Table Of ContentAN, Parshin IR. Shafarevich (Eds)
Algebraic Geometry IV
Encyclopaedia of
Mathematical Sciences
Volume 35
Fditor-in-Chiel R.¥, Gamrelidze
List of Fairs, Authors snl Translators,
2. Ghee, Ranson Asissany of Soames, Slee Mahila te
i, Waite 2. 1 oe Mou, ie Tor Seew resis INU
Css Sa, 252 Moseoes asa
8.%.Bosin, Sleloy Matias anny, (1 Yorba #2, 1 8H6 Mes
L.A Shatasish, Sek ne Mahomet ul avila 42
Pom Btncr uss
VL Page Cis et Aggbre, Depicinan ol Malema, VPM. Keds
‘eval i, ES Mone Ria
Trae Spina Blancas sna Kijbststcick Tea, Poses S00,
SSUB TA Lech Ihe Mathers
1 Vinhora Cha: of Alger, Moscow Slip Univemay, SN Moye,
Rast
Tm
yok Part
(4.8, Kall, 228 Ss), dan, OP M689 1232. USA
Contcots
Lmear Ateebeale Groups
TA Snare
ML anvaraot Theory
Va Popus 4 Visbere
Aut nes
Sujeet dex
2
I. Linear Algebraic Groups
A. Springer
Contents
Lunstascon 4
‘istaien, Comers 4
(Chapter 1. Linear Algerie Groups ver a0 Alyeeraicaly
Cloyd Fld 2
5. Rewiletions rons Algsbeane Geometry
Li. Affine Pare
12, Morphisms
18, Some Topological Propertise
Te, Tangent Spaces
1S. Progertien uf Moras
Ls NoweAdine Vi
2 Linear Algeorae iraups Basie Befinione and Propetin
ZL, The DeGaition of¢ Lizzor Algebra Group
22. Sune Basie Fists
5 GSpaccs
Sa Tue le Algsbra a aw Algbesie Group
25, Guotions
44, Struclecal epperties of Linear Algbeate Groups
2 fordsn Decurmpeniton and Kelas Resets
22, Dingemalicable Groups and Ton
33, Qno-Dimensivral Conneetal Groups
34, Connected Saivable Groups
2.5, Borehole Subgroups and Bore Sebyvurs
Ea, Rdiaab, Sermesmpls aad Reduscce Groups
1A, Reduetne Crags
A. Gone sf snk One
82, The Koct Pym and rs Root Sten,
|2), Ratie Properties of Redueive Groupe
|S4, Faistseoe nd Unnyoenese Theaters for ReVuctve Grom
45, Classifiation of Qussrsinple Treat algebraic Grows
46, Representation Theory
‘Caupte 2. Linear Algeheai Genus evo: Arhicaty Grobu els
oslo from Alyce Ganon ry
1. F-Seroceese em Allies Westie
12 PSeruscess en Arhinary Pai,
(B. bone
US, Reston of te Growl Fila
$2. EGreupa Banc Propoetis
21, Ceweruliies Sbour F-Croups
32. Queweats
23, fim:
24, Resinetaon of the Grovnd Hed
$3 Tos
sLrten
32. Tain Fine
VE Spit Tori wa Groups
Solvable Groups
BI Soleibhe Groups
52, Sections
23. Bementony Taipotert Groupe
A Proponics of Split Solvable Grup.
5, Base Resuhs ona Saabs
Redustine Crags
AD SplitRalyelive Groops
52, Panabidie Subgroups
BA Ube Small Root Sten,
Bal The Grove Gl
5. Tae Spherical‘ llling ofa Recuctive F-Giraup
Clisiisalion of Reductive F-Uroo
FA omorphisn Theorem
12. Caience
AS. Representation Theory of Groupe
(Chapter 3. Sposa Filey
$1 Lie Albros of Algshvais Gn in Chasaseris ere
UL Algtncte Susalyebias
$2 Alesbrale Groups aed Tie Giouye
Ee Leselly Compact Hise
BE. Real Lie Grou
3, Linsar Aigsoraie Gag mer Fit ies
En, Langs Theoten and iis Comsat
BE. Fuk: Gieuys ef Lie Type
23, Reproetatins nf Fhe Gass of Lis Tae
54, Linew Algebraic Groups owe Fide wh Valution
4. The Apariansne and Alfias Dyn Tagua
ys
42, The Aine Building
43, Tit Sor, Desongitons
44, Local Fills
‘onal elds
EL, Adkde Groups|
52, Ralualvt Mews
SA. Finileness Revils
BAL Galois Guhiwroas
eters
Introduction
|S ocac abeshese grvap uve an slgghtaiclle cose eld £6 suhqon9
sfayruen Glia! overlie» « anes with evtsies By olwee caries
sapiens the sets ose! of polenunial uations it the is cour
rates Thy prea ate ane tas AP he Coss a Gem alps
een,
“Ts devel ca outs gr aces aon agian goaevity The real
Ieanumal to have sore Sinaary wh at sly,
acter 1 of Ue aie event lass Bly fos the ube of Fee
bruc groups over an slpebracally coved feld& This teary culminates
‘elssticeacn afsimple neu algchrac groops. Tavs tied re explain castuD
‘he dundainenta ean and sels, oo Mustats “erm Wit concTete eames.
