Abstract
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.
Highlights
Mapping class group representations arising from Hopf algebras have been investigated extensively in the context of topological quantum field theories, Chern-Simons theories and conformal field theories and in the quantisation of moduli spaces of flat connections.In [Ly95a, Ly95b, Ly96] Lyubashenko constructed projective representations of surface mapping class groups from Hopf algebras in certain abelian ribbon categories
We show that any pivotal Hopf monoid H in a symmetric monoidal category C gives rise to actions of mapping class groups of oriented surfaces of genus g ≥ 1 with n ≥ 1 boundary components
That Cγ is a morphism of H-modules and H-comodules for the cilia that are not traversed by γ follows directly from the fact that this holds for the slide Sγ by Corollary 8.3 and it holds for adding a loop
Summary
Mapping class group representations arising from Hopf algebras have been investigated extensively in the context of topological quantum field theories, Chern-Simons theories and conformal field theories and in the quantisation of moduli spaces of flat connections. We illustrate this with the example for the mapping class group of a surface of genus 2 with one or no boundary components
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