Quantization of Galois theory, Examples and Observations

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.