Haar Measure

Abstract

Definition 1 A group is a set Γ together with an operation · : Γ × Γ → Γ obeying i) (αβ)γ = α(βγ) for all α, β, γ ∈ Γ (Associativity) ii) ∃e ∈ Γ such that αe = eα = e for all α ∈ Γ (Existence of identity) iii) ∀α ∈ Γ ∃α −1 ∈ Γ such that αα −1 = α −1 α = e (Existence of inverses) Definition 2 A topological group is a group together with a Hausdorff topology such that the maps Γ × Γ → Γ (α, β) → αβ Γ → Γ α → α −1 are continuous. A compact group is a topological group that is a compact topological space. Example 3 (Examples of compact topological groups) i) U (1) = z ∈ C |z| = 1 = e iθ 0 ≤ θ < 2π with the usual multiplication in C and the usual topology in C. That is, e iθ e iϕ = e i(θ+ϕ) , the identity is e i0 , the inverse is e iθ −1 = e −iθ and d e iθ , e iϕ = d cos θ + i sin θ, cos ϕ + i sin ϕ = (cos θ − cos ϕ) 2 + (sin θ − sin ϕ) 2 ii) SO(3) = A A a 3 × 3 real matrix, det A = 1, AA t = 1l. The product is the usual matrix multiplication , the identity is the usual identity matrix and inverses are the usual matrix inverses. The topology is given by the usual Pythagorean metric on IR 9. The requirement that AA t = 1l is equivalent to requiring that the three columns of A = [ a 1 , , a 2 , , a 3 ] ∈ SO(3) be mutually perpendicular unit vectors. That det A = a 1 × a 2 · a 3 = 1 means that the triple (a 1 , , a 2 , , a 3) is right handed. (This could be taken as the definition of "right handed".) The three columns of A are also the images AA e i of the three unit vectors pointing along the positive x, y and z axes, respectively. So SO(3) can be thought of as the set of rotations in IR 3 or as the set of possible orientations of a rigid body. Let Z(θ) =   cos θ − sin θ 0 sin θ cos θ 0 0 0 1   X(θ) =   1 0 0 0 cos θ − sin θ 0 sin θ cos θ   be rotations about the z-and x-axes, respectively. Any γ ∈ SO(3) has a unique representation of the form γ = Z(ϕ 2)X(θ)Z(ϕ 1) with 0 ≤ θ ≤ π, 0 ≤ ϕ 1 , ϕ 2 < 2π. The argument showing this is in [N]. The angles θ, ϕ 1 and ϕ 2 are called the Euler angles. The map (ϕ 1 , ϕ 2 , θ) → γ(ϕ 1 , ϕ 2 , θ) = Z(ϕ 2)X(θ)Z(ϕ 1) provides a local coordinate system on a neighbourhood of each point of SO(3). Locally, SO(3) looks like IR 3. A group that carries local coordinate systems like this is called a Lie group. iii) Any group with only finitely many elements, like the group, S n , of permutations of {1, · · · , n}. Just use the discrete topology.