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Tannaka duality for proper Lie groupoids. (English) Zbl 1196.58007
The author constructs, for each smooth manifold \(X\), a category with objects called smooth Euclidean fields over \(X\) which is the proper enlargement of the category of smooth vector fields over \(X\) and is the analogue of the notion of continuous Hilbert fields introduced by J. Dixmier and A. Douady [Bull. Soc. Math. Fr. 91, 227–284 (1963; Zbl 0127.33102)]. One has a natural notion of representation of Lie groupoids of \(\mathcal G\) on a smooth Euclidean field \(\mathcal E\) over the base manifold \(M\) of \(\mathcal G\) (Section 4), which is a monoidal category connected to the category of \(\mathcal E\) by a canonical forgetful functor. By generalizing the duality theory of T. Tannaka [Tôhoku Math. J. 45, 1–12 (1938; Zbl 0020.00904; JFM 64.0362.01)], the author gets for each proper Lie groupoid \(\mathcal G\), a “reconstructed groupoid” \(\mathcal T(\mathcal G)\) called Tannakian bidual of \(\mathcal G\) (Definition 6.1). This groupoid comes equipped with a natural candidate for a differentiable structure on its space of arrows, namely, a sheaf of algebras of continuous real valued functions stable under composition with arbitrary smooth functions of several variables. The canonical homomorphism \(\pi_{\mathcal G}: \mathcal G\to \mathcal T(\mathcal G)\) is obtained by \(\pi_{\mathcal G}(\mathcal E, \rho)=\rho(g)\) where \(\rho: {\mathbf s}^*\mathcal E\to {\mathbf t}^*\mathcal E\), and \({\mathbf s},\;{\mathbf t}\) are the source and the target of the arrow \(g\). Finally, the Tannaka duality for \(\mathcal G\) (Theorem 6,9) is shown, that is, \(\pi_{\mathcal G}\) is an isomorphism.

MSC:
58H05 Pseudogroups and differentiable groupoids
22D35 Duality theorems for locally compact groups
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