Normal Forms in Differential Galois Theory for the Classical Groups

TL;DR

This paper constructs a differential field extension where a generic Lie algebra element for classical groups is gauge equivalent to a normal form, preserving the differential Galois group, and explores specializations of coefficients.

Contribution

It introduces a method to normalize generic Lie algebra elements over an extended field while maintaining the differential Galois group, extending previous work by Seiss.

Findings

Constructed a differential field extension with the same Galois group.

Established gauge equivalence to a normal form over the extension.

Analyzed the effects of coefficient specialization on the structure.

Abstract

Let $G$ be a classical group of dimension $d$ and let $\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the Lie algebra $\mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ of $G$ obtained from parametrizing a basis of $\mathfrak{g}$ with the indeterminates $\boldsymbol{a}$. It is known (cf. work by Juan) that the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F\langle \boldsymbol{a} \rangle$ is $G(C)$. In this paper we construct a differential field extension $\mathcal{L}$ of $F\langle \boldsymbol{a} \rangle$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a matrix in normal form which we introduced in work by Seiss. We also consider specializations of the coefficients of $A(\boldsymbol{a})$.