Linearity of groups definable in o-minimal structures

Frécon, Olivier

doi:10.1007/s00029-016-0247-9

Cited by 1 publication

(1 citation statement)

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“…By [, Lemma 8], each Cartan subgroup of

G

contains

Z (G)

and

Ctn (G / Z (G)) = {H / Z (G) : H \in Ctn (G)} .

Thus, it suffices to prove (1) and (2) for

G / Z (G)

. By Frécon Theorem [, Theorem 5.15],

G / Z false(G false) = G_{1} \times \dots \times G_{m}

, where, for each

i = 1, \dots, m

G_{i}

is a definable group and there are a definable real closed field

R_{i}

and an integer

n_{i}

such that

G_{i}

is a definable subgroup of

GL (n_{i}, R_{i})

. By [, Corollary 10]

Ctn false(G / Z (G) false) = false{H_{1} \times \dots \times H_{m} : H_{i} \in Ctn (G_{i}), i = 1, \dots, m false} .

Thus, clearly it suffices to prove (1) and (2) for each

G_{i}

, and moreover we can assume that each

G_{i}

is definable in an o‐minimal expansion of

R_{i}

.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

Cartan subgroups and regular points of o‐minimal groups

Baro

Berarducci

Otero

2019

J. London Math. Soc.

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Let G be a group definable in an o‐minimal structure scriptM. We prove that the union of the Cartan subgroups of G is a dense subset of G. When scriptM is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.

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“…By [, Lemma 8], each Cartan subgroup of