Drinfeld 中心中作为 ribbon 对偶对象的特异可逆对象
The distinguished invertible object as ribbon dualizing object in the Drinfeld center
AI总结：
本文证明 pivotal 有限张量范畴的 Drinfeld 中心具有唯一 ribbon Grothendieck-Verdier 结构，其对偶对象为特异可逆对象，并由此得到 ansular 函子甚至 modular 函子。
摘要：
我们证明，一个 pivotal 有限张量范畴 $\mathcal{C}$ 的 Drinfeld 中心 $Z(\mathcal{C})$ 带有 Boyarchenko-Drinfeld 意义下的 ribbon Grothendieck-Verdier 范畴结构。用 operad 的语言表述，这使得 $Z(\mathcal{C})$ 成为 framed $E_2$-operad 上的循环代数。对偶对象的底层对象是 $\mathcal{C}$ 的特异可逆对象，它出现在著名的 Etingof-Nikshych-Ostrik Radford 同构中。在等价意义下，这是 $Z(\mathcal{C})$ 上唯一的 ribbon Grothendieck-Verdier 结构，它扩张了 $Z(\mathcal{C})$ 已经具备的典范平衡辫结构。该 ribbon Grothendieck-Verdier 结构的对偶函子与刚性对偶一致，当且仅当 $\mathcal{C}$ 在 Douglas-Schommer-Pries-Snyder 意义下是 spherical 的。我们代数结果的主要拓扑推论是，$Z(\mathcal{C})$ 给出一个 ansular 函子，事实上甚至是一个 modular 函子，无论 $\mathcal{C}$ 是否为 spherical。为了证明上述 ribbon Grothendieck-Verdier 结构的唯一性陈述，我们推导出一个七项正合序列，刻画平衡辫范畴上 ribbon Grothendieck-Verdier 结构的空间。该序列涉及平衡辫范畴的 Müger 中心的平衡版本的 Picard 群。
英文摘要：
We prove that the Drinfeld center $Z(\mathcal{C})$ of a pivotal finite tensor category $\mathcal{C}$ comes with the structure of a ribbon Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. Phrased operadically, this makes $Z(\mathcal{C})$ into a cyclic algebra over the framed $E_2$-operad. The underlying object of the dualizing object is the distinguished invertible object of $\mathcal{C}$ appearing in the well-known Radford isomorphism of Etingof-Nikshych-Ostrik. Up to equivalence, this is the unique ribbon Grothendieck-Verdier structure on $Z(\mathcal{C})$ extending the canonical balanced braided structure that $Z(\mathcal{C})$ already comes equipped with. The duality functor of this ribbon Grothendieck-Verdier structure coincides with the rigid duality if and only if $\mathcal{C}$ is spherical in the sense of Douglas-Schommer-Pries-Snyder. The main topological consequence of our algebraic result is that $Z(\mathcal{C})$ gives rise to an ansular functor, in fact even a modular functor regardless of whether $\mathcal{C}$ is spherical or not. In order to prove the aforementioned uniqueness statement for the ribbon Grothendieck-Verdier structure, we derive a seven-term exact sequence characterizing the space of ribbon Grothendieck-Verdier structures on a balanced braided category. This sequence features the Picard group of the balanced version of the Müger center of the balanced braided category.
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