On duality between étale groupoids and Hopf algebroids
type: Item
instance of: scholarly article
title: On duality between étale groupoids and Hopf algebroids
zbMATH Open document ID: 1115.22003
published in: Journal of Pure and Applied Algebra
publication date: 2007-04-26T00:00:00Z
review text: To any étale Lie groupoid \(G\) over a smooth manifold \(M\), we can associate a Hopf algebroid by taking the groupoid convolution algebra \({\mathcal C}_C^\infty(G)\) of smooth functions with compact support on \(G\) with its natural coalgebra structure. In this paper the author provides explicit conditions for identifying when a Hopf algebroid arises from a Lie groupoid in this way. In addition, the author gives a complementary construction for a Hopf algebroid \(A\) over \({\mathcal C}_C^\infty(M)\): the `associated spectral étale Lie groupoid' \({\mathcal G}_{sp}(A)\) has the property that setting \(A = {\mathcal C}_C^\infty(G)\) yields the original Lie groupoid \(G\) in a functorial way, and \({\mathcal G}_{sp}\) is a right adjoint to the functor \({\mathcal C}_C^\infty\).
reviewed by: Martin D. Crossley
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 81R60
zbMATH DE Number: 5146873
zbMATH Keywords: Lie groupoid
zbMATH Keywords: Hopf algebroid
zbMATH Keywords: etale groupoid
zbMATH Keywords: duality
author: Janez Mrčun
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.09.006
OpenAlex ID: W2170766751
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DOI: 10.1016/J.JPAA.2006.09.006
description: scientific article; zbMATH DE number 5146873