nLab bicrossed product

Idea

The bicrossed product, also known as the Zappa–Szép product, generalizes the semidirect product of groups.

This construction is essential to the quantum double construction of Drinfel’d. The bicrossed product applies more generally to monoids and other algebraic entities, see (Brin 04).

Definition

Given a pair of matched groups $H$ and $K$, the bicrossed product of groups $H\times K$ on the set $H \times K$ is given by

$$(h,k)\cdot(h',k') = (h\alpha(k,h'),\beta(k,h')k')$$

with unit $(1,1)$ and $h,h'\in H$, $k,k'\in K$, where $\alpha: K\times H\rightarrow H$, $\beta: K\times H\rightarrow K$ are left and right actions, respectively.

A pair of groups $(H,K)$ is said to be matched if there exists a left action $\alpha$ of $K$ on the set $H$ and a right action $\beta$ of the group $H$ on the set $K$ such that for all $h,h'\in H$, $k,k'\in K$, the following hold:

• $\beta(k k',h) = \beta(k,\alpha(k',h))\beta(k',h)$,
• $\alpha(k,h h') = \alpha(k,h)\alpha(\beta(k,h),h')$,
• $\alpha(k,1) = 1$,
• $\beta(1,h) = 1$.

Need to define the bicrossed product of algebras.

Properties

The bicrossed products of two groups $H$ and $K$ precisely correspond to the distributive laws between the monads $T=H \times -$ and $S=K\times -$.

References

• Christian Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer (1995) [doi:10.1007/978-1-4612-0783-2, webpage, errata pdf]

• Matthew G. Brin, On the Zappa-Szep Product [arXiv:math/0406044]