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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
##### Keywords:
proper Lie groupoid; Tannaka duality
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##### References:
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