Noether bound for invariants in relatively free algebras

TL;DR

This paper extends Noether's classical bound on degrees of invariants to relatively free algebras within weakly noetherian varieties of associative algebras, establishing a uniform degree bound for invariants under finite group actions.

Contribution

It proves a general degree bound for invariants in relatively free algebras of weakly noetherian varieties, generalizing Noether's classical result.

Findings

Invariant subalgebras are generated by elements of bounded degree.

The degree bound depends only on the variety and the group, not on the generating vector space.

The result applies to a broad class of associative algebras beyond commutative polynomial rings.

Abstract

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$ denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ of rank $d$ freely generated by the vector space $V$. It is proved that the subalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements of degree at most $b(\mathfrak{R},G)$ for some explicitly given number $b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group $G$ (but not on $V$). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants $K[V]^G$ is generated by invariants of degree at most $\vert G\vert$.