Quantization of Galois theory, Examples and Observations

TL;DR

This paper explores how Galois groups behave under quantization of differential equations, showing that linear cases remain classical while non-linear cases exhibit quantized Galois groups, supported by examples.

Contribution

It demonstrates that quantization affects Galois groups differently for linear and non-linear equations, extending the understanding of differential Galois theory in a quantized setting.

Findings

Linear q-analogues have classical Galois groups.

Non-linear equations exhibit quantized Galois groups.

Results rely on differential Galois theory for non-linear equations.

Abstract

If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We show by Examples that if we consider non-linear equations Galois group is quantized. The results depend on general differential Galois theory for non-linear differential equations developed by the second author.