Finite W-superalgebras for basic Lie superalgebras

We consider the finite $W$-superalgebra $U({\mathfrak{g}}_{\mathfrak{\text{bbf}}},e)$ for a basic Lie superalgebra ${⋙}_{\text{bbf}}=({⋙}_{\text{bbf}}{)}_{\text{bz}}+({⋙}_{\text{bbf}}{)}_{\text{bo}}$ associated with a nilpotent element $e\in ({⋙}_{\text{bbf}}{)}_{\bar{0}}$ both over the field of complex numbers $\text{bbf}=\mathbb{C}$ and over $\text{bbf}=\text{bbk}$ an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for $U({⋙}_{\text{bbf}},e)$. Then the construction of $U({⋙}_{\text{bbf}},e)$ can be understood well, which in contrast with finite $W$-algebras, is divided into two cases in virtue of the parity of $\text{dim}{\mathfrak{g}}_{\mathfrak{\text{bbf}}}(-1{)}_{\bar{1}}$. This observation will be a basis of our sequent work on the dimensional lower bounds in the super Kac-Weisfeiler property of modular representations of basic Lie superalgebras (cf. \cite[\S7-\S9]{ZS}).