Structures of (supersymmetric) classical W-algebras

In the first part of this paper, we discuss the classical W-algebra W(g,F) associated with a Lie superalgebra g and the nilpotent element F in an sl2-triple. We find a generating set of W(g,F) and compute the Poisson brackets between them. In the second part, which is the main part of this paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra W(g,f) associated with a Lie superalgebra g and an odd nilpotent element f in a subalgebra isomorphic to osp(1|2). The first construction is via the supersymmetric (SUSY) classical Becchi-Rouet-Stora-Tyutin (BRST) complex, and the second is via the SUSY Drinfeld-Sokolov Hamiltonian reduction. We show that these two methods give rise to isomorphic SUSY Poisson vertex algebras. As a supersymmetric analog of the first part, we compute explicit generators and Poisson brackets between the generators.