Classifying bicrossed products of two Sweedler's Hopf algebras

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In this paper we continue the study started recently in ABMbp by describing and classifying all Hopf algebras E that factorize through two Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products H₄ H₄. There are three steps in our approach. First, we explicitly describe the set of all matched pairs (H₄, H₄,,) by proving that, with the exception of the trivial pair, this set is parameterized by the ground field k. Then, for any λ∈ k, we describe by generators and relations the associated bicrossed product, H₁₆, λ. This is a 16-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra E factorizes through H₄ and H₄ if and only if E H₄ H₄ or E H₁₆, λ. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: H₄ H₄, H₁₆, ₀ and H₁₆, ₁ D(H₄), the Drinfel'd double of H₄. The automorphism group of these objects is also computed: in particular, we prove that Hopf(D(H₄)) is isomorphic to a semidirect product of groups, k× Z₂.