The bicrossed product generalizes the semidirect product of groups.
This construction is essential to the quantum double construction? of Drinfel’d.
Given a pair of matched groups and , the bicrossed product of groups on the set is given by
(h,k)\cdot(h',k') = (h\alpha(k,h'),\beta(k,h')k')
with unit and , , where , are left and right actions, respectively.
A pair of groups is said to be matched if there exists a left action of on the set and a right action of the group on the set such that for all , , the following hold:
Need to define the bicrossed product of algebras.
C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York-Berlin, 1995.