bicrossed product

The bicrossed product generalizes the semidirect product of groups.

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

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

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

with unit $(1,1)$ and $h,h\prime \in H$, $k,k\prime \in K$, where $\alpha :K\times H\to H$, $\beta :K\times H\to K$ 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 $h,h\prime \in H$, $k,k\prime \in K$, the following hold:

- $\beta (kk\prime ,h)=\beta (k,\alpha (k\prime ,h))\beta (k\prime ,h)$,
- $\alpha (k,hh\prime )=\alpha (k,h)\alpha (\beta (k,h),h\prime )$,
- $\alpha (k,1)=1$,
- $\beta (1,h)=1$.

Need to define the bicrossed product of algebras.

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)