for SU(2) M= | a b | |-b* a*| det M = |a|^2 + |b|^2 The group manifold is S3. Relationship to SU(2): X=(X1,X2,X3) S=(s1,s2,s3)=generator matrices Note that: * {si,sj}=anti-commutator=2delta(i,j) * Trace(sisj)=2delta(i,j) S*X= |X3 X1-iX2| |X1+iX2 -X3| =general 2x2 hermitian traceless matrix. properties: * g(S*X)g^=S*X for g in SU(2). * (S*X)(S*X)=X*X if X'*X'=X*X => related by a 3-D rotation. x'i=1/2 Tr(si(S*X'))=1/2 Tr(si(gS*Xg^))=Mij Xj for Mij = 1/2 Tr(si(gsjg^)) in SO3. => homomorphism from g in SU(2) -> M in SO(3). M(g) = 3-d rep of g. g in SU(2) => -g in SU(2) but M(g)=M(-g) So homomorphism 2-1. K = kernel of homeomorphism = {I2,-I2} = normal subgroup of SU(2) g1(t) = exp(.5is1t) M(g1) = |1 0 0 | |0 cos t -sin t| |0 sin t cos t| = rotation by angle t about x1 axis. X1 in standard rep = i * |0 0 0 | |0 0 -1| |0 1 0 | X2= i * |0 0 1| |0 0 0| |-1 0 0| X3= i * |0 -1 0| |1 0 0| |0 0 0| [Xi,Xj]=eijk Xk => SO3 and SU2 have the same Lie algebra irreps: There exist irreps of every dimension. Dim 1: g->1 Dim 2: g->defining matrix Dim 3: SU(2)->SO(3) matrices SU(2) has its terminology derived from use in quantum mechanics. irreps are labeled by j=0,1/2,1,.. Dim D(j+1)mn=2j+1 with m,n=-j,1-j,...,j For j=n/2, n odd irrep is faithful => not irreps of SO(3) For j=n, n integer irrep is faithful rep of SO(3) Use euler coordinates to derive irreps. D(j)mn(a,b,g)=[D(j)(0,0,g)D(g)(0,b,0)Dj(a,0,0)]mn =e^(-ima)d(j)mn(b)e^(-img) dg = sin b dbdadg Proof: show that d(g0g)=dg => jacobian = 1. d(j)mn(b)= jacobi polynomials. spherical harmonics: for l integer, Ylm(theta,phi)=((2l+1)/4Pi)^.5 * D(l)m0(phi,theta) legendre polynomials: Pl(cos theta)= D(l)00(theta) The addition theorem for legendre polynomials is easily provable with its group representation. conjugacy classes: l a 3-d vector with |l|<=2Pi for g(l)= e^(.5 s*l) |l| = C = a conjugacy class of SU(2). D(j)mn(phi,0,0)=e^(-im*phi)delta(m,n)=> X(j)=Sum(m=-j,j,e^(-im*phi)) =(Sin(j+1/2)phi)/sin(phi/2) => X(j)(0)=2j+1 as expected. Adjoint rep: The general adjoint of rep has the young diagram: ## # . . . # with m-1 rows. Dimension = m^2-1 = dimension of the group.