Lie Algebras and their Representations PDF
Preview Lie Algebras and their Representations
Lie 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. 1 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 2 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. 3 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. 4 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. 5 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. 6 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 7 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. 8 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. 9 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
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