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

Abstract

In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type e S n and S n .Moreover, we show that the variety of nilpotent elements in S n is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on S n .

Adjoint quotient maps for the restricted Lie algebrasWnandSn

“…Note that D p i η is a basis for W n−1 over B n−1 . Hence f 0 , · · · , f n−2 are annihilated by all derivations of B n−1 , this implies that f i ∈ k. Furthermore, use the same induction argument in [WCL,Lemma 4.1,4.2],…”

“…Proof. We only need to prove the surjectivity, since the map is injective ( [WCL,Lemma 5.5]). Ω 1 is an affine subspace of S n , so k[Ω 1 ] is a polynomial ring in the n − 1 coordinate functions D η → (−1) n−i η i .…”

“…where Ω η defined in (4-6). In [WCL,Proposition 5.2], the authors proved G.Ω = S n and G.Ω 0 = N(S n ) (see also Theorem 4.4). The following result is a direct consequence of Lemma 4.12.…”

The nilpotent variety of W(1;n)p is irreducible

“…Since G ′ is connected and n is irreducible, the variety N (g) is irreducible [3, p. 64]. Moreover, this conjecture holds for the Jacobson-Witt algebra W (n; 1) [6], for the Special Lie algebras S(n; 1) [13] and for the Hamiltonian Lie algebras H(2n; 1) [14]. In this paper, we are interested in the minimal p-envelope of the Zassenhaus algebra.…”