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Revista Matemática Complutense

, Volume 32, Issue 2, pp 531–558 | Cite as

Locally \(\mathbb {C}\)-Nash groups

  • Elías Baro
  • Juan de Vicente
  • Margarita OteroEmail author
Article

Abstract

We introduce the category of locally \({\mathbb {C}}\)-Nash groups, basic examples of such groups are complex algebraic groups. We prove that the latter form a full subcategory. We also show that both, abelian locally Nash and abelian locally \({\mathbb {C}}\)-Nash groups, can be characterised via meromorphic maps admitting an algebraic addition theorem; we give an invariant of such groups associated to the groups of periods of a chart at the identity. Finally, we prove that the category of simply connected abelian locally \({\mathbb {C}}\)-Nash groups coincides with that of universal coverings of the abelian complex irreducible algebraic groups (a complex version of a result of Hrushovski and Pillay in Isr J Math 85(1–3):203–262, 1994; Conflu Math 3(4):577–585, 2011).

Keywords

Semialgebraic set Nash map Locally \(\mathbb {C}\)-Nash Algebraic group 

Mathematics Subject Classification

Primary 14P10 Secondary 14P20 

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  • Elías Baro
    • 1
  • Juan de Vicente
    • 2
  • Margarita Otero
    • 2
    Email author
  1. 1.Departamento de Álgebra, Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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