Cable algebras and rings of $G_a$-invariants

TL;DR

This paper introduces cable algebras as a new framework to study rings of invariants under unipotent group actions, providing explicit constructions and relations for specific examples.

Contribution

It demonstrates that certain invariant rings are cable algebras, offering explicit generators and relations, advancing understanding of their structure.

Findings

The invariant ring for Daigle and Freudenburg's $G_a$-action is a monogenetic cable algebra.

A generating cable and relations are explicitly constructed for this ring.

The invariant ring for Roberts' $G_a$-action is also shown to be a cable algebra.

Abstract

For a field $k$, the ring of invariants of an action of the unipotent $k$-group $G_a$ on an affine $k$-variety is quasi-affine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the $G_a$-action on $A^5_k$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known $G_a$-action on $A^7_k$ due to Roberts is a cable algebra.