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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 222 / 2013

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

#### Streszczenie

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 $G$.

#### Autorzy

• 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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