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On the Construction of Simply Connected Solvable Lie Groups

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