Extensions of Yetter-Drinfel'd Hopf algebras

yeter - drinfel 'd Hopf代数是在一般Hopf代数的基础上定义的准对称一元范畴中的Hopf代数。它们作为半直积的适当推广的因子出现在普通Hopf代数理论中：如果群中含有允许子群上有缩回的子群，即从大群到子群的群同态，且子群上有恒等式，则该大群是子群与正规子群的半直积，即缩回的核。******这一事实由群论推广到Hopf代数如下：如果一个Hopf代数包含一个Hopf子代数，该Hopf子代数允许在Hopf子代数上有一个缩回，即Hopf代数从大Hopf代数到Hopf子代数的同态，而Hopf子代数限制在Hopf子代数上的恒等，则大Hopf代数可以分解为Hopf子代数与缩回的Hopf代数核的张量积。然而，在这种情况下，Hopf代数核本身通常不是Hopf代数。更确切地说，它是在Hopf子代数上的一个Yetter-Drinfel Hopf代数。这个结果被称为Radford投影定理，这就是为什么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延伸可以用第一基团对第二基团的作用和关于该作用的循环来描述。Hopf代数的扩展可以用类似的方法来描述，通过使用两个额外的结构元素，即一个共作用和关于这个共作用的对偶环。我们目前的研究目标是为Yetter-Drinfel'd Hopf代数的扩展找到一个类似的描述。我们已经取得了实质性的进展，并且可以说还需要什么：除了一个动作、一个合作、一个循环和一个双循环之外，还需要一个所谓的偏差图和一个协偏差图。有了这些结构元素，我们就可以写出乘积和副乘积的显式公式。然而，为了产生yeter - drinfel 'd Hopf代数，这些结构元素必须满足的相容性条件仍然需要确定。例如，虽然在Yetter-Drinfel'd Hopf代数的情况下以类似的方式定义了循环，但它不再自动满足它在Hopf代数情况下满足的标准循环恒等式。到目前为止，我们只知道在特殊情况下必要的兼容性条件。我们的目标是找到它们的一般形式。

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.