Orbit Closures in the Witt Algebra and Its Dual Space

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.

Restricted Cartan Type Lie Algebras and the Coadjoint Representation

“…Not so for a restricted Lie algebra L of Cartan type, where the Killing form is identically zero. Here we prove that there are no nontrivial invariants of L * , which of course represents a radical departure from the characteristic zero theory (this was proved for the simplest case L = W (1) in [5]).…”

“…Use the action of k * on χ 0 + χ 1 and take the limit as t approaches zero. Using the grading on L and the assumption p ≥ 5 one checks that, with the exception of the case L = W (1), p = 5, we have (ad y) 2 = 0 and [(ad y)(x 1 ), (ad y)(x 2 )] = 0 for all x 1 , x 2 ∈ L, which implies that g = exp(ad y) = id + ad y is an automorphism of L (if L = W (1) one can use the results on orbit representatives in [5] to prove Y = L * ). We treat the case L ∈ {W, S, H} first.…”

Annual report of Department of Thoracic Surgery at Shanghai Chest Hospital

“…[6]), there are infinitely many nilpotent orbits in the Witt algebra. Moreover, M. Mygind [7] provided a complete picture of the orbit closures in the Witt algebra and its dual space, extending the results in [14]. This paper is structured as follows.…”