Handbook of Algebra, Volume 3 (Handbook of Algebra)

1185 Pages·2003·5.37 MB·English

by  Michiel Hazewinkel

HANDBOOKOFALGEBRA
VOLUME3
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
ELSEVIER
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•
OXFORD•PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
HANDBOOK OF ALGEBRA
Volume 3
edited by
M. HAZEWINKEL
CWI,Amsterdam
2003
ELSEVIER
AMSTERDAM•BOSTON•HEIDELBERG•LONDON•NEWYORK•
OXFORD•PARIS•SANDIEGO•SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
ELSEVIERSCIENCEB.V.
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ISBN:0444512640
ISSN:15707954
(cid:2)∞ThepaperusedinthispublicationmeetstherequirementsofANSI/NISOZ39.48-1992(PermanenceofPaper).
PrintedinTheNetherlands.
Preface
Basicphilosophy
Algebra,asweknowittoday,consistsofagreatmanyideas,conceptsandresults.Area-
sonable estimate of the numberof these different“items” would be somewherebetween
50000and200000.Manyofthesehavebeennamedandmanymorecould(andperhaps
should) have a “name”, or other convenient designation. Even the nonspecialist is quite
likelytoencountermostofthese,eithersomewhereintheliteraturedistinguisedasadef-
initionoratheoremortohearaboutthemandfeeltheneedformoreinformation.Ifthis
happens, one should be able to find at least something in this Handbook;and hopefully
enough to judge whether it is worthwhile to pursue the quest at least. In addition to the
primaryinformationreferencestorelevantarticles,booksorlecturenotesshouldhelpthe
readertocompletehisunderstanding.
To make this possible we have providedan indexwhich is more extensivethan usual,
andnotlimitedtodefinitions,theoremsandthelike.
ForthepurposesofthisHandbook,algebrahasbeendefinedmoreorlessarbitrarilyas
theunionofthefollowingareasoftheMathematicsSubjectClassificationScheme:
–20 (Grouptheory)
–19 (K-theory;thiswillbetreatedatanintermediatelevel)
–18 (Categorytheoryandhomologicalalgebra;includingsomeoftheusesofcategoryin
computerscience,oftenclassifiedsomewhereinsection68)
–17 (Nonassociativeringsandalgebras;especiallyLiealgebras)
–16 (Associativeringsandalgebras)
–15 (Linearandmultilinearalgebra,Matrixtheory)
–13 (Commutativeringsandalgebras;herethereisafinelinetotreadbetweencommu-
tative algebrasand algebraicgeometry;algebraic geometryis definitely nota 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
soon.Adetailedandcomprehensiveplanwasmadeintermsoftopicswhichneededtobe
coveredandauthorstobeinvited.Thatturnedouttobeaninefficientapproach.Different
authorshavedifferentprioritiesandtowaitforthelastcontributiontoavolume,asplanned
originally,wouldhaveresultedinlongdelays.Therefore,wehaveoptedforadynamically
evolvingplan.Thisalsopermitstotakenewdevelopmentsintoaccount.
This means that articles are published as they arrive and that the reader will find in
thisthird volumearticles fromfive differentsections. The advantagesof thisscheme are
two-fold:acceptedarticles will be publishedquicklyand the outline of the series can be
allowedtoevolveasthevariousvolumesarepublished.Suggestionsfromreadersbothas
to topics to be covered and authors to be invited are most welcome and will taken into
seriousconsideration.
Thelistofthesectionsnowlooksasfollows:
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
Foramoredetailedplan(2002version),thereaderisreferredtotheOutlineoftheSeries
followingthispreface.
Theindividualchapters
Itisnottheintentionthatthehandbookasawholecanalsobeasubstituteundergraduate
orevengraduate,textbook.Thetreatmentofthevarioustopicswillbemuchtoodenseand
