Classical $\mathcal{W}$-algebras in types $A$, $B$, $C$, $D$ and $G$

A. I. Molev and E. Ragoucy

Abstract

We produce explicit generators of the classical $\mathcal{W}$-algebras associated with the principal nilpotents in the simple Lie algebras of all classical types and in the exceptional Lie algebra of type $G_2$. The generators are given by determinant formulas in the context of the Poisson vertex algebras. We also show that the images of the W-algebra generators under the Chevalley-type isomorphism coincide with the elements defined via the corresponding Miura transformations.