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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677152
• 关键词: Extensions; Yetter; Drinfel; Hopf; algebras; 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'dHOPF代数是某些准对称单体类别中的HOPF代数，这些类别是根据普通的Hopf代数定义的。它们是在普通的Hopf代数理论中作为半领产品的适当概括的因素而出现的：如果一个组包含一个子组，该子组承认缩回对亚组的缩回，即，从大组到从大组到子组的同构同源性，该子组限制了限制在子组上的身份，则大型组是半组的次数群体组成的产品和一个正常的产品，并构成了一个正常的产品，并构成了一个正常的产品，并且是一个正常的概述和一个正常的产品。 *****这一事实从群体理论概括到霍普夫代数如下：如果一个hopf代数包含一个hopf subgebra，它承认撤回了霍普夫亚gengera，即，霍普夫代数同源物是从大型霍夫布拉（Hopf）代数限制了霍普夫（Hopf subalgebra）的众多希望的人身份的概括，而越来越多的hopf cai则占据了人们的身份。被分解为Hopf亚级别的张量产品和缩回的Hopf代数核。但是，Hopf-Elgebraic内核通常在这种情况下本身并不是HOPF代数。相反，这是一个近乎的hopf代数，而不是霍普夫subgerbra。该结果被称为Radford投影定理，是为什么Yetter-drinfel'Hopf代数在普通的Hopf代数理论中发挥作用。可以通过使用两个附加的结构元素来描述HOPF代数的扩展，即相对于此配置，即共同体和双重共生。我们研究的当前目标是找到类似的描述，以扩展到Yetter-drinfel'd hopf代数。我们已经取得了长足的进步，可以说明需要什么：除了动作，共同行动，旋转和双重共生外，还需要一个所谓的偏差图和代码贴图。有了这些结构元素，我们可以写下产品和互合型的显式公式。但是，必须确定这些结构要素的兼容性条件，这些元素必须确定YoPFF代数。例如，尽管在Yetter-drinfel'Hopf代数的情况下以类似方式定义了Cocycle，但它确实不再自动满足Hopf代数案例所满足的标准合过程身份。到目前为止，我们仅在特殊情况下才知道必要的兼容性条件。我们的目标是总体上找到它们。