Extensions of Yetter-Drinfel'd Hopf algebras

Yetter-Drinfel‘d Hopf代数是关于普通Hopf代数定义的某些拟对称么半群范畴中的Hopf代数。它们出现在普通Hopf代数理论中，作为半直积适当推广的因子：如果一个群包含一个允许收缩到子群上的子群，即从大群到子群的同态限制到子群上的恒等式，则大群是子群和正规子群的半直积，即收缩的核。 这一来自群论的事实推广到Hopf代数如下：如果一个Hopf代数包含一个允许收缩到Hopf子代数上的Hopf子代数，即从大Hopf代数到限制在Hopf子代数上的恒等式的Hopf代数同态，则大Hopf代数可以分解成Hopf子代数和收缩的Hopf-代数核的张量积。然而，在这种情况下，Hopf-代数核本身一般不是Hopf代数。相反，它是Hopf子代数上的Yetter-Drinfel‘d Hopf代数。这一结果被称为Radford投影定理，这就是Yetter-Drinfel‘d Hopf代数在普通Hopf代数理论中发挥作用的原因。 一个基团对另一个基团的扩展可以由第一基团对第二基团的作用和相对于该作用的余循环来描述。Hopf代数的扩张可以用类似的方式来描述，它使用两个附加的结构元素，即上作用和关于这个上作用的对偶上循环。我们目前的研究目标是为Yetter-Drinfel‘d Hopf代数的扩张找到类似的刻划。我们已经取得了实质性的进展，可以说还需要什么：除了一个行动、一个共同作用、一个协同周期和一个双重协同周期，人们还需要一个所谓的偏差图和一个协差图。有了这些结构元素，我们就可以写出乘积和余积的显式公式。然而，为了产生Yetter-Drinfel‘d Hopf代数而必须满足的这些结构元素的相容条件仍然需要确定。例如，尽管在Yetter-Drinfel‘d Hopf代数的情况下以类似的方式定义了上循环，但它不再自动满足它在Hopf代数中所满足的标准上循环恒等式。到目前为止，我们只知道在特殊情况下的必要相容条件。我们的目标是在总体上找到它们。

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.