Van Est Research Articles

Deformations of Lie brackets and representations up to homotopy

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.

The Weil algebra and the Van Est isomorphism

This paper belongs to a series of papers devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman’s BRST model, here we introduce the Weil algebra W(A) associated to any Lie algebroid A. We then show that this Weil algebra is related to the Bott-Shulman complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of [6] on the reconstructions of multiplicative forms and of a result of [21, 9] on the reconstruction of connection 1-forms. This reveals the relevance of the Weil algebra and Van Est maps to the integration and the pre-quantization of Poisson (and Dirac) manifolds.

A p-adic analogue of the Borel regulator and the Bloch–Kato exponential map

AbstractIn this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

THE VAN EST SPECTRAL SEQUENCE FOR HOPF ALGEBRAS

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.

Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Est’s argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14]. In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the Van Est map. This extends Evens-Lu-Weinstein’s characteristic class $\theta_{L}$ [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 30]. In the last section we describe applications to Poisson geometry.

In Vitro Studies on tRNA Annealing and Reverse Transcription with Mutant HIV-1 RNA Templates

The human immunodeficiency virus type 1 (HIV-1) RNA genome encodes a semistable stem-loop structure, the U5-PBS hairpin, which occludes part of the tRNA primer binding site (PBS). In previous studies, we demonstrated that mutations that alter the stability of the U5-PBS hairpin inhibit virus replication. A reverse transcription defect was measured in assays with the virion-extracted RNA-tRNA complexes. We now extend these studies with in vitro synthesized wild-type and mutant RNA templates that were tested in primer annealing and reverse transcription assays. The effect of annealing temperature and the presence of the viral nucleocapsid protein on reverse transcription was analyzed for the templates with a stabilized or destabilized U5-PBS hairpin, and in reactions initiated by tRNA or DNA primers. The results of this in vitro assay are consistent with the in vivo findings, in that both tRNA annealing and initiation of reverse transcription are sensitive to stable template RNA structure. Reverse transcription initiated by a DNA primer is less hindered by secondary structure in the RNA template than tRNA primed reactions. The inhibitory effect of template structure on tRNA-primed reverse transcription is more pronounced in this in vitro assay compared with the in vivo material, indicating that the heat-annealed RNA-tRNA complex differs from the virion-extracted viral RNA-tRNA complex.

Une version quantique du théorème de van Est

We introduce a concept of quantum Lie algebra compatible with Woronowicz's differential calculus. We construct a cohomological theory for these quantum Lie algebras. We give a quantum version of the van Est theorem which leads to a comparison between the cohomology of a Lie algebra and the cohomology of a coalgebra.

Relation of the Van Est spectral sequence to $K$-theory and cyclic homology

Relation of the Van Est spectral sequence to $K$-theory and cyclic homology

Free Banach-Lie algebras, couniversal Banach-Lie groups, and more

The construction of free Banach-Lie algebra over a normed space enables us to build a connected separable Banach-Lie group of which any other connected separable Banach-Lie group is a quotient. New proofs are given to the result on representabil ity of any Banach-Lie algebra as a quotient of an enlargable Banach-Lie algebra (due to van Est and Swierczkowski) and to the result on representabil ity of any topological group as a quotient of a group with no small subgroups (due to successive efforts of Morris and Thompson, the author, and Sipacheva and Uspenskiϊ). 1. Introduction. Over the last 50 years a number of constructions of universal arrows (see, e.g., [Go]) to the categories of topological algebraic systems have been studied. Important contributions are those by Markov [M], Graev [Gr], and Arhangel'skiϊ [A2] on free topological groups, Mal'cev [Me] on free topological algebras, Arens and Eells [AE], Raϊkov [R], and Uspenskiϊ [U] on free Banach spaces and free locally convex spaces. By virtue of these constructions a first ever example of a non-normal Hausdorff topological group was obtained [M], and the representability of any topological group as a quotient group of a zero-dimensional group was proved [Al]. Here we apply the concept of a free complete normed Lie algebra to theory of topological and Lie groups. Our construction is an extension of the well-known construction of Arens-Eells [AE] to the case of normed Lie algebras. Our main result is that there exists a couniversal separable connected Banach-Lie group, that is, such a separable connected Banach-Lie group that any other such Banach-Lie group is its quotient Lie group. This follows from observation that any free Banach-Lie algebra is enlargable, that is, comes from an approriate Banach-Lie group. Also we give entirely new and rather transparent proofs of two earlier known results. Cohomological technique has enabled van Est and independently Swierczkowski [S2] to prove that any Banach-Lie algebra is a quotient algebra of an enlargable Banach-Lie algebra. Here we deduce the result from enlargability of free Banach-Lie algebras. In his book [Ka] Kaplansky asked whether a quotient group of a

