On the Construction of Simply Connected Solvable Lie Groups

Abstract

Let ${\omega }_{\mathfrak{g}}$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d{\omega }_{\mathfrak{g}}+\frac{1}{2}{\omega }_{\mathfrak{g}}\wedge {\omega }_{\mathfrak{g}}=0$ where $\mathfrak{g}$ is solvable. We show that the problem of finding a smooth map $\rho :M\to G$, where $G$ is an $n$-dimensional solvable Lie group with Lie algebra $\mathfrak{g}$ and left invariant Maurer-Cartan form $\tau$, such that ${\rho }^{\ast }\tau ={\omega }_{\mathfrak{g}}$ 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.

22 pages. Fixed typos from version 1, and added more details in the examples