Extreme modules over the Weyl algebra $A_n$
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.
Збережено в:
| Дата: | 1993 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1993
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/5920 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512141843365888 |
|---|---|
| author | Bavula, V. V. Бавула, В. В. Бавула, В. В. |
| author_facet | Bavula, V. V. Бавула, В. В. Бавула, В. В. |
| author_sort | Bavula, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | 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. |
| first_indexed | 2026-03-24T03:24:04Z |
| format | Article |
| 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.
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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
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I I I' I I I~ I ' I J I
[X;, x7] = [Xt, Xj] = [Xt, x7] = 0 Vi*f
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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
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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.
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BOro KOJILQa) npoCTblX A-MOAYJICH. IlpHBOAHMbie B llaHH0M nyHKTC pe3yJll,TaTbl
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aBTOMOpqm3M <J YAOBJiernopileT ycnoemo: C ) <J"(P) °i' P ¥0 °i' 11 e ~ . P C 'lJ -
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1leJieum1 C OCTaTK0M) 0TII0CHTeJibH0 c:pyHKQHH ",nJIHHbI" /: /(aX"' + ... + 13X") =
= 11 - m, a, 13 e k - - He uyJ1eeb1e, m < .. . < n.
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5l I J u.Meem Kone1.tny,o 01iuny, c11eooeame11bHO, pa3MepHocmb Kpy111111 a11ze6pbt 5l
pa01ta K - dim 5'I. = l .
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.nynett, m-map1rn1rn-10e 0THOCHTeJibJ-10 H30MOpQ)H3Ma M0/lYJiett. Tor.na '{epe3 A.(Q)
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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.
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a. = TT-s:SISO e1i(l3o), 13 = TT-m-s:Sj:Sl oi(l3o),
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➔ [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
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a). TiycTh R = Ar ;J A - JI0KaJm3aI.1;1U1 aJire6pb1 A no MYJihTHnJIHKaTHBHO
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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
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| id | umjimathkievua-article-5920 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:24:04Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/02/bf206d2ddd29c94322dc1c5828f95d02.pdf |
| spelling | umjimathkievua-article-59202020-03-19T09:21:13Z Extreme modules over the Weyl algebra $A_n$ Крайние модули над алгеброй Вейля $A_n$ Bavula, V. V. Бавула, В. В. Бавула, В. В. 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. Класифікуються деякі класи простих модулів над узагальненими алгебрами Вейля (крайні, поліноміальні, з сильним $D$-скрутом). Для цих алгебр доводяться аналоги теорем Бернштейна і Стаффорда. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5920 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1187-1197 Український математичний журнал; Том 45 № 9 (1993); 1187-1197 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5920/8523 https://umj.imath.kiev.ua/index.php/umj/article/view/5920/8524 Copyright (c) 1993 Bavula V. V. |
| spellingShingle | Bavula, V. V. Бавула, В. В. Бавула, В. В. Extreme modules over the Weyl algebra $A_n$ |
| title | Extreme modules over the Weyl algebra $A_n$ |
| title_alt | Крайние модули над алгеброй Вейля $A_n$ |
| title_full | Extreme modules over the Weyl algebra $A_n$ |
| title_fullStr | Extreme modules over the Weyl algebra $A_n$ |
| title_full_unstemmed | Extreme modules over the Weyl algebra $A_n$ |
| title_short | Extreme modules over the Weyl algebra $A_n$ |
| title_sort | extreme modules over the weyl algebra $a_n$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5920 |
| work_keys_str_mv | AT bavulavv extrememodulesovertheweylalgebraan AT bavulavv extrememodulesovertheweylalgebraan AT bavulavv extrememodulesovertheweylalgebraan AT bavulavv krajniemodulinadalgebrojvejlâan AT bavulavv krajniemodulinadalgebrojvejlâan AT bavulavv krajniemodulinadalgebrojvejlâan |