On the Construction of Simply Connected Solvable Lie Groups

摘要

Let omega(g) be a Lie algebra valued differential 1-form on a manifold M satisfying the structure equations d omega(g)+1/2 omega g boolean AND omega(g) = 0 where g is solvable. We show that the problem of finding a smooth map rho: M -> G, where G is an n-dimensional solvable 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.