Table Of ContentLie Algebras and their Representations
LecturesbyDavidStewart
NotesbyDavidMehrle
[email protected]
CambridgeUniversity
MathematicalTriposPartIII
Michaelmas2015
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 LieGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Representationsofslp2q. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 MajorResultsonLieAlgebras . . . . . . . . . . . . . . . . . . . . . . . 24
5 RepresentationsofSemisimpleLieAlgebras. . . . . . . . . . . . . . . 34
6 ClassificationofComplexSemisimpleLieAlgebras . . . . . . . . . . 60
LastupdatedMay29,2016.
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Contents by Lecture
Lecture1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Lecture2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Lecture3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Lecture4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Lecture5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Lecture6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Lecture7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Lecture8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Lecture9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Lecture10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Lecture11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Lecture12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Lecture13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Lecture14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Lecture15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Lecture16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Lecture17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Lecture18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Lecture19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Lecture20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Lecture21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Lecture22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Lecture23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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Lecture1 8October2015
1 Introduction
Therearelecturenotesonline.
We’llstartwithabitofhistorybecauseIthinkit’seasiertounderstandsome-
thingwhenyouknowwhereitcomesfrom. Fundamentally,mathematicians
wantedtosolveequations,whichisaratherbroadstatement,butitmotivates
things like the Fermat’s Last Theorem – solving x3`y3 “ z3. Galois theory
beginswithwantingtosolveequations. OneofGalois’sfundamentalideaswas
tonottrytowriteasolutionbuttostudythesymmetriesoftheequations.
SophusLiewasmotivatedbythistodothesamewithdifferentialequations.
Couldyousaysomethingaboutthesymmetriesofthesolutions?Thistechnique
isusedalotinphysics.ThisledhimtothestudyofLiegroups,andsubsequently,
Liealgebras.
Example1.1. TheprototypicalLiegroupisthecircle.
A Lie group G is, fundamentally, a group with a smooth structure on it.
The group has some identity e P G. Multiplying e by a P G moves it the
correspondingpointaroundthemanifold.Importantly,ifthegroupisn’tabelian,
thenaba´1b´1isnottheidentity. Wecallthisthecommutatorra,bs. Lettinga,b
tendtozero,wegettangentvectors X,Y andacommutatorrX,Ysbyletting
ra,bstendto0. ThesepointsofthetangentspaceareelementsoftheLiealgebra.
We’lmakethisallpreciselater. We’llclassifythesimpleLiealgebras,which
ends up using the fundamental objects called root systems. Root systems
are so fundamental to math that this course might better be described as an
introductiontorootsystemsbywayofLiealgebras.
Definition1.2. Letkbeafield. ALiealgebragisavectorspaceoverkwitha
bilinearbracketr´,´s: gˆgÑgsatisfying
(i) AntisymmetryrX,Xs“0,forallX Pg;
(ii) JacobiidentityrX,rY,Zss`rY,rZ,Xss`rZ,rX,Yss“0.
TheJacobiIdentityisprobablyeasiertothinkofas
rX,rY,Zss“rrX,Ys,Zs`rY,rX,Zss.
BracketingwithXsatisfiesthechainrule! ActuallyrX,´sisaderivation.
Definition 1.3. A derivation δ of an algebra A is an endomorphism A Ñ A
thatsatisfiesδpabq“ aδpbq`δpaqb.
Remark1.4. Fromnowon,ourLiealgebrasgwillalwaysbefinitedimensional.
Mostofthetime,k“C(butnotalways!). We’llsometimespointouthowthings
gowrongincharacteristic pą0.
Example1.5.
(i) IfV isanyvectorspace,equipV withthetrivialbracketra,bs“0forall
a,bPV.
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Lecture2 10October2015
(ii) IfAisanyassociativealgebra,equipAwithra,bs“ ab´baforalla,bP A.
(iii) Letg “ Mnˆnpkq,thenˆnmatricesoverafieldk. Thisisoftenwritten
gl pkq or glpnq when the field is understood. This is an example of an
n
associativealgebra,sodefinerA,Bs“ AB´BA.
ThereisanimportantbasisforglpnqconsistingofE for1ďi,jďn,which
ij
isthematrixwhoseentriesareallzeroexceptinthepi,jq-entrywhichis1.
