Differential Invariants in Algebra

TL;DR

This paper explores two methods for analyzing orbit spaces of algebraic Lie groups, emphasizing that the differential approach offers deeper insights into invariants and orbit structures, with applications to classical equivalence problems.

Contribution

It compares algebraic and differential methods for studying orbit spaces, highlighting the advantages of the differential approach in understanding invariants.

Findings

Differential approach provides better understanding of invariants.

Application to SL-classification of forms and affine classification of curves.

Illustrates classical equivalence problems using differential invariants.

Abstract

In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential equation. The main goal of these lectures is to show that the differential approach gives us a better understanding of structure of invariants and orbit spaces. We illustrate this on classical equivalence problems, such as $\mathrm{SL}$ - classification of binary and ternary forms, and affine classification of algebraic plane curves.