# Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras

## TL;DR

This paper demonstrates that the derived category of the fixed ring of the enveloping algebra of a semisimple Lie algebra under a finite automorphism group uniquely determines the Lie algebra and the automorphism group, using geometric properties of the Zassenhaus variety.

## Contribution

It establishes a derived invariance result linking the structure of the fixed ring to the original Lie algebra and automorphism group, utilizing geometric methods.

## Key findings

- Derived category determines the Lie algebra and automorphism group.
- Uses geometry of the Zassenhaus variety in positive characteristic.
- Shows non-existence of certain étale coverings.

## Abstract

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and let $W$ be a finite subgroup of $\mathbb{C}$-algebra automorphisms of the enveloping algebra $U(\mathfrak{g})$. We show that the derived category of $U(\mathfrak{g})^W$-modules determines isomorphism classes of both $\mathfrak{g}$ and $W.$ Our proofs are based on the geometry of the Zassenhaus variety of the reduction modulo $p\gg 0$ of $\mathfrak{g}.$ Specifically, we use non-existence of certain \'etale coverings of its smooth locus