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 $$A$$ and $$X$$, if there exist linear maps $$\alpha:A\otimes X \to X$$ turning  $$X$$ a module coalgebra over $$A$$, and $$\beta: A\otimes X\to A$$  turning $$A$$ into a right module coalgebra over  $$X$$. We call them a pair of matched bialgebras, if we set $$\alpha(a\otimes x)=a\cdot x$$ and $$\beta(a\otimes x)=a^x$$, the following conditions are satisfied

$$a\cdot (xy)=\sum_{(a),(x)}(a_{(1)} \cdot x_{(1)}) (a_{(2)}^{x_{(2)}} \cdot y)$$

$$a\cdot 1_X=\varepsilon_A(a)1_X$$

$$(ab)^x=\sum_{(b),(x)}a^{b_{(1)} \cdot x_{(1)}} b_{(2)}^{x_{(2)}}$$

$$1_A^x=\varepsilon_X(x)1_A$$

$$\sum_{(a),(x)}a_{(1)}^{x_{(1)}} \otimes a_{(2)}\cdot x_{(2)}=\sum_{(a),(x)}a_{(2)}^{x_{(2)}}\otimes a_{(1)}\cdot x_{(1)}$$

for all $$a,b\in A$$ and $$x,y\in X$$. Here the Sweedler's notation of coproduct of Hopf algebra is used.

For matched pair of Hopf algebras $$A$$ and $$X$$, there exists a unique Hopf algebra over $$X\otimes A$$, the resulting Hopf algebra is called bicrossed product of $$A$$ and $$X$$ and denoted by $$X \bowtie A$$,


 * The unit is given by $$(1_X\otimes 1_A)$$;
 * The multiplication is given by $$(x\otimes a)(y\otimes b)=\sum_{(a),(y)}x(a_{(1)}\cdot y_{(1)}) \otimes a_{(2)}^{y_{(2)}} b$$;
 * The counit is $$\varepsilon(x\otimes a)=\varepsilon_X(x)\varepsilon_A(a)$$;
 * The coproduct is $$\Delta(x\otimes a)=\sum_{(x),(a)} (x_{(1)}\otimes a_{(1)}) \otimes (x_{(2)}\otimes a_{(2)})$$;
 * The antipode is $$S(x\otimes a)=\sum_{(x),(a)}S(a_{(2)})\cdot S(x_{(2)}) \otimes S(a_{(1)})^{S(x_{(1)})}$$.

Drinfeld quantum double
For a given Hopf algebra $$H$$, its dual space $$H^*$$ has a canonical Hopf algebra structure and $$H$$ and $$H^{*cop}$$ are matched pairs. In this case, the bicrossed product of them is called Drinfeld quantum double $$D(H)=H^{*cop}\bowtie H$$.