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On duality between étale groupoids and Hopf algebroids. (English) Zbl 1115.22003

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 \({\mathcal C}_C^\infty\).

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
58H05 Pseudogroups and differentiable groupoids
81R60 Noncommutative geometry in quantum theory
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