We describe an N = 2 supersymmetric Poisson vertex algebra structure of N = 1 (resp. N = 0) classical W -algebra associated with sl(n + 1|n) and the odd (resp. even) principal nilpotent element. This N = 2 supersymmetric structure is connected to the principal sl(2|1)-embedding in sl(n + 1|n) superalgebras, which are the only basic Lie superalgebras that admit such a principal embedding.
In the first part of this paper, we generalize the Dirac reduction to the extent of non-local Poisson vertex superalgebra and non-local SUSY Poisson vertex algebra cases. Next, we modify this reduction so that we explain the structures of classical W-superalgebras and SUSY classical W-algebras in terms of the modified Dirac reduction.
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