• Corpus ID: 246035317

Classification of Consistent Systems of Handlebody Group Representations

@inproceedings{Muller2022ClassificationOC,
  title={Classification of Consistent Systems of Handlebody Group Representations},
  author={Lukas Muller and Lukas Woike},
  year={2022}
}

• L. Muller, Lukas Woike
• Published 19 January 2022
• Mathematics

The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategoryM (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in M. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.