an:06155941
Zbl 1271.16032
Delvaux, L.; Van Daele, A.; Wang, S. H.
Bicrossproducts of algebraic quantum groups
EN
Int. J. Math. 24, No. 1, Paper No. 1250131, 48 p. (2013).
00315786
2013
j
16T05 16T20 17B37
multiplier Hopf algebras; integrals; algebraic quantum groups; bicrossproducts; smash products
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 \(\dots\).
Lo??c Foissy (Calais)