Bicrossed product of Hopf algebra

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

[math]\displaystyle{ a\cdot (xy)=\sum_{(a),(x)}(a_{(1)} \cdot x_{(1)}) (a_{(2)}^{x_{(2)}} \cdot y) }[/math]

[math]\displaystyle{ a\cdot 1_X=\varepsilon_A(a)1_X }[/math]

[math]\displaystyle{ (ab)^x=\sum_{(b),(x)}a^{b_{(1)} \cdot x_{(1)}} b_{(2)}^{x_{(2)}} }[/math]

[math]\displaystyle{ 1_A^x=\varepsilon_X(x)1_A }[/math]

[math]\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)} }[/math]

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

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

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

Drinfeld quantum double

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