On Levi subgroups and the Levi decomposition for groups definable in $o$-minimal structures

Volume 222 / 2013

Annalisa Conversano, Anand Pillay Fundamenta Mathematicae 222 (2013), 49-62 MSC: 03C64, 22E15. DOI: 10.4064/fm222-1-3

Abstract

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups $G$ definable in an $o$-minimal expansion of a real closed field. With a rather strong definition of ind-definable semisimple subgroup, we prove that $G$ has a unique maximal ind-definable semisimple subgroup $S$, up to conjugacy, and that $G = R\cdot S$ where $R$ is the solvable radical of $G$. We also prove that any semisimple subalgebra of the Lie algebra of $G$ corresponds to a unique ind-definable semisimple subgroup of $G$.

Authors

  • Annalisa ConversanoInstitute of Natural and Mathematical Sciences
    Massey University
    P/bag 102-904 NSMC
    Auckland, NZ
    e-mail
  • Anand PillayDepartment of Pure Mathematics
    University of Leeds
    LS2 9JT Leeds, UK
    e-mail

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