MS01 LieGroupsandLieAlgebras ProblemSheet11. ShowthatSU(n)isagroup.2. Grouphomomorphismsa) Let: G1G2beagrouphomomorphism. Showthatim()isasubgroupofG2andthatker() := 1(e2),thepre-imageoftheidentitye2ofG2,isasubgroupofG1.b) LetAut(G)thesetofallgroupautomorphismsofG. Showthatthisisagainagroup.3. Semi-directproductsLet Hand Nbe groups, and let : H Aut(N) be a group homomorphism. The semi-directproductHNisthegroupofallorderedpairs(h, n)H Nwithcompositionandinversedenedby(h, n) (h, n) := (h h, n h(n)) , (h, n)1:= (h1, h1(n1)) .(Here,histheautomorphismofNassociatedbytotheelementh H. Theordinarydirectproductarisesfromthechoiceh= idNforallh H.)a) ShowthatHNisagroup.b) ForH= O(n)andN= Rn,ndanobvious: H Aut(N). ThiscanbeusedtoshowthattheeuclideangroupE(n)isisomorphictoO(n)Rn.4. MatrixnormForanymatrixA Matn(C),wedene||A|| :=n

i,j=1|Aij|.Showthat||A||=|| ||A||forall C, that||A + B||||A|| + ||B||and||AB||||A|| ||B||forallA, B Matn(C).Since we also have that ||A|| = 0 i (if and only if)A = 0, these relations in particularshowthat||A||denesanormonthespaceofmatrices.5. ThecircleasamanifoldIncasewegettothedenitionofamanifold,argue(orideallyshow)thatS1isindeedadierentiablemanifold.(As a topology, one can use the one induced fromR2: a subset U S1is open i U= DS1forsomeopensubsetDof R2.)