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 H and K, the bicrossed product of groups H×K on the set H×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) and h,hH, k,kK, where α:K×HH, β:K×HK are left and right actions, respectively.

A pair of groups (H,K) is said to be matched if there exists a left action α of K on the set H and a right action β of the group H on the set K such that for all h,hH, k,kK, the following hold:

  • β(kk,h)=β(k,α(k,h))β(k,h),
  • α(k,hh)=α(k,h)α(β(k,h),h),
  • α(k,1)=1,
  • β(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)