Some remarks on representations up to homotopy
type: Item
instance of: scholarly article
title: Some remarks on representations up to homotopy
zbMATH Open document ID: 1352.22002
DOI: 10.1142/S0219887816500249
author: Giorgio Trentinaglia
author: Chenchang Zhu
published in: International Journal of Geometric Methods in Modern Physics
publication date: 2016-04-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.02686
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 6573992
zbMATH Keywords: groupoids
zbMATH Keywords: representations up to homotopy
zbMATH Keywords: Peter-Weyl theorem
MaRDI profile type: Publication
arXiv ID: 1511.02686
cites work: Representations up to homotopy of Lie algebroids
cites work: Representations up to homotopy and Bott's spectral sequence for Lie groupoids
cites work: Tensor products of representations up to homotopy
cites work: Hodge cycles, motives, and Shimura varieties
cites work: Algebraic Quantum Field Theory in Curved Spacetimes
cites work: The Novikov conjecture for hyperbolic foliations
cites work: Central extensions and physics
review text: The notion of representation up to homotopy of a topological or Lie groupoid was introduced and studied by \textit{C. A. Abad} and \textit{M. Crainic} [J. Reine Angew. Math. 663, 91--126 (2012; Zbl 1238.58010)]. In the paper under review, this concept is explored for its links with classical representation theory. It is shown that such `homotopy representations' separate points on any locally trivial bundle of compact abelian groups (torus bundle) and that any locally trivial bundle of compact groups admits enough homotopy representations. In the second part of the paper, it is shown that, for any compact group, \(G\), the category of finite dimensional (ordinary) representations of \(G\) can be recovered from the dg-category of homotopy representations of \(G\) using simple standard methods.
reviewed by: Timothy Porter
description: scientific article; zbMATH DE number 6573992