Extensions of Yetter-Drinfel'd Hopf algebras

Yetter-Drinfel'd Hopf 代数是某些准对称幺半群范畴中的 Hopf 代数，这些范畴是相对于普通 Hopf 代数定义的。它们出现在普通 Hopf 代数理论中，作为半直积的适当推广的因素：如果一个群包含一个允许撤消到子群上的子群，即从大群到子群的群同态限制子群上的恒等式，那么大群是子群和正规子群的半直积，即撤消的核。 ******这个来自群论的事实概括为 Hopf 代数如下：如果 Hopf 代数包含一个允许回缩到 Hopf 子代数上的 Hopf 子代数，即，从大 Hopf 代数到 Hopf 子代数的 Hopf 代数同态限制了 Hopf 子代数上的恒等式，则大 Hopf 代数可以分解为 Hopf 的张量积子代数和回缩的 Hopf 代数核。然而，在这种情况下，Hopf 代数核本身通常不是 Hopf 代数。相反，它是 Hopf 子代数上的 Yetter-Drinfel'd Hopf 代数。这个结果被称为 Radford 投影定理，是 Yetter-Drinfel'd Hopf 代数在普通 Hopf 代数理论中发挥作用的原因。******一个群对另一个群的扩展可以通过第一群对第二群的作用以及关于该作用的余循环来描述。 Hopf 代数的扩展可以通过使用两个附加结构元素以类似的方式描述，即一个共同作用和关于该共同作用的对偶余循环。我们当前研究的目标是为 Yetter-Drinfel'd Hopf 代数的扩展找到类似的描述。我们已经取得了实质性进展，可以说还需要什么：除了作用、相互作用、余循环和对偶余循环之外，还需要所谓的偏差图和共偏差图。有了这些结构元素，我们就可以写出积和余积的明确公式。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兼容性条件仍然需要确定。例如，尽管在 Yetter-Drinfel'd Hopf 代数的情况下以类似的方式定义了余循环，但它不再自动满足它在 Hopf 代数情况下所满足的标准余循环恒等式。到目前为止，我们只知道特殊情况下必要的兼容性条件。我们的目标是总体上找到它们。

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.