm cu 3JieMeina a s nH.ue AP06H. T orAa a= a*n{a0 j 0 E O(a)}' r /lC B COMH())K lffCJl b a() E K... UXOil51T BCe nep a3JIO)KHMhle 3JlCMCIIThl , np1rnarv1e)Kaum e op61ne 0 , a• e 'if -- ormo3Ha,mo onpeue11em1u fi 3neMeHT. TiycTh V(p, a ) - KpanJOCTb BXO)KJJ,emrn /J E :M" B 1-1eCOKpa-r11My10 ilP06h a . Jle~1Ma 5. n ycmb a T- 0 e k., Tozoa pewe11ue h e k_ ypa811emu <J(h) = a h cyuiecnwyem mou}a u mo,1bKO mmoa, 1':0lcla cyuiecnwyem pewe11ue ft* e ']J* np u Cl = o.* U iJA.,'1, 11l060i1 op6unlbl Q E (~:a): <a, O> :~ L .,, v(cr1(p ) , a)= 0. IE(£. npu ,nwM p ewe11ue 1u1eem auo h = h *TIOEO(a) n pEO pri.P ), zoe L ( p ) = - L ;~o v(cr- '(p ), a ). Ee;rn p ew e,we cyu{ecmoyem, m o 0 1-1 0 npu1-1ao11eJKum K011bl()' ']) fl lOlOCI u mO,lbK.O mmoa, KOlOa 0/IJI ll l06oi1 op 6umbl O e O(a) u ,uo6ow p e CXa): L(p);:;,; 0. C.rte;.J,CTBHC ( KpHTCpHi:i npOCTOTLI MO/.U,'JIR Wj). fl O,lli/10),IUCl/lb Hblil Jl ­ MOO)'/lb WJJ I cr(a)-npocmoii mOloa u mO/lbKO mozoa, Ko1oa O/lfl .11 io6ow 1-1eo6pamu­ MOW a' :tc O e '1J nwKoto, t.tmo a' I O'(a) , 1-1e abuw ,1.1111emc11 npeOb1oyu{a11 AeMMCI npu a= a' If IlpHBeACM CHO!tcrna ITOJIIUIOMHaJibHbl X A-MO/lyJlett I . w - 110,1UIIOMUG,1bli b/Ll A -.\100_)'1lb ¢::) w "' wf := ®7 wf; 0/IJI HeK.omopow oiJ110311at.tHO onpeoe11e1111020 eeKmopa f = (J1, . • . j~), f; e K[H;] ,J; I cr,~1), 1 si s 11. 2. Wr npocmoii A -MOOy /1.b ¢::) wf,-npocmoii Jl,-,\,00)'/lb 0/IJI 11106020 1 s i s f l. 3. n o11U1-IO.\tlla/lbHbte MOO)'/IU Wf II W g U30A10pc/Jllbl mOlOa LI mO/lbKO fllOlOa , Kozoa f = g , m. e. J; = g1 Oil.fl 111060w l s i s 11. 4. Ec11u 11pocmpa1tcmao WMOMopc/Ju3M06 HomA(W1, Wg) :tc 0, mo 01-10 oOHOMep ­ ,w u nopoJKiJaemc11 <MoJKe1-111eM <p = ® <p1, zoe <p ;: w1; -> W8 , - e.-10JKe m1e Jl;­ Mooy11ea. 'I'OlOll HomA(Wp Wg) = 0. 5. n o/lUHOMUGllbHblll J,W0)'1lb Wj cooep)KUm /taUMe/tbWUii (om11ocume11b/lO 8K./I. IO'teHUJI) 1te1ty11eooti noOMOO)'llb, KomopblU mo)Ke JW/IJtemc.fl flOIIWIOMltallbllb/.M M00)'/101 Wg, zoe wg, - HGUMCHbl«UU llOOMOO)'/lb (3 wf , 0 /IJI 111060w i. 6 . .ll,06oi1 no11UJtOMUllllbHblLt MOO)'llb Hepa3110)K,LIM u UMeem KOHeLtH)'/0 0/IUH)'. l(o,uname,u,emeo. 1. CneAyeT H3 COOTHOlllCHHtt Y ,X; = a 1 11 X I Y1 = cr,<a,) 11 <f.HlKTOpH3.TlbHOCTH KOJlbI~a MHOrO'lJieHOB HM ITOJICM. 2. 5IBJl.llCTC~ CJICt{CTBHCM TeopeMbl 7. 