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.
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.
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.
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.
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.
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.
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.
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.
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).
It has been 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 therefore natural …
It has been 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 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 \cite{COSsolvable} 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 $G$ is Lie isomorphic to a group definable in an $o$-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 $G$ has a "good Levi descomposition", we prove that in order for $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in \cite{COSsolvable}.
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>,
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of …
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f -generic” or $dfg$ group we mean a definable group in a saturated model with a global f -generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o -minimal groups in the p -adic context. In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f -generic type, etc. In this paper we will show that every definable f -generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o -minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f -generic types coincide with almost periodic types in the p -adic case.
The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group …
The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group admits a global f-generic type which is over a small submodel. This definable is a dual concept to finitely satisfiable generic, and a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context.
In this paper we will show that every $f$-generic group in $\Q$ is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over $\Q$. This is analogous to the $o$-minimal context, where every connected torsion free group in $\R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open $f$-generic subgroup of a $f$-generic group has finite index, and every $f$-generic type of a $f$-generic group is almost periodic, which gives a positive answer on the problem raised in \cite{P-Y} of whether $f$-generic types coincide with almost periodic types in the $p$-adic case.
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and …
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.
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.
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.
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).
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably …
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an …
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.
A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role …
A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0-subgroups. Along the way we give structural theorems for solvable groups, linear groups, and extensions of definably compact by torsion-free definable groups.
We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and …
We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real …
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real field the definable subsets of R' share many of the nice structural properties of semialgebraic sets. For example, definable subsets have only finitely many connected components, definable sets can be stratified and triangulated, and continuous definable maps are piecewise trivial (see [5]). In this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions, and use this result to obtain o-minimality. We were led to this while studying work of Ressayre [13] and several of his ideas emerge here in simplified form. However, our treatment is formally independent of the results of [16], [17], [9], and [13].
Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie …
Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie Algebras Universal Enveloping Algebra Compact Lie Groups Finite-Dimensional Representations Structure Theory of Semisimple Groups Advanced Structure Theory Integration Induced Representations and Branching Theorems Prehomogeneous Vector Spaces Appendices Hints for Solutions of Problems Historical Notes References Index of Notation Index
We give an unconditional proof of the André-Oort conjecture for arbitrary products of modular curves.We establish two generalizations.The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined …
We give an unconditional proof of the André-Oort conjecture for arbitrary products of modular curves.We establish two generalizations.The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group.The second includes the Manin-Mumford conjecture for abelian varieties defined over Q.Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink's relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-Oort-Manin-Mumford type by Pila.
We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionally
We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionally
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment …
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.
As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. …
As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure
Soient G un groupe de Lie simplement connexe, 0 son algèbre de Lie.L'application exponentielle (cf.[3]) est une application analytique de 9 dans G. Si G est niipotent, cette application est …
Soient G un groupe de Lie simplement connexe, 0 son algèbre de Lie.L'application exponentielle (cf.[3]) est une application analytique de 9 dans G. Si G est niipotent, cette application est même un isomorphisme de la variété analytique g sur la variété analytique G; mais il est loin d'en être ainsi en général.Nous étudions dans ce Mémoire le cas où G est résoluble.Dans le cas où G est un groupe complexe, l'application exponentielle ne peut être un iso< morphisme que si G est niipotent; si G est réel, elle peut être un isomorphisme dans des cas plus étendus, et par exemple s'il existe dans G une suite de sous-groupes distingués décroissants tels que les quotients successifs soient de dimension i. Dans tous les cas, l'application exponentielle possède des propriétés qui, on'le verra, la rapproche d'un isomorphisme.
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).