# On automorphisms of enveloping algebras

## TL;DR

This paper explores the relationship between automorphisms of universal enveloping algebras and Lie algebra automorphisms, introducing a new invariant derived from reduction mod infinitely large primes and establishing connections with derived categories.

## Contribution

It introduces a canonical construction of a Lie algebra invariant over an extended field and links automorphisms of enveloping algebras to Lie algebra automorphisms, providing new structural insights.

## Key findings

- The invariant $rak{g}_ ext{infty}$ is determined by the derived category of modules.
- A canonical homomorphism from automorphisms of $rak{U}rak{g}$ to automorphisms of $rak{g}$ is constructed.
- Finite automorphism groups of $rak{U}rak{g}$ are subgroups of Lie algebra automorphisms.

## Abstract

Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras $\mathfrak{g}_{\infty}$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a nontrivial semi-simple automorphism. As a corollary we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$