5. JlocraTO'·lHO ITOKa:xtTb, 'ITO )lJlJl mo6oro w ic- 0 e Wj ITOJJ,MOAYJib Au CO/lCp­ )KHT He1-1yneaott 3JICMCHT BHAa @ ll;, r.ue u, e WJ;- Ana.norH'lHO AOKa3aTCJibCTBY TeOpCMbl 7 (cJiy'fatt 1) npel].CTaBHM w a BHAC (2): ISSN 0041-6053. YKp . Mam . JKJP" ·· /993, 111. 45. N' 9 1196 B. B. 6ABYJIA Lm n w= u- @w - u -eW.1 we®Wt 1 J } ' ] I' J 2 k 0 ITocKOJibKY .H/1.pO oneparnpa b =cr1(u1)/1 - u1X 1 B npoCTpaucrne W11 pamrn Ku 1, TO B c1rny ycJiomrA i), ii) bw = L;• buj@ wj ~ 0 H /l.0Ka3crreJrbCTBO 3aBepwaeM HH/1.YKQHett no m H II. 6. ITOCKOJlbKY ITOJIHHOMHaJl.bHblA MOJWJib ue HMeeT D-KpytJeHH.H, TO JU06oA ueuyJieBOA r0M0MOp(pH3M ITOJIHH0MI-IaJlbHbIX M0AYJiett .HBJI.HeTCM BJIO)KeHlfeM , CJie/1,0BaTeJibH0, IT0J1HH0MH3JlbH bit!. M0AY Jib uepa3JI0)KHM, a KOHetJH0CTb /1,JlHHbl CJie/1,yeT H3 CBOttCTBa Ill H TeOpeMbl 7. 4. TaK KaK w 1MO/l.YJib K0He•rnoA /1,JlHHbl H 6e3 D -KpytJellHj/. TO ero KOJlhl~0 3H/l,OMOPQ)H3MOB .HBJI.HeTC.H TeJIOM T Ha/1, ITOJieM K. B ClfJIY aiire6pautJeCKOtl. 3aMKH)'TOCTH non.ii K H Kone•rnonopo)Kt1em10CTH ai1r-e6phl A T = K. 3. CJ1e/l,yeT HJ cBottcrn 4 H IT4'. I1CTOpHtJecKH nepBblMH npHMepaMH IT0JIHH0MHaJibHbl X M0AYJiett .HBJI.HIOTC.H A,.- MOJIYJIH W,.,=-A,./A,.(cJi-A.1,· ·· ·a,,-A,.)=(K[X1, · ···x,,1, (d; - A.;)(F)=dF/dX; - cpopMaJibHa.H np0H3D0/l.llaH, X;(F) = X ,F - YMHO)Ke1me), A = 0,. 1 ~ 0, ... , A,.~ 0) e K 11 H rrpoc fhle sl(2}-MOAYJIH YttneKKepa W,._(µ), µ ~ 0, A e K [9], rrpoc­ TpaucTBO rrpe/l,CT3BJieHH.H K0T0pb!X panuo K0 Jlh11,y MHOr0tJJiell0B K[h]: Hh11 = h11+1, Xh11 = µ(h - l)". Yh 11 = -µ- 1(h + l)11(h -A)(h +A+ 1), 11 ~ 0. ,llpyrue npHMepbl II0JIHH0MHaJlbHblX s l(2)-MOAyJiett npHBeneHbl n [10]. E. K. CKJIHHHH [11], B tJaCTH0CTH, paccMaTpHBaJ1 K0HetJ110Mep111,1e A-MOLWJIH, pea­ JIH3Y.H HX KaK cpaKTOp-M0/1.YJib II0JlHII0MHaJl.bH0ro. 11. Mo/1.yJH,HaJI c·1·pyKTypa A. CrncpcpopA [ 12] rrocTaBHJt no npoc o uaxo)K­ JJ.eHHH 6oJiee wttpoKoro KJiacca K0J1eu, tJeM a1ll'e6pb1 BeAJrn na11 110J1eM xapaKTe­ pHCTHKH ny Jib, /1,JIH KOT0pblX BepHbl pe3y Jib TUTbl, H3Jl0)KeUIIbie B § 3 H3 [ 1 2 ]. OKa3bIBaeTC5f, npHMepoM TaKoro KJIUCCa K0J1e1.1; MOl' YT 6bITb rrpOCTbie 0606w.em-1b1e aJ1re6pb1 BeAJI.H, paccMaTp1maeMh1e a AUHH0lt pa6<.ne . ITpe)KAe 'feM ccpopMyJ111- poBaTb pe3yJihTaTbl, npuBe/~eM Oil.HO onpe/l,eJ1e1me. ITyCTb R - HeTepona 06JiaCTb C TeJI0M 'faCTHblX Q H M nopO)K/1,eHHhIA: JieBbn1 R-M0/1.YJib. OnpeAem-1M paur M0LJ.yJrn M TaK: Q @RM"' Q(s). TeopeMa 9. flycmb 060611{eHHOJ1 a,ne6pa Beii11.J1 A-npocma. ToziJa - KOBe4B0- rank M = s: 1. Jbo6oii npaebll1 uiJea11 113 A 11opo)KiJaemcJ1 iJeyMR 3110,efmiaMu. Eo11ee mow, ec11u npaebtii uoea11 I = aA + bA + cA u cl ~ 0 e A , mo cy11{ecnl8yiom f u g e A maKue, ttmo I= (a + cfcf)A + (b + cgd)A . 