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On the modules of a Drinfel’d double multiplier Hopf (*-)algebra. (English) Zbl 1081.16040
Let \(\langle A,B\rangle\) be a pairing of multiplier Hopf algebras with canonical multiplier \(W\in M(A\otimes B)\), such that \((a\otimes 1)W(1\otimes b)\) and \((1\otimes b)W(a\otimes 1)\) are in \(A\otimes B\) for any \(a\in A\), \(b\in B\). Let \(D=A\times B^{cop}\) be the Drinfeld double associated to this pairing, and let \(R\) be an algebra with a non-degenerate product.
It is proved that: (1) \(R\) is a left \(A\)-module (algebra) if and only if \(R\) is a right \(B\)-comodule (algebra); (2) \(R\) is a left unital \(D\)-module if and only if \(R\) is a left unital \(B\)-module and a right \(B\)-comodule, with the two structures satisfying a certain compatibility condition; (3) \(R\) is a left \(D\)-module algebra if and only if \(R\) is a left \(B^{cop}\)-module algebra and a right \(B\)-comodule algebra.
Also results for correspondence of morphisms associated to the structures in (1), (2), (3) are proved, and versions of these results in the *-case are given. In particular, for a finite dimensional Hopf algebra \(H\), it is obtained that the modules over the Drinfeld double \(D(H)\) are just the \(H\)-crossed bimodules, a result of S. Majid [Commun. Algebra 19, No. 11, 3061-3073 (1991; Zbl 0767.16014)].

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
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