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 $D(H)=H^{*cop}\bowtie H$.