Toward quantization of Galois theory

TL;DR

This paper explores the possibility of developing a Galois theory where the Galois group is a quantum group, extending classical concepts to non-commutative and quantum algebraic structures.

Contribution

It proposes a framework for Galois theory with quantum groups as Galois groups, especially for linear equations over Hopf algebras with constant base fields.

Findings

Existence of non-commutative Picard-Vessiot rings.

Development of asymmetric Tannaka theory.

Examples suggesting potential for non-linear equations.

Abstract

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. For non-linear equations there are examples that might make us optimistic.