On the Construction of Simply Connected Solvable Lie Groups

Mark E. Fels¹

¹Dept. of Mathematics and Statistics, Utah State University, Logan, UT 84322, U.S.A.

Journal of Lie Theory, Tome 27 (2017) no. 1, pp. 193-215

Résumé

\def\g{{\frak g}} Let $\omega_\g$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d\omega_\g+{1\over2}\omega_\g\wedge\omega_\g=0$, where $\g$ is a solvable real Lie algebra. We show that the problem of finding a smooth map $\rho\colon M\to G$, where $G$ is an $n$-dimensional solvable real Lie group with Lie algebra $\g$ and left invariant Maurer-Cartan form $\tau$, such that $\rho^* \tau= \omega_\g$ can be solved by quadratures and the matrix exponential. In the process, we give a closed form formula for the vector fields in Lie's third theorem for solvable Lie algebras. A further application produces the multiplication map for a simply connected $n$-dimensional solvable Lie group using only the matrix exponential and $n$ quadratures. Applications to finding first integrals for completely integrable Pfaffian systems with solvable symmetry algebras are also given.

Détail

Classification : 22E25, 58A15, 58J70, 34A26
Mots-clés : Solvable Lie algebras, solvable Lie groups, Lie's third theorem, first integrals