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