柄体群表示的相容系统的分类
Classification of Consistent Systems of Handlebody Group Representations
AI总结：
本研究围绕柄体群表示的相容系统分类问题，以柄体模operad为核心工具，证明其在对称幺半双范畴上的代数（ansular函子）等价于自对偶平衡辫子代数，线性框架下对应ribbon Grothendieck-Verdier范畴，还给出柄体对应向量空间的公式并可应用于顶点算子代数。
摘要：
柄体群的分类空间构成一个模operad。柄体operad上的代数给出了与粘合相容的柄体群表示系统。我们证明，取值于任意对称幺半双范畴$\u27e8M\u27e9$（我们将其命名为ansular函子）的柄体模operad上的代数，等价于$\u27e8M\u27e9$中的自对偶平衡辫子代数。在线性框架下特殊化后，这证明了有限维向量空间上的柄体群表示的相容系统等价于Boyarchenko-Drinfeld意义下的ribbon Grothendieck-Verdier范畴。此外，它还通过Lyubashenko余积的推广，给出了分配给任意柄体的向量空间的具体公式。我们的主要结果可用于从满足温和有限性条件的顶点算子代数得到ansular函子。这包括表示范畴具有非正合幺半积的顶点算子代数的例子。
英文摘要：
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 $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. 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.
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