Explicit generators in rectangular affine $\mathcal{W}$-algebras of type $A$

Tomoyuki Arakawa and Alexander Molev

Abstract

We produce in an explicit form free generators of the affine $\mathcal{W}$-algebra of type $A$ associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.