Toward differentiation and integration between Hopf algebroids and Lie algebroids
Ardizzon, Alessandro
El Kaoutit Zerri, Laiachi
Saracco, Paolo
(co)commutative Hopf algebroids
Affine groupoid schemes
Differen- tiation and integration
Kähler module
Lie–Rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
Finite dual
Tannaka reconstruction
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.
2023-05-18T11:23:57Z
2023-05-18T11:23:57Z
2023
info:eu-repo/semantics/article
A. Ardizzoni, L. El Kaoutit, P. Saracco. Diff and Int for Hopf and Lie Algebroids. Publ. Mat. 67 (2023), 3–88. [DOI: 10.5565/PUBLMAT6712301]
https://hdl.handle.net/10481/81658
10.5565/PUBLMAT6712301
eng
http://creativecommons.org/licenses/by-nc-nd/4.0/
info:eu-repo/semantics/openAccess
Attribution-NonCommercial-NoDerivatives 4.0 Internacional
Universidad Autónoma de Barcelona, Dpto. de Matemáticas. España