A definability criterion for connected Lie groups

2022-03-22

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be … It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G $G$ is Lie isomorphic to a group definable in an o $o$ -minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G $G$ has its Levi subgroups with finite center, we prove that in order for G $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452).

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A definability criterion for linear Lie groups

2019-10-24

Alf Onshuus , Sacha Post

It is known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to … It is known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to ask when does a Lie group is Lie isomorphic to a group definable in such expansion. Conversano, Starchenko and the first author answered this question in \cite{COSsolvable} in the case where the group is solvable. We give here a criterion in the case where the group is linear. More precisely if $G$ is a linear Lie group it is isomorphic to a group definable in an $o$-minimal expansion of the reals if and only if its solvable radical is isomorphic to such group.

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

2011-11-09

Annalisa Conversano , Anand Pillay

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 … 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.

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

2011-01-01

Annalisa Conversano , Anand Pillay

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 … 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.

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Definability of continuous isomorphisms of groups definable in o-minimal expansions of the real field

2023-01-01

Alf Onshuus

In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer … In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer to as "definable groups"). It is known (\cite{Pi88}) that any group definable in an o-minimal expansion of the real field is a Lie group, and in \cite{COP} a complete characterization of when a Lie group has a "definable group" which is \emph{Lie isomorphic} to it was given. We continue the analysis by explaining when a Lie homomorphism between definable groups is a definable isomorphism. Among other things, we prove that in any o-minimal expansion $\mathcal R$ of the real field we can add a function symbol for any Lie isomorphism between definable groups to the language of $\mathcal R$ preserving o-minimality, and that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.

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Real Lie groups and o-minimality

2021-09-22

Annalisa Conversano , Alf Onshuus , Sacha Post

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group … We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group definable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Subscript exp"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>exp</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {R}_{\exp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which any Lie automorphism is definable.

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Real Lie groups and o-minimality

2021-01-01

Annalisa Conversano , Alf Onshuus , Sacha Post

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field.

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

2013-01-01

Annalisa Conversano , Anand Pillay

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 … 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 <i>ind-definable semisimple subgroup</i>,

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SOLVABLE LIE GROUPS DEFINABLE IN O-MINIMAL THEORIES

2016-04-28

Annalisa Conversano , Alf Onshuus , Sergei Starchenko

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field. In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

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Solvable Lie groups definable in o-minimal theories

2015-01-01

Annalisa Conversano , Alf Onshuus , Sergei Starchenko

In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field. In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

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Solvable Lie groups definable in o-minimal theories

2015-06-26

Annalisa Conversano , Alf Onshuus , Sergei Starchenko

In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field. In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

A Jordan-Chevalley decomposition beyond algebraic groups

2022-01-01

Annalisa Conversano

A generalization of the Jordan-Chevalley decomposition of linear algebraic groups to definable groups in o-minimal structures is considered. As an application, we give a description of the commutator subgroup of … A generalization of the Jordan-Chevalley decomposition of linear algebraic groups to definable groups in o-minimal structures is considered. As an application, we give a description of the commutator subgroup of a definable group G, and show that when G has definable Levi subgroups, [G, G] is definable. Along the way, we identify crucial central subgroups, study disconnected groups, definable p-groups and 0-groups, abstractly compact subgroups, the Weyl group, and discuss a couple of other structure decompositions: 1) a compact-torsionfree decomposition and 2) a Levi decomposition. It turns out that definable p-groups are solvable, like finite p-groups, but they are not necessarily nilpotent.

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Lie-like decompositions of groups definable in o-minimal structures

2009-01-01

Annalisa Conversano

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study … There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2021-08-19

Annalisa Conversano

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GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

2008-01-01

Pantelis E. Eleftheriou

In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure … In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure theorem for the special case of semi-linear groups, exemplifying their relation with real Lie groups.

On the homotopy type of definable groups in an o-minimal structure

2011-02-24

Alessandro Berarducci , Marcello Mamino

We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 … We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 → G00 → G → G/G00 → 1, where G00 is the ‘infinitesimal subgroup’ of G and G/G00 is a compact real Lie group. We show that given a connected open subset U of G/G00, there is a canonical isomorphism between the fundamental group of U and the o-minimal fundamental group of its preimage under the projection p: G→ G/G00. We apply this result to show that there is a natural exact sequence 1 → G 00 → G ~ → G / G 00 ~ → 1 , where G ~ is the (o-minimal) universal cover of G, and G / G 00 ~ is the universal cover of the real Lie group G/G00. We also prove that, up to isomorphism, each finite covering H → G/G00, with H a connected Lie group, is of the form H/H00→ G/G00 for some definable group extension H→G. Finally we prove that the (Lie-)isomorphism type of G/G00 determines the definable homotopy type of G. In the semisimple case a stronger result holds: G/G00 determines G up to definable isomorphism. Our results depend on the study of the o-minimal fundamental groupoid of G and the homotopy properties of the projection G→ G/G00.

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2019-01-01

Annalisa Conversano

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite … We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2019-04-22

Annalisa Conversano

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite … We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.

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Cartan subgroups and regular points of o‐minimal groups

2019-04-01

Elías Baro , Alessandro Berarducci , Margarita Otero

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M … Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M 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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Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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A definability criterion for connected Lie groups

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