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When is the automorphism group of an affine variety nested?

Alexander Perepechko, Andriy Regeta

Abstract

For an affine algebraic variety $X$, we study the subgroup $\mathrm{Aut}_{\text{alg}}(X)$ of the group of regular automorphisms $\mathrm{Aut}(X)$ of $X$ generated by all the connected algebraic subgroups. We prove that $\mathrm{Aut}_{\text{alg}}(X)$ is nested, i.e., is a direct limit of algebraic subgroups of $\mathrm{Aut}(X)$, if and only if all the $\mathbb{G}_a$-actions on $X$ commute. Moreover, we describe the structure of such a group $\mathrm{Aut}_{\text{alg}}(X)$.

When is the automorphism group of an affine variety nested?

Abstract

For an affine algebraic variety , we study the subgroup of the group of regular automorphisms of generated by all the connected algebraic subgroups. We prove that is nested, i.e., is a direct limit of algebraic subgroups of , if and only if all the -actions on commute. Moreover, we describe the structure of such a group .

Paper Structure

This paper contains 8 sections, 11 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivations and group actions
  4. Ind-groups
  5. Lie algebras of ind-groups
  6. The case: ${\mathcal{U}}(X)$ is not abelian
  7. The case: ${\mathcal{U}}(X)$ is abelian
  8. Conclusion

Key Result

Theorem \oldthetheorem

Given an affine variety $X$, let $\operatorname{Aut}_{\operatorname{alg}}(X)$ be the subgroup of $\operatorname{Aut}(X)$ generated by all connected algebraic subgroups. The following conditions are equivalent:

Theorems & Definitions (33)

  • Conjecture \oldthetheorem: P.--Zaidenberg, Feb.'13
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Conjecture \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 23 more