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

TL;DR

This paper extends classical Lie theory concepts of Levi subgroups and Levi decomposition to groups definable in o-minimal structures, establishing existence and uniqueness results for these analogues.

Contribution

It introduces suitable definitions and proves the existence and uniqueness of maximal semisimple subgroups and the Levi decomposition in the o-minimal setting.

Findings

Existence of a unique maximal ind-definable semisimple subgroup

Decomposition of G as RS where R is solvable radical

Correspondence between semisimple subalgebras and semisimple subgroups

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 suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS 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.