On Sophus Lie's Fundamental Theorems. II

’ Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darrnstadt, Germany, FRG and Tulane University, New Orleans, La. 70118, U.S.A. 2 Louisiana State University, Baton Rouge, La. 70803, U.S.A. Communicated by Prof. W.T. van Est at the meeting of February 27, 1984

On Sophus Lie's fundamental theorems. I

’ Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darrnstadt, Germany, FRG and Tulane University, New Orleans, La. 70118, U.S.A. 2 Louisiana State University, Baton Rouge, La. 70803, U.S.A. Communicated by Prof. W.T. van Est at the meeting of February 27, 1984

Functionally regular spaces

Referring to the p π 8 A (Functionally regular) axiom of Van Est and Freudenthal [W. T. Van Est and H. Freudenthal, Trenning durch Stetigkeit Funktionen in topologischen Raumen, Indagationes Math. 15, 359–368 (1951)] the following are included in the results. Functionally regular spaces with weak topology of point countable type or pseudocompact or such that the weak topology is the semi-regularization can be densely and C ∗-embedded in a functionally regular space that is closed in every functionally regular space it is embedded in (FR closed). A functionally regular space can be C ∗-embedded as a closed and nowhere dense subset of an FR closed space iff its weak topology is realcompact. Any functionally regular space may be C ∗-embedded as a nowhere dense subset of a FR-closed space. A functionally regular space is C-embedded in every functionally regular space it is embedded in iff its weak topology is almost compact. Further results are obtained on the C-embedding of pseudocompact subsets.

Leaf invariants for foliations and the Van Est isomorphism

Leaf invariants for foliations and the Van Est isomorphism

On C- and C∗-embeddings

Referring to the separation axioms of Van Est and Freudenthal [ W. T. van Est and H. Freudenthal, Trenning durch Stetigkeit Funktionen in topologischen Räumen, Indagationes Math. 15, 359–368 (1951)] the following are included in the results. A set is C-embedded in every space satisfying p τ S q that it is embedded in iff it is closed in every space satisfying p τ S q that it is embedded in. A pseudocompact set X is C-embedded in a p τ S A space Y which it is embedded in iff X with its weak topology is C-embedded in Y with its weak topology. Consequently a space X is C-embedded in every p τ S A space that it is embedded in iff given two disjoint zero sets in X at least one is closed in every p τ S q space that it is embedded in and furthermore a closed pseudocompact space with pseudocompact zero sets is C-embedded in a p τ S A space Y if Y with its weak topology satisfies A τ S B. A C-embedded subset is closed in a space X if X satisfies p τ S A and every point is a G δ . A C ∗-embedded subset is closed in a space X if X is sequential T 1 and hereditarily weakly normal and completely regular.

A Definition of Separation Axiom

Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition.

Quasicompactness and functionally Hausdorff spaces

In [1] Van Est and Freudenthal introduced and studied several new separation axioms for a topological space. One of these was thepτsq axiom: Given distinct pointspandqofX, there exists a real continuous functionfonXwithf(p) ≠f(q). They observed that thepτsqaxiom lies strictly between the Hausdorff and completely regular axioms, and that it neither implies nor is implied by theT3axiom.

