On the Construction of Simply Connected Solvable Lie Groups

TL;DR

This paper demonstrates how to explicitly construct simply connected solvable Lie groups from their Lie algebra data using quadratures and matrix exponential, providing practical formulas and applications in differential systems.

Contribution

It introduces a method to construct solvable Lie groups explicitly via quadratures and matrix exponential, including a closed form for Lie's third theorem vector fields.

Findings

Explicit construction of solvable Lie groups from Lie algebra using quadratures.

Closed form formula for vector fields in Lie's third theorem for solvable algebras.

Application to find multiplication maps and first integrals in differential systems.

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 $\rho^* \tau= \omega_\mathfrak{g}$ 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.