Structure of connected nested automorphism groups

A nested group is an increasing union of algebraic groups. It is well known that any algebraic subgroup of the automorphism group \(\mathrm{Aut}(X)\) of an affine variety \(X\) is closed with respect to the ind-topology. The closedness of connected nested subgroups in \(\mathrm{Aut}(X)\) is an open...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Author: Perepechko, Alexander
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Set theory
Subgroups
Topology
Online Access:Get full text
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Summary:A nested group is an increasing union of algebraic groups. It is well known that any algebraic subgroup of the automorphism group \(\mathrm{Aut}(X)\) of an affine variety \(X\) is closed with respect to the ind-topology. The closedness of connected nested subgroups in \(\mathrm{Aut}(X)\) is an open question (Kraft--Zaidenberg'2022, arXiv:2203.11356). In this paper, we describe maximal nested unipotent subgroups of \(\mathrm{Aut}(X)\) by generalizing the one of triangular automorphisms of \(\mathbb{A}^n\). We show that if an abstract subgroup of \(\mathrm{Aut}(X)\) consists of unipotent elements, then it is closed if and only if it is nested. This implies that a connected nested subgroup of \(\mathrm{Aut}(X)\) is closed. We also extend the recent description of maximal commutative unipotent subgroups (Regeta--van Santen'2024, arXiv:2112.04784), offering a direct construction method and relating them to our description.
ISSN:2331-8422