Yangian characters and classical $W$-algebras

A. I. Molev and E. E. Mukhin

Abstract

The Yangian characters (or $q$-characters) are known to be closely related to the classical $W$-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the $W$-algebras from the characters of the Kirillov-Reshetikhin modules associated with multiples of the first fundamental weight in types $B$ and $D$ and of the fundamental modules in type $C$. We also give an independent derivation of the character formulas for these representations in the context of the RTT presentation of the Yangians. In all cases the generators of the $W$-algebras correspond to the recently constructed elements of the Feigin-Frenkel centers via an affine version of the Harish-Chandra isomorphism.