Contractions, Hopf algebra extensions and cov. differential calculus

Abstract

We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with the contracted algebras and the requirements for their existence are examined as well.