Bicrossed product of Hopf algebra
In quantum group and Hopf algebra, the bicrossed product is a process to create new Hopf algebras from the given ones. It's motivated by the Zappa–Szép product of groups. It was first discussed by M. Takeuchi in 1981, and now a general tool for construction of Drinfeld quantum double. Bicrossed product 

Consider two bialgebras  and , if there exist linear maps   turning   a module coalgebra over , and   turning  into a right module coalgebra over  . We call them a pair of matched bialgebras, if we set  and , the following conditions are satisfied

for all  and . Here the Sweedler's notation of coproduct of Hopf algebra is used. For matched pair of Hopf algebras  and , there exists a unique Hopf algebra over , the resulting Hopf algebra is called bicrossed product of  and  and denoted by ,

 The unit is given by ; 
 The multiplication is given by ;
 The counit is ;
 The coproduct is ;
 The antipode is . Drinfeld quantum double 

For a given Hopf algebra , its dual space  has a canonical Hopf algebra structure and  and  are matched pairs. In this case, the bicrossed product of them is called Drinfeld quantum double . References 

Hopf algebras