Extensions of Yetter-Drinfel'd Hopf algebras

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Yetter-Drinfel'd Hopf algebras are Hopf algebras in certain quasisymmetric monoidal categories that are defined with respect to an ordinary Hopf algebra. They arise in the theory of ordinary Hopf algebras as factors in the appropriate generalization of semidirect products: If a group contains a subgroup that admits a retraction onto the subgroup, i.e., a group homomorphism from the large group to the subgroup that restricts to the identity on the subgroup, then the large group is a semidirect product of the subgroup and a normal subgroup, namely the kernel of the retraction. This fact from group theory generalizes to Hopf algebras as follows: If a Hopf algebra contains a Hopf subalgebra that admits a retraction onto the Hopf subalgebra, i.e., a Hopf algebra homomorphism from the large Hopf algebra to the Hopf subalgebra that restricts to the identity on the Hopf subalgebra, then the large Hopf algebra can be decomposed into a tensor product of the Hopf subalgebra and the Hopf-algebraic kernel of the retraction. However, the Hopf-algebraic kernel is in this situation in general not itself a Hopf algebra. Rather, it is a Yetter-Drinfel'd Hopf algebra over the Hopf subalgebra. This result, which is known as the Radford projection theorem, is the reason why Yetter-Drinfel'd Hopf algebras play a role in the theory of ordinary Hopf algebras. An extension of one group by another can be described by an action of the first group on the second group and a cocycle with respect to this action. An extension of Hopf algebras can be described in a similar way by using two additional structure elements, namely a coaction and a dual cocycle with respect to this coaction. The current goal of our research is to find a similar description for extensions of Yetter-Drinfel'd Hopf algebras. We have already made substantial progress and can say what is needed in addition: Besides an action, a coaction, a cocycle, and a dual cocycle, one needs a so-called deviation map and a codeviation map. With these structure elements, we can write down explicit formulas for product and coproduct. However, the compatibility conditions for these structure elements that have to be satisfied in order to yield a Yetter-Drinfel'd Hopf algebra still need to be determined. For example, although the cocycle is defined in an analogous fashion in the case of Yetter-Drinfel'd Hopf algebras, it does no longer automatically satisfy the standard cocycle identity that it satisfies in the Hopf algebra case. So far, we know the necessary compatibility conditions only in a special case. Our goal is to find them in general.