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,[1] and now a general tool for construction of Drinfeld quantum double.[2][3]
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 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
^Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621