论文标题
编织的交换代数和有限张量类别的deligne猜想的同义不变性
Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories
论文作者
论文摘要
在有限张量类别中找到代数$ \ mathbb {t} \ in \ mathcal {c} $ in \ in \ mathcal $ \ mathcal {c} $自然而然地带有编织的交换代数$ \ mathsf {c {t} $ in z（\ nathcal clintfecal in z（\ nathcalcal clintf） $ \ Mathcal {C} $。实际上，任何有限的张量类别都具有至少两个这样的代数，即单型单元$ i $和规范端$ \ int_ {x \ in \ Mathcal {c}} x \ otimes x^\ vee $。使用编织的作业理论，我们证明，对于任何此类代数$ \ mathbb {t} $同型不变性，即从$ i $到$ i $到$ \ mathbb {t} $的派生形态空间，自然而然地伴随着差异分析的$ e_2 $ -a_2 $ -algebra的结构。这样，我们获得了有限张量类别的同源代数中的差异分级$ e_2 $代数的丰富来源。我们使用此结果来证明DeLigne的$ E_2 $ - 结构在有限张量类别的Hochschild Cochain Complex上是由规范端，其乘法和其非交叉的半编织而引起的。借助DeLigne的$ E_2 $ - 结构的新的，更明确的描述，我们可以将有限张量类别类别的Ext代数的Farinati-Solotar支架提升到Cochain级别的$ E_2 $结构。此外，我们证明，对于单型关键有限张量类别，将EXT代数纳入Hochschild Cochains是框架$ e_2 $ algebras的单态性，从而完善了Menichi的结果。
It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit $I$ and the canonical end $\int_{X\in\mathcal{C}} X\otimes X^\vee$. Using the theory of braided operads, we prove that for any such algebra $\mathbb{T}$ the homotopy invariants, i.e. the derived morphism space from $I$ to $\mathbb{T}$, naturally come with the structure of a differential graded $E_2$-algebra. This way, we obtain a rich source of differential graded $E_2$-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's $E_2$-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's $E_2$-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an $E_2$-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed $E_2$-algebras, thereby refining a result of Menichi.
