Finite W-superalgebras for basic Lie superalgebras

We consider the finite W-superalgebra U (g(F), e) for a basic Lie superalgebra g(F) = (g(F))((0) over bar) + (g(F))((1) over bar) associated with a nilpotent element e is an element of (g(F))((0) over bar) 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 (g(F), e). Then the construction of U (g(F), e) can be understood well, which in contrast with finite W-algebras, is divided into two cases in virtue of the parity of dim g(F)(-1)((1) over bar). 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]). (C) 2015 Elsevier Inc. All rights reserved.