and ts gle same ide ofthe restate peat
Tere ae several moval he aerial vf de chaptes[B2. [lh
iSp2 i nze Ue incensed rencer eo fn ce deta liso Ts era.
‘Chxpier 2 discsses the relive tagvis. hye 4 Feld of defn ones
Tow play This i emg a suns 9f sch tht the polyol eqeaisns
9 the Best Line can be ake to bve evelcoane mF is eb User
Tesuied for exo. Hone wists 1 deal wil ati melact questvns iat
ingalgebuve 00
‘he erent thee de nat exis. monoseaphs sovssing Us thoorg. which
reakes i Tee secon. Cave trict to proent u eahsten: pean elon the
same lise ae i Cher I
Te Chaprer aria fsrrtatpaicnnc af he tlie tay, fv patie
lon oud dewion foley fit, lal ane global elds The ai ot the
trap el show ot the ery a algebrne groupe vy ort abso,
sch peal el
Dene isa gre aburckete mere Kisaane of Heaney |
have semeives boon aie sheehy.*
Tg gelecenies the eal the sul £0 Mave Ibe ptemeum of ean
oroplee, Ha Chops ath the Help of her eae il be le eae
Ure esa are ether detil, of ees todo se
‘A elsrenea ip he ate to 1, 2.3.4 rsp, 2.345 reco wo. 2.34 ol Chapter
ff fae satne Chapter,
Historical Comments
1 ager ages etn re full
is thigh have bos iesonphrate 0 cane
yyy trod tiow U9
1 bist wie of anterior devel
ven or another, the chcory ef near elgsbraie groups, er we ave nly
ee Thal hoor,
Fist thre is the study of concrete Unsar groups, Gas alteady intndinced
the group PGL,i0y] of Eucaomal averse esr mpi elas bier 1?)
Df the nine fd FE, An exteosins study othe geese! Hincareroupe ove suc
1 field fin ang Aitension) ara related “chases” arcu ive Orta? ones)
‘Resmmats he. Joedan in Tstiin hihnk "Trai dss setting) This was
tinued hy TB. Bigkwsv ansata 1911 ane hy 1. Diedonné ajouad 1950
These nuthees sly granettcorctieal qnsstnns. wach as Ae detorannatind oF
wom byes fo esa wm
1 lundieark in thie dovslopnisel & C. Cheealess paper “Sar osstaias
ssstape gly” FF ahoky Moth 1, 1955, LAG im when Ts thz01¥ ra
npewcrce. He cis acs, ov aay simple“ alyelna ove) Hae Sones Feld,
{corresponding linzor group over uy fila F und ie discusses Laer grou
"ioorerzal progesties ‘Phe standard aascical groups arespecal cues
Theifemars. fnndan’s hoek — mcteinned akase ~ canine 9 version at
Jordan's xauital fers a tteces ‘he Tort: decnpesition in linet alps beat
‘oupy see | 3.1pisa desea
Linear elgebric groups aver the fed of coaplen aumnwer apprar 22
Pears work Ghos then alineur diersntiat equations iaronm 1485, se
nic agit” rytlionsditersalslesFinetrase es groupes algeoriques de
famatiags”. Qaueres TT 1-139), An exaupls ofthe questions studied by him
sh flensieg Crier 9 de hownngeneou iar eel agua oh
nthe carples place
‘vith polynomal egeticiass 2, One knows thit the sveqyebets ho.omarphne
sslions form a ndinoasional comple vectoe sptse, Hf ny fy) 924 Bai
Tel obs sid ld of mecamsphie fant inne suai hy adeine
ing td fal C2 of econ Funetio the Janta the’ deivalives Denote
bere the group of i lincan sutamung hams uff eh eumvute with die
tion. This the Calois group of che equation. Vicon aim ie la develop @
‘Galou theory Ij € 6 there ens vouples nurabers sch Us
a-— ¥ salt
and Brean! shows tha the mutes i fel] = GLI form» incur ulyeaie
