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Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

Jospeh H. Silverman

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.

Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

Abstract

Let be a finite-dimensional semisimple Lie algebra of rank over an algebraically closed field of characteristic . Associated to is a family of polynomial folding maps having the property that has topological degree and We derive formulas for the leading terms of the folding maps on associated to the Lie algebras , , and , and we use these formulas to compute the affine automorphism group of each folding map.

Paper Structure

This paper contains 5 sections, 10 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Folding Maps for the Lie Algebra ${\mathcal{A}}_2$
  3. Folding Maps for the Lie Algebra ${\mathcal{B}}_2$
  4. Folding Maps for the Lie Algebra ${\mathcal{G}}_2$
  5. Geometric Properties of $2$-Dimensional Folding Maps

Key Result

Theorem 1.4

MR909112MR880608 Let ${\mathcal{L}}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $N$ be the rank of ${\mathcal{L}}$, i.e., the dimension of a Cartan subalgebra. Then for each $n\ge0$ there is an associated polynomial map called the $n$th folding mapFor ${\mathcal{B}}_2$, ${\mathcal{G}}_2$, and ${\mathcal{F}}_4$, there are additi

Figures (1)

  • Figure 4.1: The first few $G_n$ polynomials

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Theorem 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • proof
  • Remark 1.9
  • ...and 22 more