Invariants of finite groups acting on (free) skew fields
Harm Derksen, Jurij Volčič
TL;DR
The paper develops a noncommutative invariant theory for finite group actions on skew fields, proving that invariant skew subfields are finitely generated in general and, for free skew fields under linear group actions with char(k) ∤ |G|, are themselves free with rank |G|(m-1)+1. It provides explicit constructions and fundamental relations governing invariants, and shows that freeness can fail for nonlinear actions (notably Z2), leading to non-free invariant subfields and non-rational centralizers that refute conjectures about centralizers. It furthermore connects these algebraic results to a noncommutative analogue of Noether’s problem and Lüroth’s problem, and offers concrete algorithms for constructing generators and verifying freeness. The work highlights a sharp dichotomy between linear and nonlinear group actions on free skew fields, with implications for the structure of invariants, centralizers, and computational approaches in noncommutative invariant theory.
Abstract
Let $M$ be a finitely generated skew field over a ground field $k$, and let $G$ be a finite group of $k$-linear automorphisms of $M$. This paper investigates finite generation of the skew subfield $M^G$ of $G$-invariants in $M$, and relations between the generators. The first main result shows that $M^G$ is finitely generated. Stronger conclusions hold when $M$ is a free skew field, i.e., the universal skew field of fractions of a free algebra. The second main result is the solution of the free Noether problem for non-modular linear group actions: if $G$ acts linearly on the free skew field $M$ on $m$ generators and the characteristic of $k$ does not divide $|G|$, then $M^G$ is the free skew field on $|G|(m-1)+1$ generators. In contrast, a nonlinear action of $Z_2$ on the free skew field $M$ on two generators is presented such that $M^{Z_2}$ is not a free skew field, resolving the free Lüroth problem. This action also exposes a non-scalar element of $M$ whose centralizer is not a rational field, refuting a conjecture of P. M. Cohn from 1978.