On the cohomology of local groups

Most known homology theories (e.g. the homology of modules, rings, groups, sheaves, …) have been found to be special cases of a general theory proposed by M. Barr and J. Beck [1], [2]. The aim of this paper is to show that the cohomology of a local group, as defined by W. T. van Est [4], also fits the scheme of Barr and Beck. At the same time it will be shown that local group cohomology is a relative derived functor in the sense of S. Eilenberg and J. C. Moore [3].

Algebraic cohomology of topological groups

A general cohomology theory for topological groups is described, and shown to coincide with the theories of C. C. Moore [12] and other authors. We also recover some invariants from algebraic topology. This article contains proofs of results announced in [151. We consider algebraic cohomology groups of topological groups, which are shown to include the invariants considered by Van Est [6], Hochschild and Mostow [7], C. C. Moore [12], and Tate (see [5]). We identify some of these groups as invariants familiar from algebraic topology. Let G be a topological group. A topological G-module is an abelian topological group A together with a map G x A -4 A satisfying the usual relations g(a + a) = ga + ga', (gg ')a = g(g 'a), la = a. The category of topological G-modules and equivariant homomorphisms is a quasiabelian category in the sense of Yoneda [16], and hence we get Ext functors just as in an abelian category. A proper short exact sequence will be a sequence O ) A -4 B -A C 0 of topological G-modules which is exact as a sequence of abstract groups and such that A has the subspace topology induced by its embedding in B, and such that u be an open map. For any G-module A we define the algebraic cohomology groups HZ(G, A) to be the ith Ext group Ext2 (Z, A), where Z denotes the group of integers with the discrete topology and trivial G-action. There is another set of short exact sequences we might have chosen which also give the category of topological G-modules the structure of a quasi-abelian S-category in the sense of Yoneda. We might have demanded that in addition to being exact in the previous sense, there be a map s: C -* B such that the composition u o s be the identity on C. If G is locally compact we recover the continuous theory, which is discussed in [5], [6], and [7]. If G is not locally compact it must be shown that cochains are effaceable, i.e. that for any cocycle c: GC -n A there is a short exact sequence 0 -O A B -4 C -0 such that r 0 c is the coboundary of a Received by the editors September 24, 1971. AMS (MOS) subject classifications (1970). Primary 22A05. 83 Copyrdit 0) 1973, American Mathematical Society This content downloaded from 157.55.39.253 on Sat, 11 Jun 2016 06:04:56 UTC All use subject to http://about.jstor.org/terms

The existence of a vector of weight 0 in irreducible Lie groups without centre

R. Bott and H. Samelson raised the question (communicated to me by W. T. van Est) whether it is true that every irreducible linear, semi-simple compact Lie group with trivial centre possesses a vector of weight 0. In the following I shall give a proof of this conjecture and of its converse. In the meantime R. Bott wrote me that HarishChandra also proved this conjecture. His proof is quite different from the present one. Let G be an irreducible linear, semi-simple compact Lie group.

Dense imbedding of topological groups

The present article originated from the problem of determining when a continuous representation of a Lie group into another is open. Important cases of the problem have been discussed by A. Malcev2 and the author,3 and some of them have been extended to the case of more general groups by the author and H. Yamabe.4 Recently W. T. van Est obtained new results concerning the same problem on Lie groups.6 Here I shall give an extension of his essential result to a more general case in a simpler way. Let G be a locally compact connected group and let A (G) be the group of all continuous automorphisms of G. Let A (G) be topologized by the notion of uniform convergence in the wider sense. Now let I(G) be the subgroup of A (G) composed of all inner automorphisms of G. We shall call G a (CA) group6 if I(G) is a closed subgroup of A(G).