On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Gabriel Andreas Dill
View graph of relations

Details

Original languageEnglish
Number of pages15
JournalTransformation Groups
Early online date26 Jul 2022
Publication statusE-pub ahead of print - 26 Jul 2022

Abstract

Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

ASJC Scopus subject areas

Cite this

On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0. / Dill, Gabriel Andreas.
In: Transformation Groups, 26.07.2022.

Research output: Contribution to journalArticleResearchpeer review

Dill GA. On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0. Transformation Groups. 2022 Jul 26. Epub 2022 Jul 26. doi: 10.48550/arXiv.2107.14667, 10.1007/s00031-022-09748-2
Download
@article{d8595a72da34404e81663d427c812164,
title = "On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0",
abstract = "Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.",
author = "Dill, {Gabriel Andreas}",
note = "Funding Information: Open Access funding enabled and organized by Projekt DEAL. This work was supported by the Swiss National Science Foundation through the Early Postdoc.Mobility grant no. P2BSP2_195703. I thank the Mathematical Institute of the University of Oxford and my host there, Jonathan Pila, for hosting me as a visitor for the duration of this grant. During the final revisions of this manuscript, I was employed by the Leibniz Universit{\"a}t Hannover on the ERC grant of Ziyang Gao. Gabriel Dill has received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement n 945714).",
year = "2022",
month = jul,
day = "26",
doi = "10.48550/arXiv.2107.14667",
language = "English",
journal = "Transformation Groups",
issn = "1083-4362",
publisher = "Birkhauser Boston",

}

Download

TY - JOUR

T1 - On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0

AU - Dill, Gabriel Andreas

N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL. This work was supported by the Swiss National Science Foundation through the Early Postdoc.Mobility grant no. P2BSP2_195703. I thank the Mathematical Institute of the University of Oxford and my host there, Jonathan Pila, for hosting me as a visitor for the duration of this grant. During the final revisions of this manuscript, I was employed by the Leibniz Universität Hannover on the ERC grant of Ziyang Gao. Gabriel Dill has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n 945714).

PY - 2022/7/26

Y1 - 2022/7/26

N2 - Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

AB - Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

UR - http://www.scopus.com/inward/record.url?scp=85134811721&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2107.14667

DO - 10.48550/arXiv.2107.14667

M3 - Article

AN - SCOPUS:85134811721

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

ER -