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A Cartier-Gabriel-Kostant structure theorem for Hopf algebroids. (English) Zbl 1269.16025

Let \(M\) be a smooth manifold and let \(\mathcal C^\infty_c(M)\) be the algebra of smooth real functions with compact support on \(M\). Let \(G\) be an étale Lie groupoid on \(M\). The authors show that if \(G\) acts on a bundle of Lie groups \(B\) over \(M\), then \(G\) acts on the universal enveloping algebra \(\mathcal U(b)\) of the Lie algebroid \(b\) associated to \(B\), and the corresponding twisted tensor product \(G\times\mathcal U(b)\) is a Hopf \(\mathcal C^\infty_c(M)\)-algebroid. An extension of the Cartier-Gabriel-Kostant structure theorem is given for Hopf algebroids. Thus it is proved that a Hopf \(\mathcal C^\infty_c(M)\)-algebroid \(A\) satisfying a certain local condition is isomorphic to a twisted tensor product \(G\times\mathcal U(b)\), where \(G\) is the spectral étale Lie groupoid over \(M\) associated to \(A\), and \(b\) is a certain \(G\)-bundle of Lie algebras over \(M\).

MSC:

16T05 Hopf algebras and their applications
17B35 Universal enveloping (super)algebras
58H05 Pseudogroups and differentiable groupoids
22A22 Topological groupoids (including differentiable and Lie groupoids)
17B66 Lie algebras of vector fields and related (super) algebras
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