professionalfor that. Basically, the levelis graduateand up,and such materialas can be
foundinP.M.Cohn’sthreevolumetextbook“Algebra”(Wiley)will,asarule,beassumed.
An importantfunctionof the articles in this Handbookis to provideprofessionalmathe-
maticiansworkinginadifferentareawithsufficientinformationonthetopicinquestionif
andwhenitisneeded.
Each chapter combines some of the features of both a graduate-level textbook and a
research-level survey. Not all of the ingredients mentioned below will be appropriate in
eachcase,butauthorshavebeenaskedtoincludethefollowing:
Preface vii
– Introduction(includingmotivationandhistoricalremarks)
– Outlineofthechapter
– Basicconcepts,definitions,andresults(proofsorideas/sketchesoftheproofsaregiven
whenspacepermits)
– Commentsontherelevanceoftheresults,relationstootherresults,andapplications
– Reviewoftherelevantliterature;possiblysupplementedwiththeopinionoftheauthor
onrecentdevelopmentsandfuturedirections
– Extensivebibliography(severalhundreditemswillnotbeexceptional)
Thefuture
Of course, ideally, a comprehensive series of books like this should be interactive and
have a hypertextstructure to make finding material and navigationthrough it immediate
and intuitive. It should also incorporate the various algorithms in implemented form as
wellaspermitacertainamountofdialoguewiththereader.Plansforsuchaninteractive,
hypertext, CD-Rom-based version certainly exist but the realization is still a nontrivial
numberofyearsinthefuture.
Kvoseliai,July2003 MichielHazewinkel
KaumnenntmandieDingebeimrichtigenNamen,
soverlierensieihrengefährlichenZauber
(Youhavebuttoknowanobjectbyitspropername
forittoloseitsdangerousmagic)
E.Canetti
This Page Intentionally Left Blank
Outline of the Series
(asofJune2002)
PhilosophyandprinciplesoftheHandbookofAlgebra
ComparedtotheoutlineinVolume1thisversiondiffersinseveralaspects.
First,thereisamajorshiftinemphasisawayfromcompletenessasfarasmoreelemen-
tarymaterialisconcernedandtowardsmoreemphasisonrecentdevelopmentsandactive
areas.
Second,theplanisnowmoredynamicinthatthereisnolongerafixedlistoftopicsto
becovered,determinedlonginadvance.Insteadthereisamoreflexiblenonrigidlistthat
cananddoeschangeinresponsetonewdevelopmentsandavailabilityofauthors.
Thenewpolicyistoworkwithadynamiclistoftopicsthatshouldbecovered,toarrange
these in sections and larger groups according to the major divisions into which algebra
falls,andtopublishcollectionsofcontributionsastheybecomeavailablefromtheinvited
authors.
Thecodingbystylebelowisasfollows.
– Author(s)inbold,followedbychaptertitle:articles(chapters)thathavebeenreceived
andarepublishedorreadyforpublication.
– Chaptertitleinitalic:chaptersthatarebeingwritten.
– Chaptertitleinplaintext:topicsthatshouldbecoveredbutforwhichnoauthorhasyet
beendefinitelycontracted.
– ChaptersthatareincludedinVolumes1–3havea(x;yypp.)afterthem,where‘x’isthe
volumenumberand‘yy’isthenumberofpages.
ComparedtotheplanthatappearedinVolume1thesectionon“Representationandin-
varianttheory”hasbeenthoroughlyrevised.Thechangesofthiscurrentversioncompared
totheoneinVolume2(2000)arerelativelyminor:mostlytheadditionofsome5topics.
Section1.Linearalgebra.Fields.Algebraicnumbertheory
A. LinearAlgebra
G.P.Egorychev,VanderWaerdenconjectureandapplications(1;22pp.)
V.L.Girko,Randommatrices(1;52pp.)
A.N.Malyshev,Matrixequations.Factorizationofmatrices(1;38pp.)
L.Rodman,Matrixfunctions(1;38pp.)
CorrectiontothechapterbyL.Rodman,Matrixfunctions(3;1p.)
ix