Zariski Closures of Reductive Linear Differential Algebraic Groups

TL;DR

This paper investigates the Zariski closures of linear differential algebraic groups (LDAGs), providing a Tannakian framework to characterize and classify these closures, especially for reductive and simple LDAGs, with implications for differential Galois theory.

Contribution

It offers a Tannakian characterization of Zariski closures of LDAGs, extending classical results and aiding in algorithms for computing differential Galois groups.

Findings

Zariski closures of reductive LDAGs are all isomorphic for minimal representations

A Tannakian description of simple LDAGs is provided

The results extend classical Cassidy theorems and impact differential Galois group computations

Abstract

Linear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all isomorphic. In addition, we give a Tannakian description of simple LDAGs. This substantially extends the classical results of P. Cassidy and, we hope, will have an impact on developing algorithms that compute differential Galois groups of the above equations and factoring partial differential operators.