Lie foliations transversely modeled on nilpotent Lie algebras

Abstract

In this paper, we study on the following two problems of realization for Lie foliations. (1) Which pair of Lie algebras $$(\mathfrak {g},\mathfrak {h})$$ can be realized as a Lie $$\mathfrak {g}$$ -foliation in a closed manifold with the structure Lie algebra $$\mathfrak {h}$$ ? (2) Which pair $$(\mathfrak {g},m)$$ can be realized as a Lie $$\mathfrak {g}$$ -flow in a closed manifold with the structure Lie algebra $${\mathbb {R}}^m$$ ? We give a complete answer to (1) in the case where $$\mathfrak {g}$$ is a nilpotent Lie algebra and give a complete answer to (2) in the case where $$\mathfrak {g}$$ is a nilpotent Lie algebra which has a rational structure.