[roup over C2 ssomorshie to G He adits to be tbe Ret to veo nae ike
“Ugeeric gro 1p"
‘This Galas theory a Gteralgebrgized tad Gurther mwlopel by Rit
‘aroun 831) aud Koken, The work 3 se ler of 1948 fees poper “Or:
fertuin coacepis in the theory of agebraie mall groupe, Aux, of Math, 38,
TFL 780) contain renlls sich are now bisne once athe Iboory of hess
algehraie groups. sch asthe properties ofthe denity compomer 2.2.2) and
hs Tie Kein heaven 3-41 hie states that a soneeeted soleabinoar
lgobeuic group ca. be Isangul-zod. This extend a result of ie ioe eng es
salable Lie sigebros, In contnss ¢otheluter stl the Lit-Keoktnn shearer
‘us in any eharactrin.
A. Wells werk on Jooobiaas of algbrase curs foe his book “Varidés
abslienaeser sours algibriques, 8K) led hin “0 wstndy of poner lg beac
ergs. algsbrai ratllon with a group stisctoe aon bs rnorisne 1 the
Setar of algebraie geometry Te inert Was imac in abstion vies
onmeetod algeuane gonps which az projetve etc fin otich the 0s
euccure qulomatiall conmtative, Classe abelin voreties ove
‘were studied hy wansseadentl raehow which yo nick to Kier
‘We and others exabshed somewhat later besis genera facts abou: uo
ies oan avbitaryaigslv ne arouy hy agen wbyroup. They are indie
perl ing aiont or the ago ofeach ae gers
The theury of fear sebrac props oats oval by A. Hore in 1936
[rCraupes tins algéorques". an. of Math 6420-82). ie work as come
plevea hy Chovalloy(*CTaaicasian de groupes de Tie algriques”. Seminar
Kecole Nerials Suméieose, lish 1983). TH Rah= ark the inBuches. oF
[oletn’s urs aided ator, cel? ssile, Antec esse cme is
(ie analogy wit the thew yf Liege inset” fan An itis
approic To les algebra sous wa Lealzebia bnsutable in charxenrs
Uong she gloho! msthode of ateebruc grorcetey, Gurel establised basic
cealty euch as senhigacy thsoven for owimal inane Borel subgroups
ILSS4.1.3 51) Ju aba theve he prove fixe pain! thagten (1431 hick
sereniias the Li-Kolehi theerem,matiogel before, Chevalley show
Svaleques of rxulsertatshed in Le thsory se the hela ofthe Lie
Sir he alvaina ith glob! mothnds lor sxaniphe reals abeut radicals, so
4726) The min tonal his Sennaire ie chat ths eassiicain cf imple Hisar
feshaie groups aver an algebraically chyed field of any shurasterists, icons
Dele analons tothe sins Tation of smple Lie wlgches ave the Held!
omplex nambors.
Tne work of Borel wz Caevele tbe ntueae of sal sess eum
sheory Wf Tie ups has heer songierle Caisse next, Chapler Vo the
sebcle ia seven uf he work wf Bel ed Chea
Firally moalionsbould ie eage of sone geueralzations of lyebeus wroups
sich wetiave not ~ er hardly ~tenekod upon Fist bere ase tbe group chert,
Studied estessivaly 97 Greulrdiack and bis eolahsrators (M_ Destazre and
A. Grothenit Schénes e grupey Jacl Nts ia Bath. mes. 18, 152 18
167, In ths ule bey appeue in only afew Blucos More seent generalizations
uy the gnantum groups, Which ze alsbniisgecuge in “nia-commautaite ge
omerzy” Wohase ool yin the definition ga 21. We have no: sid anything
bout “lations ducensina:”eleebeic gras, sich ax Kas-beody soups
