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

authored by

Gabriel Andreas Dill

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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Transformation Groups

No. of pages

15

ISSN

1083-4362

Publication date

26.07.2022

Publication status

E-pub ahead of print

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory, Geometry and Topology

Electronic version(s)

https://doi.org/10.48550/arXiv.2107.14667 (Access: Open)
https://doi.org/10.1007/s00031-022-09748-2 (Access: Open)