  We construct new examples of left bialgebroids and Hopf algebroids, arising
from noncommutative geometry. Given a first order differential calculus
$\Omega$ on an algebra $A$, with the space of left vector fields
$\mathfrak{X}$, we construct a left $A$-bialgeroid $B\mathfrak{X}$, whose
category of left modules is isomorphic to the category of left bimodule
connections over the calculus. When $\Omega$ is a pivotal bimodule, we
construct a Hopf algebroid $H\mathfrak{X}$ over $A$, by restricting to a
subcategory of bimodule connections which intertwine with both $\Omega$ and
$\mathfrak{X}$ in a compatible manner. Assuming the space of 2-forms
$\Omega^{2}$ is pivotal as well, we construct the corresponding Hopf algebroid
$\mathcal{D}\mathfrak{X}$ for flat bimodule connections, and recover
Lie-Rinehart Hopf algebroids as a quotient of our construction in the
commutative case. We use these constructions to provide explicit examples of
Hopf algebroids over noncommutative bases.

Quantum Algebra;Rings and Algebras