Nilpotent Orbits in the Witt AlgebraW₁

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.…”

Orbit Closures in the Witt Algebra and Its Dual Space

“…Thus every element of degree −1 is in the orbit of e −1 + ae p−2 for some a. It remains only to show that e −1 + ae p−2 and e −1 + be p−2 are in the same orbit if and only if a = b, and here one can use exactly the same method as in the proof of Proposition 3.4 in [7] (it is also a consequence of the proof of case 1 in Theorem 2.2, starting on page 10). Similarly, one can mimic the proof of Theorem 4.1 in said paper to show that G(e −1 + ae p−2 ) has trivial stabilizer in G, which implies…”

“…Cases 3, 3', 4 and 4' were taken care of in [7] as these, along with Ge −1 and 0, account for the nilpotent orbits. The proofs of 1 and 2, as well as the corresponding dimension statements, are very similar, but we include them here anyway for the sake of completeness.…”

“…Proof of case 1 in Theorem 2.2. The statement for a = 0 was proved in [7], so assume a = 0. For an arbitrary λ ∈ k we define:…”

The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn

“…(3) In [19], the authors proved that there are infinitely many nilpotent orbits in the Jacobson-Witt algebra W n . To avoid confusion, for ¹f 1 ; : : : ; f n 1 º Â K (see (4.1)) with deg f i 1, 8i , we use the notation 0 f 1 ;f 2 ;:::;f n 1 as the representative element of 0 : Fix D 0 D 0 f 1 ;:::;f n 1 2 0 ; and define D 0 D ¹g 2 G j g.D 0 / 2 0 º.…”