Simple Lie Algebras over Fields of Positive Characteristic: II. Classifying the Absolute Toral Rank Two Case (De Gruyter Expositions in Mathematics)

392 Pages·2009·2.54 MB·English

by  Helmut Strade

de Gruyter Expositions in Mathematics 42
Editors
V.P.Maslov, Academy of Sciences, Moscow
W.D.Neumann, Columbia University, New York
R.O.Wells, Jr., International University, Bremen

Simple Lie Algebras over Fields
of Positive Characteristic
II. Classifying the Absolute
Toral Rank Two Case
by
Helmut Strade
≥
Walter de Gruyter · Berlin · New York
Author
HelmutStrade
E-mail:[email protected]
MathematicsSubjectClassification2000:17-02,17B50,17B20,17B05.
Keywords:SimpleLiealgebras,Liealgebrasofpositivecharacteristic,
dividedpoweralgebras,Cartanprolongation,recognitiontheorems.
(cid:2)(cid:2) Printedonacid-freepaperwhichfallswithintheguidelines
oftheANSItoensurepermanenceanddurability.
ISSN 0938-6572
ISBN 978-3-11-019701-3
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TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
10 ToriinHamiltonianandMelikianalgebras . . . . . . . . . . . . . . . . 3
10.1 DeterminingabsolutetoralranksofHamiltonianalgebras . . . . . . 3
10.2 MoreonH.2I.1;2//.2/ . . . . . . . . . . . . . . . . . . . . . . . 13
Œp(cid:141)
10.3 2-dimensionaltoriinH.2I1Iˆ.(cid:28)//.1/ . . . . . . . . . . . . . . . . 39
10.4 SemisimpleelementsinH.2I1Iˆ.1// . . . . . . . . . . . . . . . 45
Œp(cid:141)
10.5 Melikianalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.6 SemisimpleLiealgebrasofabsolutetoralrank1and2 . . . . . . . 106
10.7 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11 1-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
11.1 Liealgebrasofabsolutetoralrank1 . . . . . . . . . . . . . . . . . 121
11.2 1-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.3 Representationsofdimension<p2 . . . . . . . . . . . . . . . . . . 131
11.4 MoreonH.2I1/.2/ . . . . . . . . . . . . . . . . . . . . . . . . . . 138
11.5 LowdimensionalrepresentationsofH.2I1/.2/ . . . . . . . . . . . . 150
12 Sandwichelementsandrigidtori . . . . . . . . . . . . . . . . . . . . . 162
12.1 Derivingidentities . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
12.2 Sandwichelements . . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.3 Rigidroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12.4 Rigidtori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
12.5 Trigonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
13 Towardsgradedalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 242
13.1 Thepentagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
13.2 Anupperbound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
13.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
13.4 MoreonHamiltonianroots . . . . . . . . . . . . . . . . . . . . . . 278
13.5 Switchingtori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
vi Contents
13.6 Goodtriples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
13.7 Ontheexistenceofgoodtoriandgoodtriples . . . . . . . . . . . . 303
14 Thetoralrank2case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
14.1 Norootisexceptional . . . . . . . . . . . . . . . . . . . . . . . . . 313
14.2 S isnotofCartantype . . . . . . . . . . . . . . . . . . . . . . . . 326
14.3 Gradedcounterexamples . . . . . . . . . . . . . . . . . . . . . . . 354
15 SupplementstoVolume1 . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Introduction
Thisthreevolumemonographon“SimpleLieAlgebrasoverFieldsofPositiveChar-
acteristic”presentsmajormethodsonmodularLiealgebras,alltheexamplesofsim-
pleLiealgebrasoveralgebraicallyclosedfieldsofcharacteristicp (cid:21) 5andthecom-
pleteproofoftheClassificationTheoremmentionedintheintroductionofVolume1.
The first volume contains the methods, examples and a first classification result. It
turnedoutduringtheworkonthereproductionoftheclassificationproofthatonehas
topayforareasonablecompletenessbyextendingthetextconsiderably. Sothewhole
workisnowplannedasathreevolumemonograph. Thissecondvolumecontainsthe
proofoftheClassificationTheoremforsimpleLiealgebrasofabsolutetoralrank2.
