Representations of Lie Algebras: An Introduction Through gl(n)

166 Pages·2012·2.212 MB·English

by  Anthony Henderson

Representations of Lie Algebras
ThisboldandrefreshingapproachtoLiealgebrasassumesonlymodestprerequisites
(linear algebra up to the Jordan canonical form and a basic familiarity with groups
andrings),yetitreachesamajorresultinrepresentationtheory:thehighest-weight
classificationofirreduciblemodulesofthegenerallinearLiealgebra.Theauthor’s
exposition is focused on this goal rather than on aiming at the widest generality,
and emphasis is placed on explicit calculations with bases and matrices. The book
beginswithamotivatingchapterexplainingthecontextandrelevanceofLiealgebras
and their representations and concludes with a guide to further reading. Numerous
examplesandexerciseswithfullsolutionsareincluded.
Basedontheauthor’sownintroductorycourseonLiealgebras,thisbookhasbeen
thoroughlyroad-testedbyadvancedundergraduateandbeginninggraduatestudents
andisalsosuitedtoindividualreaderswantinganintroductiontothisimportantarea
ofmathematics.
https://doi.org/10.1017/CBO9781139236126
AUSTRALIANMATHEMATICALSOCIETYLECTURESERIES
Editor-in-Chief:ProfessorC.Praeger,SchoolofMathematicsandStatistics,
UniversityofWesternAustralia,Crawley,WA6009,Australia
Editors:
ProfessorP.Broadbridge,SchoolofEngineeringandMathematicalSciences,LaTrobeUniversity,
Victoria3086,Australia
ProfessorM.Murray,SchoolofMathematicalSciences,UniversityofAdelaide,SA5005,Australia
ProfessorC.E.M.Pearce,SchoolofMathematicalSciences,UniversityofAdelaide,
SA5005,Australia
ProfessorM.Wand,SchoolofMathematicalSciences,UniversityofTechnology,
Sydney,NSW2007,Australia
1 IntroductiontoLinearandConvexProgramming,N.CAMERON
2 ManifoldsandMechanics,A.JONES,A.GRAY&R.HUTTON
3 IntroductiontotheAnalysisofMetricSpaces,J.R.GILES
4 AnIntroductiontoMathematicalPhysiologyandBiology,J.MAZUMDAR
5 2-KnotsandtheirGroups,J.HILLMAN
6 TheMathematicsofProjectilesinSport,N.DEMESTRE
7 ThePetersenGraph,D.A.HOLTON&J.SHEEHAN
8 LowRankRepresentationsandGraphsforSporadicGroups,
C.E.PRAEGER&L.H.SOICHER
9 AlgebraicGroupsandLieGroups,G.I.LEHRER(ed.)
10 ModellingwithDifferentialandDifferenceEquations,
G.FULFORD,P.FORRESTER&A.JONES
11 GeometricAnalysisandLieTheoryinMathematicsandPhysics,
A.L.CAREY&M.K.MURRAY(eds.)
12 FoundationsofConvexGeometry,W.A.COPPEL
13 IntroductiontotheAnalysisofNormedLinearSpaces,J.R.GILES
14 Integral:AnEasyApproachafterKurzweilandHenstock,
L.P.YEE&R.VYBORNY
15 GeometricApproachestoDifferentialEquations,
P.J.VASSILIOU&I.G.LISLE(eds.)
16 IndustrialMathematics,G.R.FULFORD&P.BROADBRIDGE
17 ACourseinModernAnalysisanditsApplications,G.COHEN
18 Chaos:AMathematicalIntroduction,J.BANKS,V.DRAGAN&A.JONES
19 QuantumGroups,R.STREET
20 UnitaryReflectionGroups,G.I.LEHRER&D.E.TAYLOR
21 LecturesonRealAnalysis,F.LÁRUSSON
AustralianMathematicalSocietyLectureSeries:22
Representations of Lie Algebras
An Introduction Through gl
n
ANTHONY HENDERSON
SchoolofMathematicsandStatistics
UniversityofSydney
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,
NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107653610
(cid:2)c AnthonyHenderson2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Henderson,Anthony,1976-author.
