Twisted forms of differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras

TL;DR

This paper classifies all twisted forms of differential Lie algebras over c(t) linked to complex simple Lie algebras, using descent theory and Hopf-Galois theory to understand their structure.

Contribution

It provides a complete classification of twisted forms of differential Lie algebras over c(t) associated with complex simple Lie algebras, answering a question posed by A. Pianzola.

Findings

All twisted forms are classified via torsors and Hopf-Galois theory.

The work extends descent theory to the differential algebra context.

Provides explicit descriptions of the twisted forms.

Abstract

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras. Hopf-Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.