Simple smooth modules over the Lie algebras of polynomial vector fields

Abstract Let and be the Witt Lie algebras. Clearly, is a proper subalegbra of . Surprisingly, we prove that simple smooth modules over are exactly the simple modules over studied by Rudakov (no need to take completion). Then, we find an easy and elementary way to classify all simple smooth modules over . When the height or , any nontrivial simple smooth ‐module is isomorphic to an induced module from a simple smooth ‐module . When and , any such module is the unique simple quotient of the tensor module for some simple ‐module , where is a particular simple module over the Weyl algebra . We further show that a simple ‐module is a smooth module if and only if the action of each of particular vectors in is locally finite on .