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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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