# Idea

The bicrossed product generalizes the semidirect product of groups.

This construction is essential to the quantum double construction? of Drinfel’d.

# Definition

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

(1)$\left(h,k\right)\cdot \left(h\prime ,k\prime \right)=\left(h\alpha \left(k,h\prime \right),\beta \left(k,h\prime \right)k\prime \right)$(h,k)\cdot(h',k') = (h\alpha(k,h'),\beta(k,h')k')

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

A pair of groups $\left(H,K\right)$ 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 $h,h\prime \in H$, $k,k\prime \in K$, the following hold:

• $\beta \left(kk\prime ,h\right)=\beta \left(k,\alpha \left(k\prime ,h\right)\right)\beta \left(k\prime ,h\right)$,
• $\alpha \left(k,hh\prime \right)=\alpha \left(k,h\right)\alpha \left(\beta \left(k,h\right),h\prime \right)$,
• $\alpha \left(k,1\right)=1$,
• $\beta \left(1,h\right)=1$.

Need to define the bicrossed product of algebras.

# References

C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York-Berlin, 1995.

Revised on May 14, 2009 03:28:17 by Toby Bartels (138.23.203.139)