38 research outputs found
On the equivarant de-Rham cohomology for non-compact Lie groups
Let be a connected and non-necessarily compact Lie group acting on a
connected manifold . In this short note we announce the following result:
for a -invariant closed differential form on , the existence of a closed
equivariant extension in the Cartan model for equivariant cohomology is
equivalent to the existence of an extension in the homotopy quotient.Comment: 5 pages. Minor corrections. To appear in DG
Higher holonomies: comparing two constructions
We compare two different constructions of higher dimensional parallel
transport. On the one hand, there is the two dimensional parallel transport
associated to 2-connections on 2-bundles studied by Baez-Schreiber, Faria
Martins-Picken and Schreiber-Waldorf. On the other hand, there are the higher
holonomies associated to flat superconnections as studied by Igusa, Block-Smith
and Arias Abad-Schaetz. We first explain how by truncating the latter
construction one obtains examples of the former. Then we prove that the
2-dimensional holonomies provided by the two approaches coincide.Comment: comments are welcome
The Weil algebra and the Van Est isomorphism
This paper belongs to a series 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 associated to
any Lie algebroid . We then show that this Weil algebra is related to the
Bott-Shulman-Stasheff 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 Bursztyn et.al. on the reconstructions of multiplicative forms and of
a result of Weinstein-Xu and Crainic 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.Comment: 28 pages. Final version, to appear in "Annales de l'Institut Fourier
Tensor products of representations up to homotopy
We study the construction of tensor products of representations up to
homotopy, which are the A-infinity version of ordinary representations. We
provide formulas for the construction of tensor products of representations up
to homotopy and of morphisms between them, and show that these formulas give
the homotopy category a monoidal structure which is uniquely defined up to
equivalence.Comment: 42 pages, 2 figure
The A∞ de Rham Theorem and Integration of Representations up to Homotopy
We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an version of de Rham's theorem due to Gugenheim [15]. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa [17] and Block-Smith [6], to the case of certain differential graded manifold
Representations up to homotopy of Lie algebroids
We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [3