
Journal of Lie Theory 27 (2017), No. 1, 193215 Copyright Heldermann Verlag 2017 On the Construction of Simply Connected Solvable Lie Groups Mark E. Fels Dept. of Mathematics and Statistics, Utah State University, Logan, UT 84322, U.S.A. mark.fels@usu.edu [Abstractpdf] \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 MaurerCartan 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. Keywords: Solvable Lie algebras, solvable Lie groups, Lie's third theorem, first integrals. MSC: 22E25; 58A15, 58J70, 34A26 [ Fulltextpdf (395 KB)] for subscribers only. 