Super formal Daboux-Weinstein theorem and finite W superalgebra

Abstract

Let $\vvv=\vvv_{\bar{0}}+\vvv_{\bar{1}}$ be a ${\mathbb{Z}}_2$-graded (super) vector space with an even ${\mathbb{C}}^{\times }$-action and $χ\in \vvv_{\bar{0}}^{*}$ be a fixed point of the induced action. In this paper we will prove a equivariant Daboux-Weinstein theorem for the formal polynomial algebras $\hat{A}=S[\vvv_{\bar{0}}]^{\wedge_χ}\otimes \bigwedge(\vvv_{\bar{1}})$. We also give a quantum version of the equivariant Daboux-Weinstein theorem. Let $⋙={⋙}_{\bar{0}}+{⋙}_{\bar{1}}$ a basic Lie superalgebra of type I and $e\in {⋙}_{\bar{0}}$ be a nilpotent element. We will use the equivariant quantum Daboux-Weinstein theorem to realize the finite $W$ superalgebra $\mathcal{U}(⋙,e)$. An indirect relation between finite U(g,e) and U(g_{\bar{0}} ,e) is presented. Finally we will use this realization to study the finite dimensional representations of $\mathcal{U}(⋙,e)$.

21pages, final version