On Gelfand-Kirillov conjecture for some W-algebras

Abstract

Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a W-algebra then it holds for the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$. This together with a result of A. Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for some $W$-algebras attached to some nilpotent orbits in Lie algebras of types $B_n(n\ge 3)$, $D_n(n\ge 4)$, $E_6,E_7,E_8$, $F_4$.

References in corpus (1)

• Modular Lie algebras and the Gelfand-Kirillov conjecture