On the Gelfand–Kirillov conjecture for the W-algebras attached to the minimal nilpotent orbits

Abstract

Abstract. Consider the W-algebra H attached to the minimal nilpotent orbit in a simple Lie algebra 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 U(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 the minimal nilpotent orbit in Lie algebras of types Bn (n ≥ 3), Dn (n ≥ 4), E 6 , E 7 , E 8 , and F 4 .

Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

“…Let g be a simple finite-dimensional Lie algebra over an algebraically closed field F of characteristic 0. We denote by U(g) the universal enveloping algebra of g. To any nilpotent element e ∈ g one can attach an associative (and noncommutative as a general rule) algebra U(g, e) which is in a proper sense a "tensor factor" of U(g), see [Los2,Theorem 1.2.1], [Pet,Theorem 2.1]. The notion of W-algebra can be traced back to the work [Lyn], see also [Kos].…”