# A generalization of Veldkamp's theorem for a class of Lie algebras

## TL;DR

This paper extends Veldkamp's classical theorem to a broader class of Lie algebras that originate from algebraic Lie algebras with specific invariance properties, providing new insights into their enveloping algebra centers.

## Contribution

It generalizes Veldkamp's theorem to Lie algebras arising as reductions modulo large primes with particular invariance conditions.

## Key findings

- Generalized the description of the center of enveloping algebras for new Lie algebra classes.
- Solved the derived isomorphism problem for these enveloping algebras.
- Established conditions under which the symmetric invariants form a polynomial algebra.

## Abstract

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo $p\gg 0$ from an algebraic Lie algebra $\mathfrak{g},$ such that $\mathfrak{g}$ has no nontrivial semi-invariants in $Sym(\mathfrak{g})$ and $Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras.