Handbook of algebra

543 Pages·2006·2.407 MB·English

by  M. Hazewinkel

HANDBOOKOFALGEBRA
VOLUME4
ManagingEditor
M.HAZEWINKEL,Amsterdam
EditorialBoard
M.ARTIN,Cambridge
M.NAGATA,Okayama
C.PROCESI,Rome
R.G.SWAN,Chicago
P.M.COHN,London
A.DRESS,Bielefeld
J.TITS,Paris
N.J.A.SLOANE,MurrayHill
C.FAITH,NewBrunswick
S.I.AD’YAN,Moscow
Y.IHARA,Tokyo
L.SMALL,SanDiego
E.MANES,Amherst
I.G.MACDONALD,Oxford
M.MARCUS,SantaBarbara
L.A.BOKUT’,Novosibirsk
HANDBOOK OF ALGEBRA
Volume 4
edited by
M. HAZEWINKEL
CWI,Amsterdam
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•OXFORD
PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
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Preface
Basicphilosophy
Algebra,asweknowittoday(2005),consistsofagreatmanyideas,conceptsandresults.
A reasonable estimate of the number of these different “items” would be somewhere be-
tween 50000 and 200000. Many of these have been named and many more could (and
perhaps should) have a “name”, or other convenient designation. Even a nonspecialist is
quitelikelytoencountermostofthese,eithersomewhereinthepublishedliteratureinthe
formofanidea,definition,theorem,algorithm,...somewhere,ortohearaboutthem,of-
teninsomewhatvagueterms,andtofeeltheneedformoreinformation.Insuchacase,if
theconceptrelatestoalgebra,thenoneshouldbeabletofindsomethinginthisHandbook;
atleastenoughtojudgewhetheritisworththetroubletotrytofindoutmore.Inaddition
totheprimaryinformationthenumerousreferencestoimportantarticles,books,orlecture
notesshouldhelpthereaderfindoutmore.
As a further tool the index is perhaps more extensive than usual, and is definitely not
limitedtodefinitions,(famous)namedtheoremsandthelike.
ForthepurposesofthisHandbook,“algebra”ismoreorlessdefinedastheunionofthe
followingareasoftheMathematicsSubjectClassificationScheme:
–20 (Grouptheory)
–19 (K-theory; this will be treated at an intermediate level; a separate Handbook of
K-theorywhichgoesintofarmoredetailthanthesectionplannedforthisHandbook
ofAlgebraisunderconsideration)
–18 (Categorytheoryandhomologicalalgebra;includingsomeoftheusesofcategoryin
computerscience,oftenclassifiedsomewhereinsection68)
–17 (Nonassociativeringsandalgebras;especiallyLiealgebras)
–16 (Associativeringsandalgebras)
–15 (Linearandmultilinearalgebra,Matrixtheory)
–13 (Commutativeringsandalgebras;herethereisafinelinetotreadbetweencommu-
tative algebras and algebraic geometry; algebraic geometry is definitely not a topic
thatwillbedealtwithinthisHandbook;therewill,hopefully,onedaybeaseparate
Handbookonthattopic)
–12 (Fieldtheoryandpolynomials)
–11 Thepartofthatalsousedtobeclassifiedunder12(Algebraicnumbertheory)
–08 (Generalalgebraicsystems)
–06 (Certainparts;butnottopicsspecifictoBooleanalgebrasasthereisaseparatethree-
volumeHandbookofBooleanAlgebras)
v
vi Preface
Planning
Originally (1992), we expected to cover the whole field in a systematic way. Volume 1
wouldbedevotedtowhatisnowcalledSection1(seebelow),Volume2toSection2,and
so on.A quitedetailedandcomprehensiveplanwasmadeintermsof topicsthatneeded
to be covered and authors to be invited. That turned out to be an inefficient approach.
Differentauthorshavedifferentprioritiesandtowaitforthelastcontributiontoavolume,
asplannedoriginally,wouldhaveresultedinlongdelays.Insteadthereisnowadynamic
evolvingplan.Thisalsopermitstotakenewdevelopmentsintoaccount.
Chaptersarestillbyinvitationonlyaccordingtothethencurrentversionoftheplan,but
the various chapters are published as they arrive, allowingfor faster publication.Thus in
this Volume4 of the Handbookof Algebra the reader will find contributionsfrom 5 sec-
tions.
As the plan is dynamic suggestions from users, both as to topics that could or should
becovered,andauthors,aremostwelcomeandwillbegivenseriousconsiderationbythe
boardandeditor.
Thelistofsectionslooksasfollows:
Section1: Linearalgebra.Fields.Algebraicnumbertheory
Section2: Categorytheory.Homologicalandhomotopicalalgebra.Methodsfromlogic
(algebraicmodeltheory)
Section3: Commutativeandassociativeringsandalgebras
Section4: Otheralgebraicstructures.Nonassociativeringsandalgebras.Commutative
andassociativeringsandalgebraswithextrastructure
Section5: Groupsandsemigroups
Section6: Representationsandinvarianttheory
Section7: Machinecomputation.Algorithms.Tables
