Classification of Consistent Systems of Handlebody Group Representations
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Classification of Consistent Systems of Handlebody Group Representations. / Müller, Lukas; Woike, Lukas.
I: International Mathematics Research Notices, Bind 2024, Nr. 6, 2024, s. 4767-4803.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Classification of Consistent Systems of Handlebody Group Representations
AU - Müller, Lukas
AU - Woike, Lukas
N1 - Publisher Copyright: © The Author(s) 2023. Published by Oxford University Press. All rights reserved.
PY - 2024
Y1 - 2024
N2 - 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 bicategory M (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.
AB - 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 bicategory M (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.
U2 - 10.1093/imrn/rnad178
DO - 10.1093/imrn/rnad178
M3 - Journal article
AN - SCOPUS:85188614322
VL - 2024
SP - 4767
EP - 4803
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 6
ER -
ID: 389416091