## Details

Original language | English |
---|---|

Number of pages | 15 |

Journal | Transformation Groups |

Early online date | 26 Jul 2022 |

Publication status | E-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 Mor_{0}(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 Mor_{0}(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

- Mathematics(all)
**Algebra and Number Theory**- Mathematics(all)
**Geometry and Topology**

## Cite this

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**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 journal › Article › Research › peer review

*Transformation Groups*. Advance online publication. https://doi.org/10.48550/arXiv.2107.14667, https://doi.org/10.1007/s00031-022-09748-2

}

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 -