For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.Comment: uses kluwer.cls; v. 2: 25 pages, significant corrections, 3 authors; version 3: significant revision, 32 pages, corrections and added geometrical viewpoint and preliminaries on formal differential operators; version 4: final corrections and slightly improved readability; accepted in Letters in Mathematical Physic

Given a Hopf algebra [Formula: see text], Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld [Formula: see text]-module algebra [Formula: see text] gives rise to an associative [Formula: see text]-bialgebroid structure on the smash product algebra [Formula: see text]. They also exhibited an antipode map making [Formula: see text] the total algebra of a Lu’s Hopf algebroid over [Formula: see text]. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair [Formula: see text] and [Formula: see text] of braided commutative Yetter–Drinfeld [Formula: see text]-module algebras, and output is a symmetric Hopf algebroid [Formula: see text] over [Formula: see text]. This construction does not require that the antipode of [Formula: see text] is invertible.

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