论文标题
在德林菲尔德中心中，区别的可逆物体作为带偶的对象
The distinguished invertible object as ribbon dualizing object in the Drinfeld center
论文作者
论文摘要
我们证明，在Boyarchenko-drinfeld的感觉中，Drinfeld Center $ Z（\ Mathcal {C}）$的$ Z（\ Mathcal {C}）$ $ \ MATHCAL {C} $都带有色带Grothendieck-verdier类别的结构。从操作上讲，这使$ z（\ nathcal {c}）$在框架$ e_2 $ -operad上的循环代数中。二元对象的基本对象是$ \ Mathcal {c} $的可逆对象出现在著名的nikshych-oStrik的著名Radford同构中。到等效，这是$ z（\ Mathcal {C}）上的独特功能区Grothendieck-Verdier结构，扩展了已经配备了$ z（\ Mathcal {C}）$的规范平衡编织结构。当且仅当$ \ Mathcal {C} $仅在Douglas-Schommer-pries-snyder的意义上，此功能区Grothendieck-Verdier结构的双重功能与刚性双重性相吻合。我们代数结果的主要拓扑结果是$ z（\ nathcal {c}）$会产生一个Ansular functor，实际上，即使是模块化的函数，无论$ \ Mathcal {C} $是否是球形的。为了证明色带Grothendieck-Verdier结构的上述唯一性陈述，我们得出了一个七个精确的序列，该序列表征了均衡的编织类别中功能区Grothendieck-Verdier结构的空间。该序列以平衡编织类别的Müger中心平衡版本的PICARD组为特色。
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.