Firstobserve
rE ,E s“δ E ´δ E .
ij rs jr is is rj
Thisequationgivesthestructureconstantsforglpnq.
Wecancalculatethat
$
’’’’&0 ti,ju‰tr,su
E i“r,j‰s
rs
rE ´E ,E s“
ii jj rs ’’’’%´Ers j“r,i‰s
2E i“r,j“s
rs
(iv) If Aisanyalgebraoverk,Der A Ă End AisaLiealgebra,thederiva-
k k
tionsof A. For α,β P DerA,definerα,βs “ α˝β´β˝α. Thiswillbea
validLiealgebrasolongasrα,βsisstilladerivation.
Definition1.6. Asubspaceh Ă gisaLiesubalgebraifhisclosedunderthe
Liebracketofg.
Definition1.7. DefinethederivedsubalgebraDpgq“xrX,Ys| X,Y Pgy.
Example1.8. Animportantsubalgebraofglpnqisslpnq, slpnq :“ tX P glpnq |
trX “0u. ThisisasimpleLiealgebraoftype A . Infact,youcancheckthat
n´1
slpnqisthederivedsubalgebraofglpnq,
slpnq“rglpnq,glpnqs“Dpglpnq.
Example 1.9. Lie subalgebras of glpnq which preserve a bilinear form. Let
Q: VˆV Ñkbeabilinearform. ThenwesayglpVqpreservesQifthefollow-
ingistrue:
QpXv,wq`Qpv,Xwq“0
forall v,w P V. RecallthatifwepickabasisforV,wecanrepresent Q bya
matrix M. ThenQpv,wq“vTMw. ThenXpreservesQifandonlyif
vTXTMw`vTMXw“0,
ifandonlyif
XTM`MX “0.
Recall that a Lie algebra g is a k-vector space with a bilinear operation
r´,´s: gˆgÑgsatisfyingantisymmetryandtheJacobiidentity.
Wehadsomeexamples,suchasg “ glpVq “ End pVq. Ifyoupickabasis,
k
thisisMnˆnpVq. Givenanyassociativealgebra,wecanturnitintoaLiealgebra
withbracketrX,Ys“ XY´YX.
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Lecture2 10October2015
Example1.10. AnotherexampleisifQ: VˆV Ñ kisabilinearform,theset
of X P glpVq preserving Q is a Lie subalgebra of glpVq. Taking a basis, Q is
representedbyamatrix MwithQp(cid:126)v,w(cid:126)q“(cid:126)vTMw(cid:126). XpreservesQifandonlyif
XTM`MX “0.
The most important case is where Q is non-degenerate, i.e. Qpv,wq “
0@wPV ifandonlyifv“0.
Example1.11. Considerthebilinearformwhere
„
0 I
n
M“
´I 0
n
M represents an alternating form and the set of endomorphisms of V “ k2n
is the symplectic Lie algebra (of rank n) and denoted spp2nq. If X P glpVq is
written „
A B
X “
C D
CheckthatXpreserves Mifandonlyif
„
A B
X “
C ´AT
withB,Csymmetricmatrices. Abasisforthisconsistsofelements
• H “ E ´E
i,i`n ii i`n,i`n
• E ´E
ij j`n,i`n
• E `E
i,j`n j,i`n
• E `E
i`n,j j`n,i
for1ďi,jďn. ThisisasimpleLiealgebraoftypeC forchark‰2.
n
Example1.12. Therearealso
• orthogonalLiealgebrasoftypeD “sop2nq,preserving
n
„
0 I
n
M“ .
I 0
n
• orthogonalliealgebrasoftypeB “sop2n`1q,preserving
n
» fi
0 I
n
– fl
M“ I 0 .
n
1
Example1.13. b istheborelalgebraofuppertriangularnˆnmatricesand
n
n isthenilpotentalgebraofstrictlyuppertriangularnˆnmatrices.
n
Definition1.14. Alinearmap f: gÑhbetweentwoLiealgebrasg,h,isaLie
algebrahomomorphismif fprX,Ysq“rfpXq, fpYqs.
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Lecture2 10October2015
Definition1.15. Wesayasubspacejisasubalgebraofgifjisclosedunderthe
Liebracket. AsubalgebrajisanidealofgifrX,YsPjforallX Pg,Y Pj.
