Abstract We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of …
Abstract We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group not necessarily abelian over a sufficiently saturated real closed field ; namely, that the o‐minimal universal covering group of is an open locally definable subgroup of for some ‐algebraic group (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group over , we describe as a locally definable extension of subgroups of the o‐minimal universal covering groups of commutative ‐algebraic groups (Theorem 3.4).
We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application …
We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group $G$ not necessarily abelian over a sufficiently saturated real closed field $R$; namely, that the o-minimal universal covering group $\widetilde{G}$ of $G$ is an open locally definable subgroup of $\widetilde{H\left(R\right)^{0}}$ for some $R$-algebraic group $H$ (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group $G$ over $R$, we describe $\widetilde{G}$ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative $R$-algebraic groups (Theorem 3.4)
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence …
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group $G$ definable in $R$ there are a connected $R$-algebraic group $H$, a definable injective map $\phi$ from a generic definable neighborhood of the identity of $G$ into the group $H\left(R\right)$ of $R$-points of $H$ such that $\phi$ acts as a group homomorphism inside its domain. This result is used in [2] to prove that the o-minimal universal covering group of an abelian connected definably compact group definable in a sufficiently saturated real closed field $R$ is, up to locally definable isomorphisms, an open connected locally definable subgroup of the o-minimal universal covering group of the $R$-points of some $R$-algebraic group.
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence …
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group $G$ definable in $R$ there are a connected $R$-algebraic group $H$, a definable injective map $\phi$ from a generic definable neighborhood of the identity of $G$ into the group $H\left(R\right)$ of $R$-points of $H$ such that $\phi$ acts as a group homomorphism inside its domain. This result is used in [2] to prove that the o-minimal universal covering group of an abelian connected definably compact group definable in a sufficiently saturated real closed field $R$ is, up to locally definable isomorphisms, an open connected locally definable subgroup of the o-minimal universal covering group of the $R$-points of some $R$-algebraic group.
We continue the analysis of definably compact groups definable in a real closed field $\mathcal{R}$. In [3], we proved that for every definably compact definably connected semialgebraic group $G$ over …
We continue the analysis of definably compact groups definable in a real closed field $\mathcal{R}$. In [3], we proved that for every definably compact definably connected semialgebraic group $G$ over $\mathcal{R}$ there are a connected $R$-algebraic group $H$, a definable injective map $ϕ$ from a generic definable neighborhood of the identity of $G$ into the group $H\left(R\right)$ of $R$-points of $H$ such that $ϕ$ acts as a group homomorphism inside its domain. The above result and our study of locally definable covering homomorphisms for locally definable groups combine to prove that if such group $G$ is in addition abelian, then its o-minimal universal covering group $\widetilde{G}$ is definably isomorphic, as a locally definable group, to a connected open locally definable subgroup of the o-minimal universal covering group $\widetilde{H\left(R\right)^{0}}$ of the group $H\left(R\right)^{0}$ for some connected $R$-algebraic group $H$.
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a …
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable …
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable groups. We also present various results on denable modules and actions, denable extensions and
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating …
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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 .
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> …
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a …
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg …
This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in …
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in M n that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some M k with the topology induced by M k . Part of this result holds for the wider class of so-called invariant groups: each invariant group G in M n has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as M n .
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
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).
SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie …
SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie algebras rather than topological groups, topological algebras, and so on.The method of study is also purely algebraic but is the replica of an algebraic process which has been widely used in topology, thus the words "toplogical methods" could be replaced by "algebraic methods suggested by algebraic topology."These purely algebraic theories do, however, have several applications in topology.The algebraic process borrowed from topology is the following.Consider a sequence of abelian groups {C q ) and homomorphisms 8,such that ôô = 0.In each group C q two subgroups are distinguished Z q = kernel of 8:C q -* 0 +1 , B q = image of ô:C*~l ~» C q
Abstract By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free …
Abstract By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.
In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable …
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:
It is the aim of this work to study product structures on four dimensional solvable Lie algebras.We determine all possible paracomplex structures and consider the case when one of the …
It is the aim of this work to study product structures on four dimensional solvable Lie algebras.We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal.These results are applied to the case of Manin triples and complex product structures.We also analyze the three dimensional subalgebras.
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the …
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable …
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable groups. We also present various results on denable modules and actions, denable extensions and
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.
In this short note we point out two errors in our paper "Covers of groups definable in o-minimal structures" [Illinois J. Math.49 (2005), 99-120], and we show how to correct …
In this short note we point out two errors in our paper "Covers of groups definable in o-minimal structures" [Illinois J. Math.49 (2005), 99-120], and we show how to correct these errors.
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.
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
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a …
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms. The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups. The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.
We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of …
We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R = ℝ, this result was claimed in [5], but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably …
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i