论文标题
映射Hopf Monoids和Ribbon图的分类组动作
Mapping class group actions from Hopf monoids and ribbon graphs
论文作者
论文摘要
我们表明，对称单体类别中的任何关键HOPF MONOID $ H $ $ \ MATHCAL {C} $都会产生映射属于$ g \ geq 1 $的级别表面的类别的动作，其中$ n \ geQ 1 $ n \ geq 1 $边界组件。这些映射的类小组动作由小组同构为$ H $以上的某些Yetter-Drinfeld模块的自动形态组提供。它们与嵌入式功能区图中的边缘载玻片相关联，将和弦图中的和弦载玻片概括。我们在产生dehn twist和定义关系方面对这些映射类小组动作进行了具体描述。对于$ \ Mathcal {C} $有限完整且合并的情况，我们还通过在Yetter-Drinfeld模块结构下强加不变性和共同变量来获得映射封闭表面的类组的动作。
We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g \geq 1$ with $n \geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $\mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.
