On duality between étale groupoids and Hopf algebroids

Abstract

For any étale Lie groupoid G over a smooth manifold M , the groupoid convolution algebra C c ∞ ( G ) of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra C c ∞ ( M ) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C c ∞ ( M ) we construct the associated spectral étale Lie groupoid G sp ( A ) over M such that G sp ( C c ∞ ( G ) ) is naturally isomorphic to G . Both these constructions are functorial, and C c ∞ is fully faithful left adjoint to G sp . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid C c ∞ ( G ) of an étale Lie groupoid G .