Abstract
We analyze the structure of a Hopf algebra H that has a Hopf subalgebra H " and a left H "-module coalgebra projection onto H ". In this situation H ≅ H " ⊗ Q for Q = H / H "+H, and the Hopf algebra structure on H can be recovered from suitable structures on Q, among others an in general nonassociative multiplication. The construction of H from H " and Q generalizes Radford biproduct, double crossproducts, and certain bicrossproducts. Further examples are Hopf algebras with a triangular decomposition, like all quantized enveloping algebras. In an appendix, we improve a standard criterion for a bicrossproduct Aα ⋈τB of two Hopf algebras to be a Hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are.
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