Orbit Closures in the Witt Algebra and its Dual Space

TL;DR

This paper computes orbit closures in the Witt algebra and its dual space over fields of characteristic p > 3, revealing the structure of invariants and extending techniques to various orbit heights.

Contribution

It provides explicit calculations of orbit closures in the Witt algebra and its dual, and demonstrates the triviality of the invariant algebra under the automorphism group.

Findings

Orbit closures in the Witt algebra are explicitly determined.

The algebra of invariants in the dual space is shown to be trivial.

Techniques are extended to all orbit heights except p-1.

Abstract

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p-1 (in which case we only obtain a conditional statement) in the dual space W^* under the induced action of G. As a corollary we prove that the algebra of invariants k[W^*]^G is trivial.