Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

TL;DR

This paper studies the automorphism groups of specific polynomial maps called folding maps, derived from semisimple Lie algebras, and provides explicit formulas and computations for these groups in certain cases.

Contribution

It introduces a family of polynomial folding maps associated with semisimple Lie algebras and computes their affine automorphism groups explicitly in key examples.

Findings

Formulas for leading terms of folding maps for types A2, B2, G2

Explicit computation of automorphism groups for these folding maps

Establishment of properties like topological degree and commutativity

Abstract

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps $$\textsf{F}_{n}:\mathbb{A}^N\to\mathbb{A}^N\quad\text{for}\quad n\ge1$$ having the property that $\textsf{F}_{n}$ has topological degree $n^N$ and $$\textsf{F}_{m}\circ\textsf{F}_{n}=\textsf{F}_{n}\circ\textsf{F}_{m}\quad\text{for all}\quad m,n\ge1.$$ We derive formulas for the leading terms of the folding maps on $\mathbb{A}^2$ associated to the Lie algebras $\mathcal{A}_2$, $\mathcal{B}_2$, and $\mathcal{G}_2$, and we use these formulas to compute the affine automorphism group of each folding map.