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 $α : A \otimes X \to X$ turning $X$ a module coalgebra over $A$ and $β : A \otimes X \to A$ turning $A$ into a right module coalgebra over $X$ , we call them a pair of matched bialgebras. If we set $α (a \otimes x) = a \cdot x$ and $β (a \otimes x) = a^{x}$ , the following conditions are satisfied

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

$a \cdot 1_{X} = ε_{A} (a) 1_{X}$

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

$1_{A}^{x} = ε_{X} (x) 1_{A}$

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

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 $X ⋈ A$ ,

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 $ε (x \otimes a) = ε_{X} (x) ε_{A} (a)$ ;
The coproduct is $Δ (x \otimes a) = \sum_{(x), (a)} (x_{(1)} \otimes a_{(1)}) \otimes (x_{(2)} \otimes a_{(2)})$ ;
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 $H^{* c o p}$ are matched pairs. In this case, the bicrossed product of them is called Drinfeld quantum double $D (H) = H^{* c o p} ⋈ H$ .

References

↑ Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621
↑ Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0783-2, ISBN 9780387943701
↑ Majid, Shahn (1995), Foundations of quantum group theory, Cambridge University Press, doi:10.1017/CBO9780511613104, ISBN 9780511613104

[1] Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621

[2] Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0783-2, ISBN 9780387943701

[3] Majid, Shahn (1995), Foundations of quantum group theory, Cambridge University Press, doi:10.1017/CBO9780511613104, ISBN 9780511613104