RepresentationsofLiealgebras:anintroductionthroughgln/AnthonyHenderson,Schoolof
MathematicsandStatistics,UniversityofSydney.
pages cm.–(AustralianMathematicalSocietylectureseries;22)
ISBN978-1-107-65361-0(pbk.)
1. RepresentationsofLiealgebras. I. Title.
QA252.3.H46 2012
512(cid:2).482–dc23
2012021841
ISBN978-1-107-65361-0Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface pagevii
Notationalconventions ix
1 Motivation:representationsofLiegroups 1
1.1 Homomorphismsofgenerallineargroups 1
1.2 Multilinearalgebra 3
1.3 Linearizationoftheproblem 7
1.4 Lie’stheorem 10
2 DefinitionofaLiealgebra 13
2.1 Definitionandfirstexamples 13
2.2 Classificationandisomorphisms 15
2.3 Exercises 18
3 BasicstructureofaLiealgebra 21
3.1 Liesubalgebras 21
3.2 Ideals 24
3.3 QuotientsandsimpleLiealgebras 27
3.4 Exercises 31
4 ModulesoveraLiealgebra 33
4.1 Definitionofamodule 33
4.2 Isomorphismofmodules 38
4.3 Submodulesandirreduciblemodules 41
4.4 Completereducibility 44
4.5 Exercises 47
5 Thetheoryofsl -modules 50
2
5.1 Classificationofirreducibles 50
5.2 Completereducibility 55
5.3 Exercises 57
v
vi Contents
6 Generaltheoryofmodules 60
6.1 Dualsandtensorproducts 60
6.2 Hom-spacesandbilinearforms 65
6.3 Schur’slemmaandtheKillingform 68
6.4 Casimiroperators 72
6.5 Exercises 75
7 Integralgl -modules 78
n
7.1 Integralweights 78
7.2 Highest-weightmodules 86
7.3 Irreducibilityofhighest-weightmodules 90
7.4 Tensor-productconstructionofirreducibles 93
7.5 Completereducibility 99
7.6 Exercises 104
8 Guidetofurtherreading 106
8.1 ClassificationofsimpleLiealgebras 106
8.2 RepresentationsofsimpleLiealgebras 109
8.3 Charactersandbasesofrepresentations 111
Appendix Solutionstotheexercises 115
SolutionsforChapter2exercises 115
SolutionsforChapter3exercises 122
SolutionsforChapter4exercises 127
SolutionsforChapter5exercises 132
SolutionsforChapter6exercises 137
SolutionsforChapter7exercises 144
References 153
Index 154
Preface
Theaimofthisbook
Why another introduction to Lie algebras? The subject of this book is one of the
areas of algebra that has been most written about. The basic theory was unearthed
morethanacenturyagoandhasbeenpolishedinalongchainoftextbookstoasheen
ofclassicalperfection.Experts’shelvesaregracedbythethreevolumesofBourbaki
[1]; for students with the right background and motivation to learn from them, the
expositions in the books by Humphreys [10], Fulton and Harris [6], and Carter [2]
could hardly be bettered; and there is a recent undergraduate-level introduction by
ErdmannandWildon[4].Sowhereistheneedforthisbook?
TheanswercomesfrommyownexperienceinteachingcoursesonLiealgebrasto
Australianhonours-levelundergraduates(seetheAcknowledgementssection).Such
coursestypicallyconsistof24one-hourlectures.Atmyownuniversitythealgebraic
background knowledge of the students would be: linear algebra up to the Jordan
canonicalform,thebasictheoryofgroupsandrings,therudimentsofgrouprepre-
sentationtheory,andalittlemultilinearalgebrainthecontextofdifferentialforms.
From that starting point, I have found it difficult to reach any peak of the theory
byfollowingtheconventionalroute.Mydefinitionofapeakincludestheclassifica-
tion of simple Lie algebras, the highest-weight classification of their modules, and
thecombinatoricsofcharacters,tensorproducts,andcrystalbases;by‘theconven-
tional route’ I mean the path signposted by the theorems of Engel and Lie (about
solvability), Cartan (about the Killing form), Weyl (about complete reducibility),
andSerre,asinthebookbyHumphreys[10].Followingthatpathwithoutskipping
proofsalwaysseemedtorequiremorethan24lectures.