WehavealreadymentioneddetailsoutliningtheproofoftheClassificationTheo-
remintheintroductionofthefirstvolume. Thereforewewilljustrecallverybriefly
some strategy in order to place the content of this volume into the whole picture.
Already in the early work on simple Lie algebras over the complex numbers people
determined,asageneralprocedure,1-sectionsL.˛/WDP L .H/withrespect
i2Fp Pi˛
to a toral CSA H, described their representations in the spaces i2Fp LˇCi˛.H/,
and determined 2-sections. The breakthrough paper [B-W88] made this procedure
workformodularLiealgebrasaswell(ifthecharacteristicisbiggerthan7). Itturned
out,however,thatthemanymoreexamplesandtherichnessoftheirstructuresmade
thingsmuchmoreinvolved. Imaginethatintheclassicalcaseonlysl.2/˚H \ker˛
occursasthe1-sectionL.˛/,whileinthemodularcasetheclassicalalgebrasl.2/,the
smallest Witt algebra W.1I1/ and the smallest Hamiltonian algebra H.2I1/.2/ have
absolute toral rank 1 (by Corollaries 7.5.2 and 7.5.9). Hence each such algebra is a
1-section of itself. There are nonsplit radical extensions of these Cartan type Lie al-
gebras,anditisapriorinotclearwhichofthesecanoccuras1-sectionsofsimpleLie
algebras. Moreover,therepresentationtheoriesofsuchextensionsareveryrich,and
thereforetheverydetailsofthesetheoriescanhardlybedescribed. Lessinformation
is, fortunately, sufficient for the Classification Theory. Namely, it is sufficient and
possible to describe semisimple quotients LŒ˛(cid:141) WD L.˛/=radL.˛/ with respect to
certaintoriT (wehavetodecidewhichtoriwetakeintoconsiderationthough),andit
isalsopossibletodescribetheT-semisimplequotientsof2-sectionsintermsofsim-
pleLiealgebrasofabsolutetoralrank1and2byBlock’sTheoremCorollary3.3.6. If
oneknowsthesimpleLiealgebrasofabsolutetoralranknotbiggerthan2oneisable
2 Introduction
todescribeallsuchsemisimplequotientsof2-sectionsintheseterms. Inthissecond
volumewewillprovethefollowing
Theorem. Every simpleLie algebraover analgebraically closed fieldof character-
isticp >3havingabsolutetoralrank2isexactlyoneofthefollowing:
(a) classicaloftypeA ,B orG ;
2 2 2
(b) therestrictedLiealgebrasW.2I1/,S.3I1/.1/,H.4I1/.1/,K.3I1/;
thenaturallygradedLiealgebrasW.1I2/,H.2I.1;2//.2/;
H.2I1Iˆ.(cid:28)//.1/,H.2I1Iˆ.1//;
(c) theMelikianalgebraM.1;1/.
Since we classified the simple Lie algebras of absolute toral rank 1 in Chapter 9 of
Volume1,theresultofthissecondvolumewillprovidesufficientinformationonthe
2-sectionsofsimpleLiealgebraswithrespecttoadequatetori.
TheproofoftheClassificationTheoremforsimpleLiealgebrasofabsolutetoral
rank2iscompletelydifferentfromwhatwehavedoneinChapter9ofVolume1and
what one has to do in Volume 3 for the general case. In fact, when writing this text
I have changed some of the original items, so that even more the description in the
introductionofthefirstvolumedoesnotcorrectlydescribethepresentprocedure. Let
me say a few words about the sources for the proofs of this volume and the citation
policy. The breakthrough paper [B-W88] gave the general procedure and provided
manyideasforthesolutionoftheabsolutetoralrank2case. Inthepresentexposition
I stressed the point of using sandwich elements and graded algebras in combination
with the Block–Weisfeiler description of these. The major contribution of sandwich
elementmethodsisduetoA.A.PREMET. Mostoftheothermaterialcanbefoundin
the papers [P-S97]–[P-S01]. I will not quote these results in detail. If the reader is
interestedintheoriginalsourcesheshouldlookinto[B-W88],[Pre85]–[Pre94]and
[P-S97]–[P-S01].
