KingsCollegeLondonUniversityOfLondonThispaperispartofanexaminationoftheCollegecountingtowardstheawardofadegree.ExaminationsaregovernedbytheCollegeRegulationsundertheauthorityoftheAcademicBoard.ATTACHthispapertoyourscriptUSINGTHESTRINGPROVIDEDCandidateNo: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DeskNo: . . . . . . . . . . . . . . . . . . . . . . .MScExamination7CCMMS01 (CMMS01) Lie groups and Lie algebras (mockexam)Summer2010TimeAllowed: TwoHoursThispaperconsistsoftwosections,SectionAandSectionB.SectionAcontributeshalfthetotalmarksforthepaper.AnswerallquestionsinSectionA.AllquestionsinSectionBcarryequalmarks,butifmorethantwoareattempted,thenonlythebesttwowillcount.NOcalculatorsarepermitted.TURNOVERWHENINSTRUCTED2010 c KingsCollegeLondon- 2 - 7CCMMS01(CMMS01)A1. 20points(i) Dene: a)Liealgebra,b)group,c)Liealgebrahomomorphism,d)grouphomomorphism,e)subgroup,andf)normalsubgroup.(ii) LetGandHbegroups. Showthatthekernel of ahomomorphism:G HisanormalsubgroupofG(youcanusewithoutproofthemainpropertiesofhomomorphisms).(iii) State the denition of the property semisimple of a Lie algebra in termsof solvableideals. Showthat if gis semisimple, thenit does not haveabelianidealsotherthan {0}.(iv) Consider a vector space V spannedover Rbythree independent ele-mentsx, y, z. Suppose[] isabilinearproductonV satisfying[vw] =[wv] v, w V and[xy] = z,[xz] = y,[yz] = 0. ShowthatV withthisproductisaLiealgebra.A2. 15points(i) StatethedenitionofamatrixLiegroup.(ii) Showthat theset of matrices O(n) = {A Mat(n; C) | ATA=1}isamatrixLiegroupforanypositiveintegern(dontforgettoshow, inparticular,thatitisagroup).(iii) IsO(n)compact?Proveyourclaim.SeeNextPage- 3 - 7CCMMS01(CMMS01)A3. 15points(i) The group SU(2) is the group of all 22 unitary matrices of determinant1. ShowthattheLiealgebraofSU(2)issu(2) = {A Mat(2; C) : A= A, Tr(A) = 0},i.e. it is the set of all 2 2 anti-hermitian traceless matrices (you can usewithoutproofthepropertiesoftheexponential).(ii) Let G be a matrix Lie group with Lie algebra g. Suppose any element of Gisoftheformegforsomeg g. SupposealsothattheBaker-Campbell-Hausdorf formulafor egeg

converges for anyg, g

g. Showthat if ghasanideal h, thenGhasanormal subgroupH=exp h(youcanusewithoutprooftherelationbetweenAdandad ,andthebasicpropertiesoftheexponential).(iii) Considersl(2; C), thealgebraof 2by2tracelesscomplexmatrices. Leth sl(2; C)bethesubalgebrah =

0 aa 0

, a C

.GiventhathisaCartansubalgebraof sl(2; C), ndtherootsandthecorrespondingrootspacedecomposition.B4. (i) DeterminethecenterZ(H)oftheHeisenberggroupH,andshowthatitisisomorphicto R. IsZ(H)asubgroupofH? Isitanormalsubgroup?Prove your claims. Show that the quotient group Q = H/Z(H) is abelian,andisomorphicto RR. Showthattheredoesnotexistanysemi-directproductQZ(H)isomorphictoH.(ii) Show that the exponential mapping from the Lie algebra of the HeisenberggrouptotheHeisenberggroupisbijective.(iii) DescribethegroupSU(2)asasubsetof R4.SeeNextPage- 4 - 7CCMMS01(CMMS01)B5. (i) UsingtheBCHformulalog(eXeY) = X +

10dtlog(ead Xetad Y)1 (ead Xetad Y)1(Y )andthepropertiesoftheexponential,showthatlog(eAeB) =u1 euA +Binthecasewhere[A, B] = uAforsomecomplexnumberu. Discusswhathappensinthecasewhereu 0. Hint: rstusethepropertiesof theexponentialtoproveeAeB= eB+uAeA.(ii) Intwodimensions, thePoincarealgebrahas threegenerators: theen-ergyH, themomentumPandtheboostoperatorB. Theysatisfythecommutationrelations[P, H] = 0, [B, P] = H, [B, H] = P.Usingtheformulaobtainedin(i),evaluatetheoperatorUineixPeixHeB= eUwherex, arerealnumbers.B6. A derivation Don a Lie algebra L is a linear map D gl(L) with the propertythatD([xy]) = [D(x)y] + [xD(y)]forallx, y L.(i) Provethatthecommutatoroftwoderivations,[D1, D2],isaderivation.(ii) Provethatthekernel ofaderivationker(D)= {x L:D(x)=0}isasubalgebra. Isitalsoanideal?(iii) Given a derivation D on L, prove that the bilinear form(x, y) = (x, D(y)),where is the Killingform(x, y) =Tr(ad x ad y), is anti-symmetric,(x, y) = (y, x).FinalPage