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

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Summary

Introduction

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

Pivotal Hopf monoids
It satisfies the diagrammatic identities p
Modules and comodules
Hopf modules and Yetter-Drinfeld modules
Background on ribbon graphs
Ribbon graphs
Chord diagrams
Mapping class groups
Presentation in terms of Dehn twists
Presentation in terms of chord slides
Left pentagon relation
Hopf monoid labelled ribbon graphs
Mapping class group actions by edge slides
The torus and the one-holed torus
Slides and twists
Slides along face paths
Adding edges
Adding an edge γ to a face path γ in Γ corresponds to the morphism
Twists along closed face paths
Mapping class group action by Dehn twists
Concluding Remarks

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