By Karl-Heinz Fieseler

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For R such subgroups are either trivial or of the form Zω with some ω ∈ R \ {0}, hence G ∼ = R or G ∼ = R/Zω ∼ = S1 . For C there is a further possibility for ker γ, namely ker γ = Λ, a lattice. So either G ∼ = C or G ∼ = C∗ or G ∼ = C/Λ. Note that in the real case R ∼ = R>0 via the exponential function, while C ∼ = R>0 × S1 resp. C/Λ ∼ = S1 × S1 as real Lie groups. On the other ˜ as complex Lie groups (or even as complex manifolds) iff hand C/Λ ∼ = C/Λ ˜ = αΛ with a nonzero complex number α ∈ C∗ .

If [Y, Z] = X we obtain the vector product multiplication table of the vectors in an orthonormal basis of R3 . More explicitly there is an isomorphism ∼ = g −→ so3 (R) with 0 0 0 0 0 1 0 1 0 X → 0 0 1 , Y → 0 0 0 , Z → −1 0 0 . 0 −1 0 −1 0 0 0 0 0 Using complex matrices we have even an isomorphism ∼ = g −→ su2 ⊂ sl2 (C) with X→ i 0 0 −i ,Y → 0 1 −1 0 ,Z → 0 i i 0 . Note that su2 ∼ = so3 (R) is not isomorphic to sl2 (R) since in the former algebra no derivation ad(X), X ∈ g \ {0} is diagonalizable.

4. , a continuous map F : [0, 1] × [0, 1] −→ Y with the following properties: F (0, t) = α(0) = β(0), F (1, t) = α(1) = β(1) and Ft (s) := F (s, t) satisfies F0 = α, F1 = β. 5. 1. To be homotopic is an equivalence relation on the set of paths from a given point x ∈ Y to another given point y ∈ Y . We denote [γ] the equivalence class (homotopy class) of the path γ. If τ : [0, 1] −→ [0, 1] is a continuous map with τ (0) = 0, τ (1) = 1 (a ”reparametrization“), then γ ◦ τ ∼ γ. 53 2. Given paths α, β : [0, 1] −→ Y , such that β(0) = α(1), we define the concatenation αβ : [0, 1] −→ Y by (αβ)(s) = α(2s) , if 0 ≤ s ≤ 12 β(2s − 1) , if 12 ≤ s ≤ 1 3.