A Lie Group

These notes introduce SU (2) as an example of a compact Lie group. The definition of SU (2) is SU (2) = A A a 2 × 2 complex matrix, det A = 1, AA * = A * A = 1l In the name SU (2), the " S " stands for " special " and refers to the condition det A = 1 and the " U " stands for " unitary " and refers to the conditions AA * = A * A = 1l. The adjoint matrix A * is the complex conjugate of the transpose matrix. That is, α β γ δ * = ¯ α ¯ γ ¯ β ¯ δ Define the inner product on C 2 by a 1 a 2 , b 1 b 2 = a 1 ¯ b 1 + a 2 ¯ b 2 A i,j a j ¯ b i = A a 1 a 2 , b 1 b 2 = a 1 a 2 , A * b 1 b 2 = 2 i,j=1 a j A * j,i b i Thus the condition A * A = 1l is equivalent to A * A a 1 a 2 , b 1 b 2 = a 1 a 2 , b 1 b 2 for all a 1 a 2 , b 1 b 2 ∈ C 2 ⇐⇒ A a 1 a 2 , A b 1 b 2 = a 1 a 2 , b 1 b 2 for all a 1 a 2 , b 1 b 2 ∈ C 2 Hence SU (2) is the set of 2 × 2 complex matrices that have determinant one and preserve the inner product on C 2. (Recall that, for square matrices, A * A = 1l is equivalent to A −1 = A * , which in turn is equivalent to AA * = 1l.) By the polarization identity (Problem Set V, #3), preservation of the inner product is equivalent to preservation of the norm A a 1 a 2 = a 1 a 2 for all a 1 a 2 ∈ C 2 Clearly 1l ∈ SU (2). If A, B ∈ SU (2), then det(AB) = det(A) det(B) = 1 and (AB)(AB) * = ABB * A * = A1lA * = 1l so that AB ∈ SU (2). Also, if …