Transitive Lie algebras of vector fields: an overview

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late nineteenth century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in twentieth-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others. This paper gives just one, subjective overview of the subject, without trying to be exhaustive.