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Not to be quoted or reproduced without permission. o l ru elements group all for R qain httematrices the that equations constants structure b hwta h udmna representation fundamental the that Show (b) .()Let (a) 6. that conclude hwta the that Show ope conjugate complex .Let 7. and r oeaieitgr.Here integers. nonegative are elLeagbao opc yeadltIb nielo .LtI Let L. of ideal an be I let and type compact of algebra Lie components, real has form Killing the .Fn h utpiiyo ahwih ftetno product tensor the of weight each of multiplicity the Find 8. where ersnainof representation )Afiiedmninlra i ler so opc yeif type compact of is algebra Lie real finite-dimensional s degenerate. it A is Is b) it 8. whether Question determine 1, and Sheet explicitly in form A defined algebra ideals. 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Not to be quoted or reproduced without permission. sn h osrit nteCra arxdrvdi h lec the in derived matrix that Cartan show the i), on part constraints the Using with nteinrpouto n w roots two any of product inner the on hr h H scniee sacmlxLealgebra. Lie complex a as considered is RHS the where ihemsTr 06 ahmtclTio atIIPo.N. Prof. L( ones, following the and question this In 1. III Part Tripos Mathematical 2016, Term Michaelmas arxAo nt-iesoa ope ipeLeagbao algebra Lie simple complex finite-dimensional a of A matrix ieso a ot fa ottodffrn lengths. different two 5. most at of roots has dimension L .If 3. L .Dtrietewihso the of weights the Determine ambiguities? 9. w remaining any constants fix normalisation you the could b How on any operat constraints for step the constants and Determine normalisation generators including Cartan operators between step brackets all down .Wiedw ai o atnsbler fL of subalgebra Cartan a for basis a down Write 2. forms. real L .Fn e fsml ot o h arxLeagbaL algebra Lie matrix the for roots simple of set a Find with 6. solutions additional no do Why classification. hoeabssfrteteLeagbacnitn ftegener the of consisting algebra Lie the the for basis a Choose atnmti hwn hti onie ihteCra matr Cartan the with coincides it that showing matrix Cartan .Satn rmisDni iga,cntuttero syst root the construct diagram, Dynkin its from Starting 7. classification. .Satn rmteconstraint, the from Starting 4. k .Tesml i algebra Lie simple The 8. C C C ∈ ( ( ( G SO SO C i osdrpsil 3 possible Consider ii) i ycnieigtesbler fra arcsso that show matrices real of subalgebra the considering By ii) )So that Show i) )Uigtecntan * fqeto ,so htteoff-di the that show 3, question of (*) constraint the Using i) m eoe its denotes ) α o which for 4)adwiedw h orsodn tpoeaosi your in operators step corresponding the down write and (4)) (2 saro fasml ope i ler ffiiedmninsh dimension finite of algebra Lie complex simple a of root a is .So htyu ouin xas l h ak3sml Lie simple 3 rank the all exhaust solutions your that Show 0. = n ) o rirr integer arbitrary for 1)) + kα ymtis ilsadPrils xmls3 Examples Particles. and Fields Symmetries, complexification l saro are root a is , m and A × 2 A n atnmtie fteform, the of matrices Cartan 3 a iperoots simple has ij antalb o-eo n n l h loe atnmatric Cartan allowed the all find and non-zero, be all cannot A A k ji 2 . = L ersnainwt yknlbl (2 labels Dynkin with representation < ± C n α 2( ( o all for 4 ( 1 ec n h ot fteLeagba L algebras Lie the of roots the find Hence . SU ,α α, ,β α, and   (2)) m ) l 2 ) ′ β ′ ∈ hwta n ope ipeLeagbao finite of algebra Lie simple complex any that show , α G ≃ , n Z 2 m l eoe h i ler faLegroup Lie a of algebra Lie the denotes ) β ′ ( L i n igeadtoa oiieroot positive additional single a and m 6= SL n 2 C arise? 0 6= j   ( C (2 SO 1 = ( ( SU , ∗ C akt hc r oetal non-zero. potentially are which rackets ) (2 , )) mo h ipeLealgebra Lie simple the of em 2 3)addtrietecorresponding the determine and (3)) ihflo rmteJcb identity. Jacobi the from follow hich xo h i algebra Lie the of ix n rank f ators r. , . . . , )adfraCra uagbaof subalgebra Cartan a for and )) r n lotebakt between brackets the also and ors ue oehrwt h eutof result the with together tures L C gnleeet fteCartan the of elements agonal { r ( h hsnbasis. chosen SU α obey wta h nyvle of values only the that ow h , , 2)hstoinequivalent two has (2)) 0). β lersi h Cartan the in algebras e , ± α e , A ± 2 β C nteCartan the in e , ( SO ± θ θ } = 3)and (3)) B Write . G 2 α Dorey . + and β es . