Extensions of Yetter-Drinfel'd Hopf algebras

Yetter-Drinfeld Hopf 代数的推广

• 批准号: RGPIN-2017-06543
• 负责人: Sommerhäuser, Yorck
• 金额: $ 1.46万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=665599
• 关键词: Extensions; Yetter; Drinfel; Hopf; algebras

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.

Yetter-Drinfel'd Hopf代数是在某些关于普通Hopf代数定义的拟对称monoidal范畴中的Hopf代数。它们出现在普通Hopf代数的理论中，作为半直积的适当推广中的因子：如果一个群包含一个允许收缩到该子群上的子群，即，一个从大群到子群的群同态限制于子群上的恒等式，则大群是子群与正规子群的半直积，即收缩核。** 这个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Hopf子代数，它允许一个收缩到该Hopf子代数上，即，证明了一个从大Hopf代数到Hopf子代数的Hopf代数同态，并将其限制在Hopf子代数上的恒等式上，则大Hopf代数可分解为Hopf子代数与收缩的Hopf代数核的张量积.然而，在这种情况下，Hopf-代数核一般本身不是一个Hopf代数。相反，它是一个在Hopf子代数上的Yetter-Drinfel'd Hopf代数。这个结果被称为拉德福投影定理，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。一个群对另一个群的扩张可以用第一个群对第二个群的作用和关于这个作用的上循环来描述。Hopf代数的一个扩展可以用类似的方法描述，通过使用两个额外的结构元素，即一个余作用和关于这个余作用的一个对偶上圈。我们目前的研究目标是找到一个类似的描述Yetter-Drinfel'd Hopf代数的扩展。我们已经取得了实质性的进展，并且可以说还需要什么：除了作用、余作用、上循环和对偶上循环之外，还需要所谓的偏差图和协偏差图。利用这些结构元素，我们可以写出乘积和余积的显式表达式。然而，为了产生Yetter-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，上循环是以类似的方式定义的，但它不再自动满足它在Hopf代数情况下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性条件。我们的目标是找到他们一般。