Definition1.16. Thecenterofg,denotedZpgqis
Zpgq“tX Pg|rX,Ys“0@Y Pgu.
Exercise1.17. CheckthatZpgqisanidealusingtheJacobiidentity.
Proposition1.18.
(1) If f: hÑgisahomomorphismofLiealgebras,thenker f isanideal;
(2) Ifj Ă gisalinearsubspace,thenjisanidealifandonlyifthequotient
bracketrX`j,Y`js“rX,Ys`jmakesg{jintoaLiealgebra;
(3) Ifjisanidealofgthenthequotientmapg Ñ g{jisaLiealgebrahomo-
morphism;
(4) IfgandharebothLiealgebras,theng‘hbecomesaLiealgebraunder
rpX,Aq,pY,Bqs“prX,Ys,rA,Bsq.
Exercise1.19. ProveProposition1.18.
Remark1.20. ThecategoryofLiealgebras,Lie,formsasemi-abeliancategory.
It’sclosedundertakingkernelsbutnotundertakingcokernels. Therepresenta-
tiontheoryofLiealgebrasdoes,however,formanabeliancategory.
Definition1.21. Thefollowingnotionsarereallytwowaysofthinkingabout
thesamething.
(a) A representation of g on a vector space V is a homomorphism of Lie
algebrasρ: gÑglpVq.
(b) AnactionofgonavectorspaceVisabilinearmapr: gˆV ÑVsatisfying
rprX,Ys,vq“rpX,rpY,vqq´rpY,rpX,vqq.WealsosaythatVisag-module
ifthisholds.
GivenanactionrofgonV,wecanmakearepresentationofgbydefining
ρ: gÑglpVqbyρpXqpvq“rpX,vq.
Example1.22. Thisisthemostimportantexampleofarepresentation. Forany
Liealgebrag,onealwayshastheadjointrepresentation,ad: gÑglpgqdefined
byadpXqpYq“rX,Ys.Thefactthatadgivesarepresentationfollowsfromthe
Jacobiidentity.
Definition1.23. IfW isasubspaceofag-moduleW,thenW isag-submodule
ifW isstableunderactionbyg: gpWqĎW.
Example1.24.
(1) Supposejisanidealing. ThenadpXqpYq“rX,YsPjforallY Pj,sojisa
submoduleoftheadjointrepresentation.
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Lecture3 13October2015
(2) IfW ĎV isasubmodulethenV{W isanotherg-moduleviaXpv`Wq“
Xv`W.
(3) IfVisag-module,thenthedualspaceV˚ “Hom pV,kqhasthestructure
k
ofag-moduleviaXφpvq“´φpXvq.
LasttimewedevelopedacategoryofLiealgebras,andsaidwhathomomor-
phismsofLiealgebraswere,aswellasdefiningkernelsandcokernels. There
areafewmoredefinitionsthatweshouldpointout.
Definition1.25. ALiealgebraissimpleifithasnonontrivialideals.
Wealsomovedonanddiscussedrepresentations. Recall
Definition1.26. Arepresentationorg-moduleofgonVisaLiealgebrahomo-
morphismgÑglpVq.
To complete the category g-Mod of g-modules, let’s define a map of g-
modules.
Definition 1.27. Let V,W be g-modules. Then a linear map φ: V Ñ W is a
g-modulemapifXφpvq“φpXvqforallX Pg.
φ
V W
X X
φ
V W
Proposition1.28. Ifφ: V ÑW isag-modulemap,thenkerφisasubmodule
ofV
Exercise1.29. ProveProposition1.28.
Definition 1.30. A g-module V (resp. representation) is simple (resp. irre-
ducible)ifV hasnonon-trivialsubmodules.
We write V “ V ‘V if V and V are submodules with V “ V ‘V as
1 2 1 2 1 2
vectorspaces.
Howcanyoubuildnewrepresentationsfromoldones? Thereareseveral
ways.IfV,Wareg-modules,thensoisV‘Wbecomesag-moduleviaXpv,wq“
pXv,Xwq.
There’sanotherwaytobuildnewrepresentationsviathetensorproduct.
Infact,g-Modismorethanjustanabeliancategory,it’samonoidalcategory
viathetensorproduct. GivenV,W representationsofg,wecanturnthetensor
productintoarepresentation,denotedVbW,bydefiningtheactiononsimple
tensorsas
Xpvbwq“pXvqbw`vbpXwq
andthenextendinglinearly.
