Roots of the affine Cremona group

TL;DR

This paper characterizes the root vectors of the affine Cremona group with respect to the diagonal torus, showing they are exactly the locally nilpotent derivations of a specific form, answering a question by Popov.

Contribution

It provides a complete description of the root vectors of the affine Cremona group, clarifying their structure in relation to locally nilpotent derivations.

Findings

Root vectors are exactly the locally nilpotent derivations of a specific form.

Provides an answer to Popov's question about the structure of root vectors.

Clarifies the algebraic structure of automorphisms of affine space.

Abstract

Let k[x_1,...,x_n] be the polynomial algebra in n variables and let A^n=Spec k[x_1,...,x_n]. In this note we show that the root vectors of the affine Cremona group Aut(A^n) with respect to the diagonal torus are exactly the locally nilpotent derivations x^a\times d/dx_i, where x^a is any monomial not depending on x_i. This answers a question due to Popov.