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.