Section8: Appliedalgebra
Section9: Historyofalgebra
For the detailed plan (2005 version), the reader is referred to the Outline of the Series
followingthispreface.
Theindividualchapters
Itisnottheintentionthatthehandbookasawholecanalsobeasubstituteundergraduate
orevengraduate,textbook.Indeed,thetreatmentsofthevarioustopicswillbemuchtoo
dense and professional for that. Basically, the level should be graduate and up, and such
materialascanbefoundinP.M.Cohn’sthreevolumetextbook‘Algebra’(Wiley)should,
asarule,beassumedknown.ThemostimportantfunctionofthearticlesinthisHandbook
istoprovideprofessionalmathematiciansworkinginadifferentareawithasufficiencyof
informationonthetopicinquestionifandwhenitisneeded.
Each of the chapters combines some of the features of both a graduate level textbook
andaresearch-levelsurvey.Notalloftheingredientsmentionedbelowwillbeappropriate
ineachcase,butauthorshavebeenaskedtoincludethefollowing:
Preface vii
– Introduction(includingmotivationandhistoricalremarks)
– Outlineofthechapter
– Basicconcepts,definitions,andresults.(Thesemaybeaccompaniedbyproofsor(usu-
allybetter)ideas/sketchesoftheproofswhenspacepermits)
– Commentsontherelevanceoftheresults,relationstootherresults,andapplications
– Reviewoftherelevantliterature;possiblycompletewiththeopinionsoftheauthoron
recentdevelopmentsandfuturedirections
– Extensivebibliography(severalhundreditemswillnotbeexceptional)
Thepresent
Volume1appearedin1995(copyright1996),Volume2in2000,Volume3in2003.Vol-
ume5isplannedfor2006.Thereafter,weaimatonevolumeeverytwoyears(orbetter).
Thefuture
Ofcourse,ideally,acomprehensiveseriesofbookslikethisshouldbeinteractiveandhave
a hypertext structure to make finding material and navigation through it immediate and
intuitive.Itshouldalsoincorporatethevariousalgorithmsinimplementedformaswellas
permitacertainamountofdialoguewiththereader.Plansforsuchaninteractive,hypertext,
CDROM(DVD)-basedversioncertainlyexistbuttherealizationisstillanontrivialnumber
ofyearsinthefuture.
Kvoseliai,July2005 MichielHazewinkel
KaumnenntmandieDingebeimrichtigenNamen
soverlierensieihrengefährlichenZauber
(Youhavebuttoknowanobjectbyitspropername
forittoloseitsdangerousmagic)
EliasCanetti
This page intentionally left blank
Outline of the Series
(asofJuly2005)
PhilosophyandprinciplesoftheHandbookofAlgebra
ComparedtotheoutlineinVolume1thisversiondiffersinseveralaspects.
First, there is a major shift in emphasis away from completeness as far as the more
elementarymaterialisconcernedandtowardsmoreemphasisonrecentdevelopmentsand
activeareas.Second,theplanisnowmoredynamicinthatthereisnolongerafixedlistof
topicstobecovered,determinedlonginadvance.Insteadthereisamoreflexiblenonrigid
listthatcananddoeschangeinresponsetonewdevelopmentsandavailabilityofauthors.
The new policy, starting with Volume 2, is to work with a dynamic list of topics that
should be covered, to arrange these in sections and larger groups according to the major
divisionsintowhichalgebrafalls,andtopublishcollectionsofcontributions(i.e.chapters)
astheybecomeavailablefromtheinvitedauthors.
Thecodingbelowisbystyleandisasfollows.
– Author(s)inbold,followedbychaptertitle:articles(chapters)thathavebeenreceived
andarepublishedorarebeingpublishedinthisvolume.
– Chaptertitleinitalic:chaptersthatarebeingwritten.
– Chaptertitleinplaintext:topicsthatshouldbecoveredbutforwhichnoauthorhasyet
beendefinitelycontracted.
ChaptersthatareincludedinVolumes1–4havea(x;yypp.)afterthem,where‘x’isthe
volumenumberand‘yy’isthenumberofpages.
Compared to the plan that appeared in Volume 1 the section on “Representation and
invariant theory” has been thoroughly revised. The changes of this current version com-
paredtotheoneinVolume2(2000)andVolume3(2003)arerelativelyminor:mostlythe
additionofquiteafewtopics.
Editorialset-up
Managingeditor:M.Hazewinkel.
Editorial board: M. Artin, M. Nagata, C. Procesi, O. Tausky-Todd,† R.G. Swan,
P.M.Cohn,A.Dress,J.Tits,N.J.A.Sloane,C.Faith,S.I.Ad’yan,Y.Ihara,L.Small,
E.Manes,I.G.Macdonald,M.Marcus,L.A.Bokut’,Eliezer(LouisHalle)Rowen,
JohnS.Wilson,VlastimilDlab.Notethatthreeeditorshavebeenaddedstartingwith
Volume5.
Plannedpublishingschedule(asofJuly2005)
1996:Volume1(published)
2001:Volume2(published)
2003:Volume3(published)
ix