WecaniterateonmultiplecopiesofV,say,togettensorpowers
Vbr “lVoobooooVooobm¨oo¨oo¨obooooVn
rtimes
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Lecture3 13October2015
Definition 1.31. The r-th symmetric power of V, with basis e ,...,e is the
1 n
vectorspacewithbasise ¨¨¨e fori ď i ď,...,ď i . ThisisdenotedSrpVq.
i1 ir 1 2 r
TheactionofgonSrpVqis
Xpe ¨¨¨e q“ Xpe qe ¨¨¨e `e Xpe q¨¨¨e `...`e e ¨¨¨e Xpe q.
i1 ir i1 i2 ir i1 i2 ir i1 i2 ir´1 ir
Ź
Definition1.32. Ther-thalternatingpowerofag-moduleV,denoted rpVq,
isthevectorspacewithbasiste ^e ^¨¨¨^e | i ă i ă ... ă i u,ifV has
i1 i2 ir 1 2 r
basise ,...,e . Theactionisfunctionallythesameasonthesymmetricpower:
1 n
Xpe ^¨¨¨^e q“ Xpe q^e ^¨¨¨^e `...`e ^e ^¨¨¨^e ^Xpe q.
i1 ir i1 i2 ir i1 i2 ir´1 ir
Wealsohavetherulethat
e ^...^e ^...^e ^...^e “´e ^...^e ^...^e ^...^e .
i1 ij ik ir i1 ik ij ir
Exercise1.33. Whatisthedimensionofthesymmetricpowers/alternating
powers?
Example1.34. Let „
0 1
X “
0 0
andletV “ k2 withbasiste ,e u. Letg “ kX Ă glpVq. ThenVbV hasbasis
1 2
e be ,e be ,e be ,ande be .
1 1 1 2 2 1 2 2
ObserveXe “0,Xe “e . Therefore,
1 2 1
Xpe be q“0
1 1
Xpe be q“e be
1 2 1 1
Xpe be q“e be
2 1 1 1
Xpe be q“e be `e be
2 1 1 2 2 1
AsalineartransformationVbV ÑVbV,Xisrepresentedbythematrix
» fi
0 1 1 0
— ffi
—0 0 0 1ffi
X “– fl.
0 0 0 1
0 0 0 0
Ź
Abasisfor 2V iste ^e u,andhere Xpe ^e q “ Xe ^e `e ^Xe “
1 2 1 2 1 2 1 2
0^e `e ^e “0. SoXisthezeromaponthealternatingsquare.
2 1 1
Exercise1.35. Workouttheprecedingexampleforthesymmetricsquare,and
thetensorcube.
2 Lie Groups
Lotsofstuffinthissectionrequiresdifferentialgeometryandsomeanalysis.
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Lecture4 15October2015
Definition2.1. AHausdorff,secondcountabletopologicalspaceXiscalleda
manifoldifeachpointhasanopenneighborhood(nbhd)homeomorphictoan
opensubsetUofRd byahomeomorphismφ: U „ φpUqĂRN.
ThepairpU,φqofahomeomorphismandopensubsetof Miscalledachart:
given open subsets U and V of X with UXV ‰ H, and charts pU,φ q and
U
pV,φ q,wehaveadiffeomorphismφ ˝φ´1: φ pUXVqÑφ pUXVqofopen
V V U U V
subsetsofRN.
WethinkofamanifoldasaspacewhichlookslocallylikeRN forsomeN.
Example2.2.
(a) R1,S1areone-dimensionalmanifolds;
(b) S2 orS1ˆS1 aretwo-dimensionalmanifolds. Thetoruscarriesagroup
structure;S2doesnot.
Definition 2.3. A function f: M Ñ N is called smooth if composition with
the appropriate charts is smooth. That is, for U Ď M, V Ď N, and charts
φ: U Ñ RM, ψ: V Ñ RN, then ψ˝ f ˝φ´1: RM Ñ RN is smooth where
defined.
Definition 2.4. A Lie group is a manifold G together with the structure of a
groupsuchthatthemultiplicationµ: GˆGÑGandinversioni: GÑGmaps
aresmooth.
