Cable algebras and rings of Ga-invariants

Abstract

For a field $k$, the ring of invariants of an action of the unipotent $k$-group ${\mathbb{G}}_a$ on an affine $k$-variety is quasiaffine, 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 ${\mathbb{G}}_a$-action on ${\mathbb{A}}_k^5$ 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 ${\mathbb{G}}_a$-action on ${\mathbb{A}}_k^7$ due to Roberts is a cable algebra.