Affine Kac-Moody Algebras at the Critical Level and Gelfand-Dikii Algebras

Abstract

We prove Drinfeld's conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii algebra, associated to the Langlands dual algebra. The center is identified with a limit of the W-algebra, defined by means of the quantum Drinfeld-Sokolov reduction.

Vertex Algebras and Algebraic Curves

“…It is isomorphic to a limit of the Wvertex algebra W k (g) as k → ∞ (see §3.7), and because of that it is called the classical 875-32 W-algebra corresponding to g. For example, A sl 2 (D) is a limit of Vir c as c → ∞. The following result was conjectured by Drinfeld and proved in [FF3].…”

“…We illustrate it in the case of quantum Drinfeld-Sokolov reduction [FF3], which leads to the definition of W-algebras. Let g be a simple Lie algebra of rank ℓ, and n its upper nilpotent subalgebra.…”

“…This theorem is proved in [FF3,FF4] for generic k and in [dBT] for all k. The vertex algebra H 0 k (g) is called the W-algebra associated to g and denoted by W k (g). We have:…”

“…On the other hand, W −h ∨ (g) is a commutative vertex algebra, which is isomorphic to the center of V −h ∨ (g) [FF3] (see §6.5). The simple quotient of W k (g) for k = −h ∨ + p/q, where p, q are relatively prime integers greater than or equal to h ∨ , is believed to be a rational vertex algebra.…”

“…, where L g is the Langlands dual Lie algebra to g and (k + h ∨ )r ∨ = (k ′ + L h ∨ ) −1 (here r ∨ denotes the maximal number of edges connecting two vertices of the Dynkin diagram of g), see [FF3]. In the limit k → −h ∨ it becomes the isomorphism of Theorem 6.3.…”