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Bicrossproducts of algebraic quantum groups. (English) Zbl 1271.16032
Let \(A\) and \(B\) be two regular multiplier Hopf algebras, \(A\) acting on \(B\), \(B\) coacting on \(A\), making them a matched pair. Several properties of the Hopf algebra \(A\#B\) are studied. If \(A\) has left and right integrals, there exists a distinguished multiplier \(y\in M(B)\), satisfying a compatibility with the integrals of \(A\). Integrals, modular elements, scaling constant on \(A\#B\) are given when \(A\) and \(B\) are algebraic quantum groups. It is shown that the dual of \(A\#B\) is the smash product of the duals of \(A\) and \(B\). Complementary results are given for \(*\)-Hopf algebras. Several examples are studied throughout the paper: the case of a matched pair of groups, the case \(B=A^{cop}\), the case where the action or the coaction is trivial \(\dots\).

MSC:
16T05 Hopf algebras and their applications
16T20 Ring-theoretic aspects of quantum groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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