Exercises Lie groups

Exercise 1. Let G be a group, equipped with the structure of a C∞-manifold. Let μ : G×G→ G, (x, y) 7→ xy be the multiplication map. We assume that μ is smooth, i.e., C∞. (a) Show that the tangent map T(e,e)μ : TeG×TeG→ TeG of μ at (e, e) is given by (X,Y ) 7→ X + Y. Hint: use partial derivatives with respect to x and y. (b) Show that the inversion map ι : G → G, x 7→ x−1 is smooth in an open neighborhood of e and that the tangent map Teι : TeG→ TeG is given by X 7→ −X. Hint: use the implicit function theorem.