On General Extension Fields for the Classical Groups in Differential Galois Theory

TL;DR

This paper constructs a general differential extension field for classical Lie groups in differential Galois theory, paralleling Noether's classical Galois theory for finite groups, and demonstrates its generic properties.

Contribution

It introduces a new construction of a differential extension field for classical groups, analogous to Noether's approach, with explicit properties and invariants.

Findings

Constructed a differential field of transcendence degree l for classical groups

Proved the field is a Picard-Vessiot extension of the invariants field

Showed the extension satisfies generic properties for G-primitive Picard-Vessiot extensions

Abstract

Let $G$ be one of the classical groups of Lie rank $l$. We make a similar construction of a general extension field in differential Galois theory for $G$ as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential field $E$ of differential transcendence degree $l$ over the constants on which the group $G$ acts and show that it is a Picard-Vessiot extension of the field of invariants $E^G$. The field $E^G$ is differentially generated by $l$ differential polynomials which are differentially algebraically independent over the constants. They are the coefficients of the defining equation of the extension. Finally we show that our construction satisfies generic properties for a specific kind of $G$-primitive Picard-Vessiot extensions.