Simple Lie Algebras over Fields of Positive Characteristic. I. Structure Theory

551 Pages·2017·2.466 MB·English

by  Helmut Strade

HelmutStrade
SimpleLieAlgebrasoverFieldsofPositiveCharacteristic
De Gruyter Expositions
in Mathematics
Editedby
LevBirbrair,Fortaleza,Brazil
VictorP.Maslov,Moscow,Russia
WalterD.Neumann,NewYorkCity,NewYork,USA
MarkusJ.Pflaum,Boulder,Colorado,USA
DierkSchleicher,Bremen,Germany
Volume 38
Helmut Strade
Simple Lie Algebras
over Fields of Positive
Characteristic
Volume I: Structure Theory
2nd Edition
MathematicsSubjectClassification2010
17-02;17B50,17B20,17B05
Author
Prof.Dr.HelmutStrade
MarmstorferWeg124
21077Hamburg
Germany
[email protected]
ISBN978-3-11-051516-9
e-ISBN(PDF)978-3-11-051544-2
e-ISBN(EPUB)978-3-11-051523-7
Set-ISBN978-3-11-051545-9
ISSN0938-6572
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Contents
Introduction 1
1 Toralsubalgebrasinp-envelopes 17
1.1 p-envelopes 17
1.2 Theabsolutetoralrank 23
1.3 Extendedroots 30
1.4 Absolutetoralranksofparametrizedfamilies 39
1.5 Toralswitching 46
2 Liealgebrasofspecialderivations 58
2.1 Dividedpowermappings 59
2.2 Subalgebrasdefinedbyflags 73
2.3 TransitiveembeddingsofLiealgebras 79
2.4 Automorphismsandderivations 89
2.5 Filtrationsandgradations 91
2.6 MinimalembeddingsoffilteredandassociatedgradedLiealgebras 99
2.7 Miscellaneous 104
2.8 Auniversalembedding 111
2.9 Theconstructionscanbemadebasisfree 119
3 Derivationsimplealgebrasandmodules 133
3.1 Frobeniusextensions 134
3.2 Inducedmodules 138
3.3 Block’stheorems 151
3.4 Derivationsemisimpleassociativealgebras 163
3.5 Weisfeiler’stheorems 167
3.6 Conjugacyclassesoftori 176
4 SimpleLiealgebras 180
4.1 ClassicalLiealgebras 180
4.2 LiealgebrasofCartantype 184
4.3 Melikianalgebras 199
4.4 SimpleLiealgebrasincharacteristic3 209
viii Contents
5 Recognitiontheorems 217
5.1 Cohomologygroups 217
5.2 FromlocaltoglobalLiealgebras 228
5.3 Representations 252
5.4 GeneratingMelikianalgebras 258
5.5 TheWeakRecognitionTheorem 262
5.6 TheRecognitionTheorem 269
5.7 Wilson’sTheorem 272
6 Theisomorphismproblem 283
6.1 Afirstattack 283
6.2 Thecompatibilityproperty 295
6.3 Specialalgebras 299
6.4 OrbitsofHamiltonianforms 314
6.5 Hamiltonianalgebras 329
6.6 Contactalgebras 346
6.7 Melikianalgebras 349
7 StructureofsimpleLiealgebras 357
7.1 Derivations 357
7.2 Restrictedness 363
7.3 Automorphisms 372
7.4 Gradings 386
7.5 Tori 388
7.6 W(1;n) 420
8 Pairingsofinducedmodules 432
8.1 Cartanprolongation 432
8.2 Modulepairings 449
8.3 Trigonalizability 461
9 Toralrank1Liealgebras 484
9.1 Solvablemaximalsubalgebras 484
9.2 Cartansubalgebrasoftoralrank1 496
Notation 521
Bibliography 527
Index 539
Introduction
ThetheoryoffinitedimensionalLiealgebrasoverfieldsF ofpositivecharacteristicp
wasinitiatedbyE.Witt,N.Jacobson[Jac37]andH.Zasssenhaus[Zas39]. Sometime
before 1937 E.Witt came up with the following example of a simple Lie algebra of
dimensionp (forp > 3),afterwardsnamedtheWittalgebraW(1;1). Onthevector
(cid:29)
p−2
space Fe definetheLieproduct
i=−1 i
(cid:15)
[e ,e ]:= (j −i)ei+j if −1≤i+j ≤p−2,
i j
0 otherwise.
This algebra behaves completely different from those algebras we know in charac-
t eristic0. Asanexample,itcontainsauniquesubalgebraofcodimension1,namely
Fe . Italsohassandwichelements,i.e.,elementsc (cid:19)=0satisfying(adc)2 =0
i≥0 i
(forexample,ep−2). E.Witthimselfneverpublishedthisexampleorgeneralizations
ofit,whichhepresumablyknewof.Atthattimehewasinterestedinthesearchfornew
finitesimplegroups. Whenherealizedthatthesenewstructureshadonlyknownau-
tomorphismgroupsheapparentlylosthisinterestinthesealgebras. Wehaveonlyoral
andindirectinformationofWitt’sworkonthisfieldbytwopublicationsofH.Zassen-
haus[Zas39]andChangHoYu[Cha41]. Changdeterminedtheautomorphismsand
irreduciblerepresentationsofW(1;1)overalgebraicallyclosedfields. Healsomen-
tionedthatWitthimselfgavearealizationofW(1;1)intermsoftruncatedpolynomial
rings. Namely,W(1;1)isisomorphicwiththevectorspaceF[X]/(Xp)endowedwith
the product {f,g} :=fd/dx(g) −gd/dx(f) for all f,g ∈ F[X]/(Xp) under the
mappinge (cid:20)→xi+1,wherex =X+(Xp).
i
In[Jac37]N.JacobsonprovedaGaloistypetheoremforinseparablefieldexten-
sionsbysubstitutingthealgebraofderivationsfortheautomorphismgroupofafield
extension. Moreexplicitly,hewasabletoshowthatthesetofintermediatefieldsofa
fieldextensionF(c ,...,c ):F withcp ∈F isinbijectionwiththesetofthoseLie
1 n i
subalgebrasofDer F(c ,...,c ),whichareF(c ,...,c )-modulesandareclosed
F 1 n 1 n
underthep-powermappingD (cid:20)→Dp.AtthatearlytimeJacobsonalreadyintroduced
the term “restrictable” for those Lie algebras, which admit a p-mapping x (cid:20)→ x[p]
satisfyingtheequationadx[p] =(adx)p forallx. Laterhepreferredtousetheterm
“restrictedLiealgebra”forpairs(L,[p]),whensuchap-mappingisfixed. TheLie
algebrasoflinearalgebraicgroupsoverF areallequippedwithanaturalp-mapping,
hencetheycarrycanonicalrestrictedLiealgebrastructures.