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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 \(b,b'\in\widehat A\).

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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