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Multiplier Hopf algebras in categories and the biproduct construction. (English) Zbl 1163.16026
Summary: Let \(B\) be a regular multiplier Hopf algebra. Let \(A\) be an algebra with a non-degenerate multiplication such that \(A\) is a left \(B\)-module algebra and a left \(B\)-comodule algebra. By the use of the left action and the left coaction of \(B\) on \(A\), we determine when a comultiplication on \(A\) makes \(A\) into a “\(B\)-admissible regular multiplier Hopf algebra”. If \(A\) is a \(B\)-admissible regular multiplier Hopf algebra, we prove that the smash product \(A\#B\) is again a regular multiplier Hopf algebra. The comultiplication on \(A\#B\) is a cotwisting (induced by the left coaction of \(B\) on \(A\)) of the given comultiplications on \(A\) and \(B\). When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct.

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