@misc{10481/81658,
year = {2023},
url = {https://hdl.handle.net/10481/81658},
abstract = {In this paper we set up the foundations around the notions of formal
differentiation and formal integration in the context of commutative Hopf algebroids
and Lie–Rinehart algebras. Specifically, we construct a contravariant functor from
the category of commutative Hopf algebroids with a fixed base algebra to that of
Lie–Rinehart algebras over the same algebra, the differentiation functor, which can
be seen as an algebraic counterpart to the differentiation process from Lie groupoids
to Lie algebroids. The other way around, we provide two interrelated contravariant
functors from the category of Lie–Rinehart algebras to that of commutative Hopf
algebroids, the integration functors. One of them yields a contravariant adjunction
together with the differentiation functor. Under mild conditions, essentially on the
base algebra, the other integration functor only induces an adjunction at the level
of Galois Hopf algebroids. By employing the differentiation functor, we also analyse
the geometric separability of a given morphism of Hopf algebroids. Several examples
and applications are presented.},
organization = {Spanish Ministerio de Economía y Competitividad},
organization = {European Union MTM2016-77033-P.},
organization = {Fonds de la Recherche Scientifique - FNRS},
organization = {National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).},
organization = {University of Turin PRX16/00108},
publisher = {Universidad Autónoma de Barcelona, Dpto. de Matemáticas. España},
keywords = {(co)commutative Hopf algebroids},
keywords = {Affine groupoid schemes},
keywords = {Differen- tiation and integration},
keywords = {Kähler module},
keywords = {Lie–Rinehart algebras},
keywords = {Lie algebroids},
keywords = {Lie groupoids},
keywords = {Malgrange groupoids},
keywords = {Finite dual},
keywords = {Tannaka reconstruction},
title = {Toward differentiation and integration between Hopf algebroids and Lie algebroids},
doi = {10.5565/PUBLMAT6712301},
author = {Ardizzon, Alessandro and El Kaoutit Zerri, Laiachi and Saracco, Paolo},
}