Classical \(\mathcal W\)-algebras in types \(A\), \(B\), \(C\), \(D\) and \(G\)
In this paper the authors propose an explicit generating system for the classical \(\mathcal{W}\)-algebras associated to the principal nilpotent elements in types \(A\), \(B\), \(C\), \(D\) and \(G\). These generators are given as certain determinants of matrices formed by elements of differential algebras associated with the underlying Lie algebra, thus producing a realization in the context of Poisson vertex algebras. Additionally, it is shown that the image of these generators under the Chevalley-type projection coincide with the elements defined via the corresponding Miura transformation. This projection yields an isomorphism of Chevalley type to an algebra of polynomials defined as the intersection of the kernels of screening operators, providing a connection to the Harish-Chandra image of the Feigin-Frenkel center.
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