Classifying coalgebra split extensions of Hopf algebras

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split extensions of A by H4, where H4 is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A#H4 by computing explicitly two classifying objects: the cohomological group H2(H4,A) and
Crp(H4,A) := the set of types of isomorphisms of all crossed products A#H4. All crossed products A#H4 are described by generators and relations and classified: they are parameterized by the set ZP(A) of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The
groups of automorphisms of these Hopf algebras are also described.

Publication year:2013

Keywords:crossed product of Hopf algebras