Commuting homogeneous locally nilpotent derivations

TL;DR

This paper characterizes when homogeneous locally nilpotent derivations commute on affine varieties with torus actions of complexity one, using combinatorial criteria, and applies this to classify certain unipotent group actions.

Contribution

It provides a combinatorial criterion for the commutation of homogeneous locally nilpotent derivations on affine $b{T}$-varieties of complexity one, advancing the understanding of unipotent group actions.

Findings

Homogeneous locally nilpotent derivations commute iff a specific combinatorial condition holds.

The results enable classification of two-dimensional unipotent group actions on affine $b{T}$-varieties.

The paper links algebraic derivations with polyhedral divisors in the context of torus actions.

Abstract

Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb{T}$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb{K}[X]$ can be described in terms of proper polyhedral divisors corresponding to $\mathbb{T}$-variety $X$. We prove that homogeneous locally nilpotent derivations commute if an only if some combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine $\mathbb{T}$-varieties.