Exercise 10, § 9

Lie Groups and Lie Algebras, III, 398

Bourbaki marks this one as harder than the rest.

Let G be a finite-dimensional real or complex Lie group. Let

p = \dim[L(G), L(G)].

Let (G, G) be given its structure as an integral subgroup of G. There exists a neighbourhood V of e in (G, G) such that every element of V is the product of p commutators of elements of G. (Let $x_1, y_1, \ldots, x_p, y_p$ be elements of L(G) such that the $[x_i, y_i]$ form a basis of L(G). For s, t in $\mathbf{R}$ , we write

\rho_i(s, t) = (\exp sy_i)^{-1}(\exp tx_i)^{-1}(\exp sy_i)(\exp tx_i).

Apply the implicit function theorem to the mapping

(s_1, t_1, \ldots, s_p, t_p) \mapsto \rho_1(s_1, t_1) \ldots \rho_p(s_p, t_p)

of $\mathbf{R}^{2p}$ into (G, G).

Solution

No solution has been written yet.