Exercise 5, § 4

Lie Groups and Lie Algebras, III, 377

Let $(e_1, e_2, e_3)$ be the canonical basis of $\mathbf{R}^3$ . We give $\mathbf{R}^3$ the nilpotent Lie algebra structure $g$ such that $[e_1, e_2] = e_3,\ [e_1, e_3] = [e_2, e_3] = 0$ . Let $c_{ijk}$ denote the constants of structure of $g$ relative to the basis $(e_1, e_2, e_3)$ .

(a) Show that on $\mathbf{R}^3$ the differential forms

\omega_1 = dx_1,\quad \omega_2 = dx_2,\quad \omega_3 = -x_1 dx_2 + dx_3

satisfy the relations

d\omega_k = -\sum_{i < j} c_{ijk} \omega_i \wedge \omega_j \qquad (k = 1, 2, 3)

and are linearly independent at each point of

\mathbf{R}^3

(b) Deduce that there exists, on an open neighbourhood of 0 in $\mathbf{R}^3$ , a Lie group germ structure $G$ such that $L(G) = g$ and

(a_1, a_2, a_3)(x_1, x_2, x_3) = (x_1 + a_1, x_2 + a_2, x_3 + a_3 + a_1 x_2)

for

(a_1, a_2, a_3),\ (x_1, x_2, x_3)

sufficiently close to 0. Show that in fact formulae (1) define a nilpotent Lie group structure on

\mathbf{R}^3

Solution

No solution has been written yet.