On Gelfand-Kirillov conjecture for some W-algebras

TL;DR

This paper investigates the Gelfand-Kirillov conjecture for W-algebras associated with minimal nilpotent orbits in simple Lie algebras, showing it fails for certain types based on existing results.

Contribution

It establishes a link between the conjecture's validity for W-algebras and universal enveloping algebras, and demonstrates its failure for specific Lie algebra types.

Findings

The Gelfand-Kirillov conjecture for W-algebras implies its validity for universal enveloping algebras.

The conjecture fails for W-algebras associated with certain nilpotent orbits in types B, D, E, and F Lie algebras.

The results extend the understanding of the conjecture's limitations in Lie algebra theory.

Abstract

Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ 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 $\mathrm U(\frak g)$. 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 $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.