2. Jbo6ou Ko11ettH011opo.)l{oem1bu1 A-MOO)'Ab M u3oMopcpHbtll Ne A1s>, 1.iJe MOOJAb N uMeem paHl S 1. 3. flycmb M - K0He'l11011opo}Ki}e1111bu'/ A-MoiJy11b, rank M ~ 2. npeo11011o}Kt1M, 'Imo ME& A"' Ne A OA.Jl He1.,omopow A-MoiJy11.J1 N. Tozoa M"' N. ITycTh 5!1 :=K[HiJ(cr1, a 1) - OABCTerrerm l H '13 1 -JI0KaJrn3aL\H5t 5!1 HJ n. 3. EcJm L :) K - pacwttpe1me IT0JI.H K. TO o603uatJHM 5!1 (L) = L @ 5!1. JleMMa 6. flycmb 5!1 - 11pocma.J1 a11z.e6pa, bl' . .. • b, e Jl1 - 11wteiiHo 11e3a­ oucuMbte ,iao K 311eJ.1eumb1. nycmb S = Jl1 111160 '13 1 11 eb16epeM t ,;c O e S. ToziJa JI , (bl . ...• b,)tS = s<r)_ PoKa3ame.AbC11l60. O<IeBHilHO, /1,0CTaT0'I H0 paCCMO'rpen, CJiy<Iati S = 5f1 H t = 1. ,UJI.H /1.0CTaT0tJHO 6on1,woro 1mTypai11,Horo s: b,.xs e K<H, X> LJ.JI5t Bcex i. Il03TOM)'. He orpaHHtJHBa.H o6LU,H0CTH pacCy)K/1,eHHtt, M0)KH0 C'IHTaTb, 4T0 3TH BKJHO- ISSN 004/-6053. YKp . .1,10111 . )l(JP"·• 1993. m. 45 , N" 9 KPAilHI-1E MOLl,Y JIH HAJJ, AJifE6POtl BEtlJI51 An 1197 q emu1 Bb!ITQJIHRIOTCH. Ecm1 Cl - MHOr0'-1.JieH H3 K[H] CTeneHH fl, TO cr(a) - Cl HMeeT CTeneHh II - l. OTCIOAa cymeCTByeT TaKOe 1-IaTypa.rihHoe m, 'ITO (3) r·,ue \· e K H (ad X)t = Xu - uX, u e 51.1• Tiepeo603naqa,1 b;, ec;m 3TO Heo6xOAHMO, u 11.eJIM JaMeHbI BHAa bi ~ Lµ 1b;, µ; e K, µ, = 1 ( Bee 3TH onepau;utt nepecranoBO'l­ uw c ad X), MO)KHO C'H:ITaTh, 'ITO 11 1 > 112 ~ .. . . Y MnO.>Kaa (3) CJieBa ua f"1 u npm.te HHR ad Y nocTaro,rnoe '-1.HCJIO p a3, ttanpttMep 11 pa3, HMeeM (AY", 0, ... , 0) AJIH ueKOTopor-o A :le- 0 e K. TiocKOJihKY 51.1 - npocra,1 a.rire6pa, TO (51.1, 0, ... , 0) C0Aep)KHTCH B 5!.1(b 1, . . .• br)S . .[(oKaJaTeJibCTBO JaBepwaeTCR HHllYKU:Hett no r. JleMMa JJ,OKaJaHa. Terrepb AOKaJaTeJihCTB0 YTBep)l(Ae1mtt 1 - 3 ,UOCJIOBHO nOBTOpReT paccy)KAeHH.il C-racpcpopna, eCJIH n 0A BBeL\eHl-lblMH B [ 12) KOJlbl.J;aMH R, S, . . . nonpa3yMeBaTb CJieAy!Oll.(He KOJibI.J;a. a). TiycTh R = Ar ;J A - JI0KaJm3aI.1;1U1 aJire6pb1 A no MYJihTHnJIHKaTHBHO 3aMKHYTOMY MHO)Kec-rny, nopmKAeHH0MY T= {H2, . .. , Hm} llJIH 1-1eK0Toporo m. OnpeneJmM S = 5!.1(K(H2, ... , Hm)) ;JU = S n A ;J 51.1. b). TiyCTb R = AT ;J A, rAe T ={HI, ... , H n' X ,, .. .. x r_l } JJ.JIH HeKOT0poro r ~ II. ECJIH Q - n0JIHOe KOJibU0 qacnu,Ix no,D.aJire6pbl B A , nopO.>Kl].eHHOtt MH0.