Finite W-superalgebras for basic classical Lie superalgebras

TL;DR

This paper studies finite W-superalgebras associated with basic classical Lie superalgebras, establishing their PBW theorem, proposing a conjecture on minimal representations, and connecting to the super Kac-Weisfeiler conjecture in positive characteristic.

Contribution

It introduces the PBW theorem for finite W-superalgebras and formulates a conjecture on minimal dimensional representations, linking to modular representation theory.

Findings

PBW theorem established for finite W-superalgebras

Conjecture on minimal representations proposed and exemplified

Lower bounds in super Kac-Weisfeiler conjecture can be achieved

Abstract

We consider the finite W-superalgebras for a basic classical Lie superalgebra g associated with an even nilpotent element in g both over the field of complex numbers field and and over a filed of positive characteristic. We present the PBW theorem for these finite W-superalgebrfas. Then we formulate a conjecture about the minimal dimensional representations of of complex finite W-superalgebras, and demonstrate it with some examples. Under the assumption that the conjecture holds, we finally show that the lower bound of dimensions predicted in the super version of Kac-Weisfeiler conjecture formulated and proved by Wang-Zhao in [40] for the modular representations of the basic classical Lie superalgebra with any p-characters can be reached.