Noncommutative Noether's problem is almost equivalent to the classical Noether's problem

TL;DR

This paper explores the noncommutative Noether's problem, showing its near equivalence to the classical problem and providing counterexamples by linking properties of skew fields of fractions with rationality issues in algebraic fields.

Contribution

It establishes a connection between the noncommutative and classical Noether's problems, demonstrating that solutions in the noncommutative case imply stable rationality in the classical case for large characteristic fields.

Findings

If the skew field of fractions is G-invariant, then the fixed field is stably rational.

Counterexamples to the noncommutative problem are derived from known classical counterexamples.

The results hold over algebraically closed fields of sufficiently large characteristic.

Abstract

Motivated by the classical Noether's problem, J. Alev and F. Dumas proposed the following question, commonly referred to as the noncommutative Noether's problem: Let a finite group $G$ act linearly on $\mathbb{C}^n,$ inducing the action on $\text{Frac}(A_n(\mathbb{C}))$-the skew field of fractions of the $n$-th Weyl algebra $A_n(\mathbb{C}),$ then is $\text{Frac}(A_n(\mathbb{C}))^G$ isomorphic to $\text{Frac}(A_n(\mathbb{C}))?$ In this note we show that if $\text{Frac}(A_n(\mathbb{C}))^{G}\cong \text{Frac}(A_n(\mathbb{C})),$ then for any algebraically closed field $k$ of large enough characteristic, field $k(x_1,\cdots, x_n)^G$ is stably rational. This result allows us to produce counterexamples to the noncommutative Noether's problem based on well-known counterexamples to the Noether's problem for algebraically closed fields.