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.