Classifying all Hopf algebras of a given finite dimension over the complex
numbers is a challenging problem which remains open even for many small
dimensions, not least because few general approaches to the problem are known.
Some useful techniques include counting the dimensions of spaces related to
the coradical filtration, studying sub- and quotient Hopf algebras, especially
those sub-Hopf algebras generated by a simple subcoalgebra, working with the
antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to
classify Radford biproducts. In this paper, we add to the classification tools
in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras
of dimension rpq and 8p where p,q,r are distinct primes.
At the end of this paper we summarize in a table the status of the
classification for dimensions up to 100 to date.Comment: This version of the paper contains a correction on the published
version. The statement and proof of Proposition 2.17 are changed and the
proof of the results that follow from it are corrected accordingly. We thank
H.-S. Ng for kindly communicating the gap to us and for the careful reading
of our pape
