Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.