Chapter 10, the first chapter of this volume, is somewhat different from the rest.
In that chapter we determine which of the Cartan type and Melikian algebras have
absolute toral rank 2, determine automorphism groups of these algebras, describe
orbits of toral elements in the minimal p-envelope under the automorphism group,
compute the centralizers of toral elements and estimate the number of weights on
restrictedmodules. Indoingthiswedecoveralotofdetailsofthestructureofthese
algebras. Thisalreadyindicatesaweaknessofthetheory: atpresentoneneedsreally
muchinformationonthealgebrastructurestoapplysomesophisticatedarguments.
The notations in this volume and all references to Chapter 1–9 refer to the first
volume. Asageneralassumption, F alwaysdenotesanalgebraicallyclosedfieldof
characteristicp > 3(whileinthefirstvolumewealsoincludedthecasep D 3),and
allalgebrasareregardedtobealgebrasoverF.
Chapter 10
Tori in Hamiltonian and Melikian algebras
InthischapterwedeterminetheHamiltonianandMelikianalgebrashavingabsolute
toral rank not bigger than 2. For those algebras the automorphism groups are deter-
mined and conjugacy classes of tori in the minimal p-envelopes are described. We
gainstructuralinsightinrootspacesandsectionswithrespectto2-dimensionaltori.
10.1 Determining absolute toral ranks
of Hamiltonian algebras
We remind the reader to the concept of an absolute toral rank of a Lie algebra g.
Namely, let g be a Lie algebra with subalgebra q and let .g ;Œp(cid:141);(cid:19)/ be any p-
Œp(cid:141)
envelope of g. In general we suppress the notion of (cid:19), regard g as a subalgebra of
g and just say that g is a p-envelope of g. The p-envelope of q in g is the
Œp(cid:141) Œp(cid:141) Œp(cid:141)
restrictedsubalgebraofg generatedby(cid:19).q/andisdenotedbyq . ByDefinition
Œp(cid:141) Œp(cid:141)
1.2.1 the toral rank TR.q;g/ of q in g equals the maximum of dimensions of tori in
q CC.g /=C.g /. Dueto[S-F,Theorem2.4.5]thenTR.q;g/equalsthemaxi-
Œp(cid:141) Œp(cid:141) Œp(cid:141)
mumofdimt=t\C.g /fortorit(cid:26)q . Accordingly,theabsolutetoralrankofg,
Œp(cid:141) Œp(cid:141)
denotedbyTR.g/,equalsthemaximumofdimt=t\C.g /fortoriting . These
Œp(cid:141) Œp(cid:141)
definitionsdonotdependonthep-envelopechosen(Theorem1.1.7).
Supposegiscenterlessandg denotesaminimalp-envelope(Definition1.1.2).
Œp(cid:141)
Bydefinition,g iscenterlessaswell. IntheNotation1.2.5,TR.g/ D MT.g /is
Œp(cid:141) Œp(cid:141)
themaximalpossibledimensionoftoriing . Theorem1.1.7provesthattheminimal
Œp(cid:141)
p-envelopeisuniqueasarestrictedLiealgebra.
It is stated in Volume 1, p. 299 that every simple finite dimensional Lie algebra
of Cartan type can be viewed as an algebra X.mInI'/.1/ (cid:26) W.mIn/ for some
X D W;S;H;K and' 2 Aut O..m//satisfyingthecompatibilityproperty(6.2.2).
c
ThroughoutthisvolumewealwayswillwithoutfurthermentioningassumethatCar-
tantypeLiealgebrasX.mInI'/.1/ satisfythecompatibilityproperty(6.2.2).
ThealgebrasW.m/,S.m/,H.2r/andK.2rC1/carrynaturalgradings(Volume1,
p.186). Correspondingly,thecommutativealgebraO.m/andthespaceofdifferential
forms(cid:127).m/aregraded. EveryCartantypeLiealgebraX.mInI'/.1/ hasauniquely
determined subalgebra X.mInI'/.1/ of minimal codimension (Theorem 4.2.6)
.0/