Finite W-superalgebras for basic lie superalgebras

We consider the finite W-superalgebra U(gF,e) for a basic Lie superalgebra gF=(gF)0-+(gF)1- associated with a nilpotent element e∈(gF)0- both over the field of complex numbers F=C and over F=k an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for U(gF,e). Then the construction of U(gF,e) can be understood well, which in contrast with finite W-algebras, is divided into two cases in virtue of the parity of dimgF(-1)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. [43, §7-§9]).