Abstract
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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