Thesolutionadoptedinthisbookisdrastic.Ihaveabandonedthewiderclassof
simpleLiealgebras,focusinginsteadonthegenerallinearLiealgebragl ,whichis
n
almost, but not quite, simple. I have jettisoned all five of the aforementioned theo-
rems, in favour of arguments specific to gl , especially the use of explicit Casimir
n
operators.Althoughtheseomissionsmayshocktheexperts,Ihavefoundthistobe
anapproachthatismoreaccessibleandyetstillreachesonepeak:theclassification
ofgl -modulesbytheirhighestweights.
n
vii
viii Preface
Ihavestartedthejourneywithamotivatorychapter,whichgivessomeexplanation
ofwhyalgebraistscareaboutthisclassificationandalsointroducessomenecessary
multilinearalgebra.Chapters2to4coverthebasicdefinitionsofLiealgebras,homo-
morphismsandisomorphisms,subalgebras,ideals,quotients,modules,irreducibility
andcompletereducibility.Inalecturecourse,thematerialinthesefirstfourchapters
wouldtypicallytakeabout12hours;sotheelegantsl theoryinChapter5isreached
2
relativelyearly.TheninChapter6Ireturntothetheoryofmodules,coveringtensor
products,bilinearforms,Schur’slemma,andCasimiroperators.
InChapter 7thesetoolsareused todevelop thehighest-weight theory. Myhope
isthatstudentswhoreachtheendofChapter7willbeinspiredtoprogresstomore
comprehensivebooks,andChapter8isintendedasamapofwhatliesahead.
Acknowledgements
This book began life as a set of lecture notes for my Lie algebras course in the
2004 Australian Mathematical Sciences Institute (AMSI) Summer School. It was
extensively revised over the next seven years, as I taught the subject again for the
summer school and as an honours course at the University of Sydney. Most of the
exerciseswereoriginallyassignmentorexamquestions.
IwouldliketothankAMSIfortheinitialopportunitytoteachthisbeautifulsub-
ject,andthestudentsinallthoseclassesfortheirfeedback.Iwouldalsoliketothank
PramodAchar,WaiLingYee,CherylPraeger,andtheanonymousreviewersfortheir
valuablesuggestionsandencouragingcomments.
Notational conventions
Tosimplifymatters,wemakeastandingconvention:
AllvectorspacesareoverCandfinite-dimensional.
Thefinite-dimensionalityassumptionallowstheexplicitcalculationswithbasesand
matricesthatareafeatureofthebook.Then×nidentitymatrixiswritten1 ,and
n
the identity transformation of a vector space V is written 1 . The elements of the
V
vectorspaceCn arealwaysthoughtofascolumnvectors;lineartransformationsof
this particular vector space are tacitly identified with n×n matrices (multiplying
on the left of the vectors). The bases of vector spaces are considered to be ordered
setsandhencearewrittenwithoutsetbraces.Thespanoftheelementsv ,...,v is
1 k
writtenC{v ,...,v }.Theterm‘subspace’alwaysmeans‘sub-vector-space’.IfW
1 k
and W(cid:2) aresubspaces ofalarger vector space V thenW ⊕W(cid:2) denotes theirsum,
anditisimpliedthatW ∩W(cid:2) ={0}(an‘internaldirectsum’);ifW andW(cid:2)arenot
subspacesofalargervectorspaceV thenW ⊕W(cid:2) meansthe‘externaldirectsum’
{(w,w(cid:2))|w ∈ W,w(cid:2) ∈ W(cid:2)}. The same principles apply to direct sums with more
thantwosummands.
Onitsrareappearances,thesquarerootof−1iswrittenitodistinguishitfromthe
italicletteri,whichiswidelyusedforotherpurposes.Thegroupofnonzerocomplex
numbersiswrittenC×.ThesetofnonnegativeintegersiswrittenN.Othernotation
willbeexplainedasitisneeded.
ix