nLab
bicrossed product

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 HH and KK, the bicrossed product of groups H×KH\times K on the set H×KH \times K is given by

(1)(h,k)(h,k)=(hα(k,h),β(k,h)k) (h,k)\cdot(h',k') = (h\alpha(k,h'),\beta(k,h')k')

with unit (1,1)(1,1) and h,hHh,h'\in H, k,kKk,k'\in K, where α:K×HH\alpha: K\times H\rightarrow H, β:K×HK\beta: K\times H\rightarrow K are left and right actions, respectively.

A pair of groups (H,K)(H,K) is said to be matched if there exists a left action α\alpha of KK on the set HH and a right action β\beta of the group HH on the set KK such that for all h,hHh,h'\in H, k,kKk,k'\in K, the following hold: * β(kk,h)=β(k,α(k,h))β(k,h)\beta(k k',h) = \beta(k,\alpha(k',h))\beta(k',h), * α(k,hh)=α(k,h)α(β(k,h),h)\alpha(k,h h') = \alpha(k,h)\alpha(\beta(k,h),h'), * α(k,1)=1\alpha(k,1) = 1, * β(1,h)=1\beta(1,h) = 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)