Exercise2.5. Actually,thefactthattheinverseissmoothfollowsfromthefact
thatmultiplicationissmoothandbylookinginaneighborhoodoftheidentity.
Proveit!
Toavoidsomesubtletiesofdifferentialgeometry,wewillassumethat M
isembeddedinRN forsome(possiblylarge)N. Thisispossibleundercertain
tamehypothesesbyNash’sTheorem.
Example2.6.
(cid:32) (
(1) GLpnq :“ nˆnmatricesoverRwithnon-zerodeterminant . Thereis
onlyasinglechart: embeditintoRn2.
(2) SLpnq:“tgPGLpnq|detg“1u.
(3) IfQ: RnˆRn ÑRisabilinearform,then
GpQq“tgP Mpnq|Qpv,wq“Qpgv,gwqu
forallv,wPRn.
RecallthataLiegroupisamanifoldwithagroupstructuresuchthatthe
groupoperationsaresmooth. Forexample,SLpnq.
Definition2.7. LetGandHbetwoLiegroups. Thenamap f: GÑ HisaLie
grouphomomorphismif f isagrouphomomorphismandasmoothmapof
manifolds.
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Lecture4 15October2015
LetGbeaLiegroupandletG˝betheconnectedcomponentofGcontaining
theidentity.
Proposition2.8. ForanyLiegroupG,thesetG˝isanopennormalsubgroup
of G. Moreover, if U is any open neighborhood of the identity in G˝, then
G˝ “xUy.
Proof. Thefirstthingweneedtoshowisthat G˝ isasubgroup. Since G isa
manifold, its connected components are path connected. Suppose a,b P G˝.
Thenwecanfindpaths γ,δ: r0,1s Ñ G˝ with γp0q “ e “ δp0q, γp1q “ a and
δp1q“b. Thentakingthepathµpγptq,δptqqgivesapathfromtheidentitytoab.
Hence,G˝isclosedundermultiplication. Similarly,ipγptqqgivesapathfrome
toa´1,andG˝isclosedunderinverse.
WhyisG˝normal? Well,themapgÞÑ aga´1givesadiffeomorphismofG˝
withthatfixeseandthereforealsoG˝.
ByreplacingUwithUXU´1,wecanarrangethatUcontainstheinverseof
everyelementinU. NowletUn “U¨U¨¨¨U “ tu u ¨¨¨u | u PUuisopen
1 2 n i
asitiŤstheunionofopencosetsu1u2¨¨¨un´1Uoverallpu1,...,un´1q. Thenset
H “ ně0Ur ThisisaŤnopensubgroupofG0containingxUy. Itisalsoclosed
sincethesetofcosets aH “G˝zHisopenastheunionofdiffeomorphic
aRH
translatesofanopenset,soHisthecomplimentofanopenset.
TheconnectedcomponentG˝isaminimalsetthatisbothopenandclosed,
soH “G.
Definition 2.9. Any open neighborhood of the identity is called a germ or
nucleus.
Corollary2.10. If f and garetwohomomorphismsfrom G to H with G con-
nected,then f “ gifandonlyif f| “ g| foranygermofg.
U U
Definition2.11. Let MĂRN beamanifold. Thetangentspaceof pP Mis
(cid:32) ˇ (
T pMq“ vPRN ˇ thereisacurveφ: p´ε,εqÑ Mwithφp0q“ p,φ1p0q“v
p
Onecanshowthatthisisavectorspace. Scalarmultiplicationiseasy: takethe
curveφpλtq“λv. Additionfollowsbylookingatadditionincharts.
Let’ssingleoutaveryimportanttangentspacewhenwereplace Mwitha
LiegroupG.
Definition2.12. IfGisaLiegroup,thenwedenoteT pGqbygandcallitthe
e
LiealgebraofG.
Wedon’taprioriknowthatthisisactuallyaLiealgebraaswedefinedit
previously,butwecanatleastseewhatkindofvectorsliveinaLiealgebraby
lookingattheLiegroup.
ˇ
Example2.13. Let’scalculatesl “ T pSL q. Ifv“d{ ˇ gptq“ g1p0q. Bythe
n In n dt t“0
conditiononmembershipinSL ,wehavedetgptq“1“detpg ptqq. Writeout
n ij
10
