Finite W-superalgebras for basic Lie superalgebras

摘要: We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over field of complex numbers $\bbf=\mathbb{C}$ and $\bbf={\bbk}$ an algebraically closed positive characteristic. In this paper, we mainly present PBW theorem $U({\ggg}_\bbf,e)$. Then construction $U({\ggg}_\bbf,e)$ can be understood well, which in contrast $W$-algebras, is divided into two cases virtue parity $\text{dim}\,\mathfrak{g_\bbf}(-1)_{\bar1}$. This observation will basis our sequent work on dimensional lower bounds super Kac-Weisfeiler property modular representations superalgebras (cf. \cite[\S7-\S9]{ZS}).