Gelfand-Dickey Realizations of the supersymmetric classical W-algebras for gl(n+1|n) and gl(n|n)

In this paper we realize the supersymmetric classical W-algebras W(gl̄(n+1|n)) and W(gl̄(n|n)) as differential algebras generated by the coefficients of a monic superdifferential operator L. In the case of W(gl̄(n|n)) (resp. W(gl̄(n+1|n))) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator L. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra W(gl̄(n+1|n)) one needs to introduce a new concept of even supersymmetric Poisson vertex algebras.