On Levi subgroups and the Levi decomposition for groups definable in $o$-minimal structures
Volume 222 / 2013
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 $G$.
