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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