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Actions of multiplier Hopf algebras. (English) Zbl 0951.16013
A. Van Daele introduced the multiplier Hopf algebras [in Trans. Am. Math. Soc. 342, No. 2, 917-932 (1994; Zbl 0809.16047)] as a generalization of the theory of Hopf algebras. A pair $$(A,\Delta)$$ of an algebra $$A$$ over $$C$$ with a non-degenerated product and a comultiplication $$\Delta$$ on $$A$$ is called a multiplier Hopf algebra if the linear maps from $$A\otimes A$$ on itself, defined by $$a\otimes b\to\Delta(a)(1\otimes b)$$, $$a\otimes b\to(a\otimes 1)\Delta(b)$$ are bijective. In this paper, the authors extend the theory of actions of Hopf algebras to the case of multiplier Hopf algebras. If $$R$$ is an algebra over $$C$$ not necessarily with unity but with non-degenerated product, $$R$$ is said to be a left $$A$$-module algebra if $$R$$ is a unital (i.e. $$AR=R$$) left $$A$$-module and $$a(xx')=\sum(a_1x)(a_2x')$$ for all $$a\in A$$ and $$x,x'\in R$$. In this situation, the smash product, $$R\#A$$, is constructed and studied.
In the last section, a duality result is obtained. A regular multiplier Hopf algebra with integrals is said to be an algebraic quantum group. If $$A$$ is an algebraic quantum group, acting on an algebra $$R$$ and if $$\widehat A$$ is the dual of $$A$$, acting on the smash product $$R\#A$$ by means of the dual action, then the bismash product $$(R\#A)\#\widehat A$$ is isomorphic with $$R\otimes(A\diamond\widehat A)$$, where $$A\diamond\widehat A$$ is $$A\otimes\widehat A$$ with the product $$(a\otimes b)(a'\otimes b')=\langle a',b\rangle a\otimes b'$$ whenever $$a,a'\in A$$ and $$b,b'\in\widehat A$$.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S40 Smash products of general Hopf actions 16W35 Ring-theoretic aspects of quantum groups (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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