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. .So htteisospin the that Show 2. Deter adjoint. adjoint. the the of in those and representation fundamental ihemsTr 06 ahmtclTio atIIPo.N. Prof. construct matrix Cartan corresponding algebra the from Starting 1. III Part Tripos Mathematical 2016, Term Michaelmas atnsbler of subalgebra Cartan h relations the rnfrsas transforms .So httetndmninlrepresentation ten-dimensional the that Show 3. clrfil nteajitrpeetto of representation adjoint the above. in field defined scalar (*) A transformation gauge the under ne ag transformation, gauge a under ino h L the of tion o h case the For x .Fn h non-trivial the Find 5. field, ainof tation stu o n arxLegroup Lie matrix any for true is .Let 7. one the with coincides transformation gauge infinitessimal .Dcmoetefloigtno rdcsof products tensor following the Decompose particu the 4. on depend answer your Does representation. this .Cnie ag hoywoeguegroup, gauge whose theory gauge a eac Consider of 6. dimension the giving irreducibles, into copies two of opnns i) components; enda ahsaeiepoint spacetime each at defined ntefnaetladajitof adjoint and fundamental the in eciigteculn fteesaa ed otetegaug the the to fields scalar these of coupling the describing idepii omlefrcvratderivatives covariant for formulae explicit Find .So httefil-teghtensor field-strength the that Show 8. a group Lie compact a of representations finite-dimensional ∈ R 3 , G 1 B hc rnfrsas, transforms which ⊂ G 2 orsod oan to corresponds , L = Mat G C I ( SU = = 3 C N ( ymtis ilsadPrils xmls4 Examples Particles. and Fields Symmetries, ⊗ SU SO H ( 2)sbler orsodn oayro.Fn h irreduc the Find root. any to corresponding subalgebra (2)) C 3 ¯ 1 eacmatmti i ru.Asaa edi h fundament the in field scalar A group. Lie matrix compact a be ) ( / 5) sn h loih nrdcdi h etrs n th find lectures, the in introduced algorithm the Using (5)). n ii) and N A and 2 B 2 ,cekthat check ), 2 n hs eeaosato ttsi h don representat adjoint the in states on act generators these and I ersnaino mletdmninaddcmoeteten the decompose and dimension smallest of representation n hypercharge and Y 3 ⊗ ( = N G x A 3 cmoetvector -component Writing . ∈ H µ G ⊗ R → 1 A 3 epciey ec rt ongueivratLagrangia gauge-invariant down write Hence respectively. 3 2 + F . µ ′ g , φ A 1 F A ( µν A : x µν φ µ ′ hc rnfrsas, transforms which µ H ae ausi h i ler L( algebra Lie the in values takes ) F → R o arxLegroup Lie matrix a for = : g 2 [ = → G 3 Y ) exp( = φ gA , R / 1 D orsod oan to corresponds , A ′ where 3 D 3 φ ftelgts eosaecretydtrie by determined correctly are mesons lightest the of → G µ , (F) µ F ′ A 1 = µ (A) R g samti i ru.Tecrepniggauge corresponding The group. Lie matrix a is → 2 = G − 3 , gφ , D , ǫX 1 D 0 L = gφ φ L( − A of µ (A) F ν (A) with ) g F H G ( ( ( C − A x ∂ uhthat such , ∗ ) 1 ( ] 1 ) µ 2 ) component. h SU endi h lectures. the in defined ietedgnrc ftewih zero weight the of degeneracy the mine g eunitary. re and ∈ orsod oardcberepresenta- reducible a to corresponds h otadwih atcso h Lie the of lattices weight and root the ) field e a otchosen? root lar g << ǫ C 3)rpeettosit irreducible into representations (3)) − N 1 H N enda ahsaeiepoint spacetime each at defined 2 G A ,so httecorresponding the that show 1, D r h tnadbssfrthe for basis standard the are × a ewitnas, written be can µ µ (F) N o a sueta the that assume may You . φ matrix F and G .Epanwythis why Explain ). becmoet of components ible φ D A µ (A) egt fthe of weights e ( x φ ) ion. A lrepresen- al o product sor ∈ transform Mat Dorey N ( C ns ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rprinlt h n ie ntelcue.Dtriethe Determine lectures. the in given G one the to proportional eueta,we L( when that, deduce steuiu nain ne rdc n ipeLealgebra Lie simple any product inner invariant unique the is ne h ag rnfrain() hsso htteLagra the that show Thus (*). transformation gauge the under ec hwthat show Hence .Uigtefc htteKligform, Killing the that fact the Using 9. invariant. gauge is = SU ( N ). F µν G rnfrsi h don ersnaino ie, G of representation adjoint the in transforms ssml,teguefil arnindfie ntepreviou the in defined Lagrangian gauge-field the simple, is ) κ ( ,Y X, r[ad [ Tr = ) F L µν = → 4 1 g X F 2 µν ◦ Tr ′ ad N = Y [ F gF ] µν µν F g µν ∀ − ,Y X, 1 ] osato rprinlt ntecase the in proportionality of constant ngian, L( ∈ L( G pt clrmultiplication, scalar to up ) G ) usinis question s