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 H-4, where H-4 is the Sweedler's four-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A# H-4 by computing explicitly two classifying objects: the cohomological "group" H-2(H-4, A) and CRP(H-4, A) := the set of types of isomorphisms of all crossed products A# H-4. All crossed products A# H-4 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.