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,^[1] and now a general tool for construction of Drinfeld quantum double.^[2][3]

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

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

${\displaystyle a\cdot 1_{X}=\varepsilon _{A}(a)1_{X}}$

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

${\displaystyle 1_{A}^{x}=\varepsilon _{X}(x)1_{A}}$

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

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

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

• The unit is given by ${\displaystyle (1_{X}\otimes 1_{A})}$;
• The multiplication is given by ${\displaystyle (x\otimes a)(y\otimes b)=\sum _{(a),(y)}x(a_{(1)}\cdot y_{(1)})\otimes a_{(2)}^{y_{(2)}}b}$;
• The counit is ${\displaystyle \varepsilon (x\otimes a)=\varepsilon _{X}(x)\varepsilon _{A}(a)}$;
• The coproduct is ${\displaystyle \Delta (x\otimes a)=\sum _{(x),(a)}(x_{(1)}\otimes a_{(1)})\otimes (x_{(2)}\otimes a_{(2)})}$;
• The antipode is ${\displaystyle S(x\otimes a)=\sum _{(x),(a)}S(a_{(2)})\cdot S(x_{(2)})\otimes S(a_{(1)})^{S(x_{(1)})}}$.

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