>KeCTBOM T, TO MO.>KH0 uanHCaTh R = Q<Xr, ... , x,,, Yr, ... ' Y,.>. TIOJIO.>KHM S = Q (Xr , Yr)= Q[Xr , X~ 1; <Jr] - K0JibD;0 K0ChIX JI0paH0BCKHX MH0r0'IJieH0B = Tr-1<H r, . . . , H,,)[Xr. r,1; <Jr]. n\e T, - K0JibD;0 '-1.aCTHbIX aJire6pbl A(r) := ®151.;, U =Sn A =A(r)[H r+ l ' . . . , Hfll :2 51 1• ABTop Bb1pa )Kae-r 61iarw.1;ap11ocTh IO. M. 13epe3aHCKOMY, IO. A . .[(p03AY, IO. C. CaMott; 1enKO 3,1 no11e3a1,1e 06cy)Kne1-11u1 u JaMe'lamrn. l. EaoyAa B. B. 0606u1eHHblC a11re6p1,1 BcAJUI H HX npcUCTa llJICH HH: .[(He .... Ka l!U. q>H3.-MaT. HayK. - Kuen, 1990. - 111 c. 2. 6aoyAa B. B. KoHC'IIIOMep110CTb Ext-B H Tor-u npoclHX ~10uyneA l1aL1 0/lHHM KJlaCCOM anre6p // <DyHK'--IHOH. allaJJH3 Hem npHJJ. - 199 I. - 25, 01.in. 3. - c. 80 - 82. 3 . EaoyAa B. B. 06o611.1e1rn1,1e anre6p1,1 BeAJIH H HX npeucTas,1eH1u1 // Am-e6pa H a11aJJH3. - 1992. - 4, Bblfl. l. - C. 74 - 95. 4 . EaoyAa 8 . 8 . KJJaCCH<pHKaQHR npocn.ix s1(2)-MOUYJidl H KOHC'IIIOMepHOCTh MOLIYJJR pacu.mpe,rnA npocTblX sl(2)-Mony11eA II YKp. MaT. lKyp11. - 1990. - 42, N° 9. - C. I 174 - 1180. 5 . Bjork J.E. Rings of d ifferential operators. -Amsterdam : North Holland, 1979. -374 p. 6 . FeAbc/)a1uJ C. H., Maw111 10. H . r o M0 1101·u'!ecKaH a;1re6pa II H-rorH uayKH u -rexHHKH. Conp. npo6,~. MaTCMaTHKH I BI1HHTH. - 1989. - 38. - 238 c. 7. 6epwumel1u H. H . Mouy11n 11a11 KOJJbllOM llH<pcpcpe HLtHa111, 111~x o n epai-opoo. l·by'le11He cpy11t{aMClffaJibl-lbl X pe1UCHH/l ypaB11eH HA C flOCTOHIIHl~MH K03<pq>HltHe HTaMH II (l>yHKQHOH. a1taJ1H3 H ero npHn. - I 97 1. - 5 , N° 2. - C. I - I 6. 8 . EaoyAa B. B. ITpoc11,1e D[X. Y; a, u]-Mo11y1111 II YKp. ~HIT. lKYPH- - 1992. - 44. N2 12. - C. 1628 - 164-4. 9 . Block R. E. Classification of the irreducible representations o f sl (2 , ct) II Bull. Ame r. Math. Soc. - 1979. - l , N° I. - P. 247 - 250. 10. EaoyAa B. B. 81,1•1uc11e u11c /-/*(sl (2 ), M) c Koactxt>1-n1ue1-1Ta~m H npoc-roM sl (2)-Mo11y11e // <Dy HKl\HOH. a11aJIH3 H ero np HJI. - 1992. - 26, Bhlfl . I. - C. 46 -47 . 11. Sklya11i11 E. K. Func1 ional Bethe Ansatz II Integ rable and supcrintegrable system. Ed. by B. A. Kupershmidt W orld Sci ., Singapore, 199 1. 12. Szafford J . T. Module structure o f Weyl a lgebra~ II J. London Math . Soc. - 1978. - 18, N" 2. - P. 429-442. ITony 'le1-10 04. 07. 91 ISSN 0041 -6053. Yhp. MlWI. JK)'J)lt .• 1993. 11! . ./5. N" 9 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011