标题： 振荡器表示的插值和张量范畴中的Azumaya代数 显示英文标题

标题： Interpolation of the oscillator representation and Azumaya algebras in tensor categories

摘要： 设 $\mathfrak{C}$ 是一个对称张量范畴，令 $A$ 是 $\mathfrak{C}$ 中的一个Azumaya代数。 假设某个不变量 $\eta(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ 为零，并固定某种符号选择，我们证明存在一个通用的张量函子 $\Phi \colon \mathfrak{C} \to \mathfrak{D}$，使得 $\Phi(A)$ 可分解。 当我们使用此方法时，$\mathfrak{C}=\underline{\mathrm{Rep}}(\mathbf{Sp}_t(\mathbf{F}_q))$是有限辛群的插值范畴，$A$是$\mathfrak{C}$中的某种扭曲群代数，我们证明分裂范畴$\mathfrak{D}$是 S. Kriz 的振荡表示的插值范畴。这种构造比之前的构造有诸多优势；例如，它适用于非半单的情况。它还为这种情况带来了一些概念上的清晰度：Kriz 范畴的存在性与$\mathfrak{C}$的布劳尔群的非平凡性有关。 显示英文摘要

摘要： Let $\mathfrak{C}$ be a symmetric tensor category and let $A$ be an Azumaya algebra in $\mathfrak{C}$. Assuming a certain invariant $\eta(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor $\Phi \colon \mathfrak{C} \to \mathfrak{D}$ for which $\Phi(A)$ splits. We apply this when $\mathfrak{C}=\underline{\mathrm{Rep}}(\mathbf{Sp}_t(\mathbf{F}_q))$ is the interpolation category of finite symplectic groups and $A$ is a certain twisted group algbera in $\mathfrak{C}$, and we show that the splitting category $\mathfrak{D}$ is S. Kriz's interpolation category of the oscillator representation. This construction has a number advantages over previous ones; e.g., it works in non-semisimple cases. It also brings some conceptual clarity to the situation: the existence of Kriz's category is tied to the non-triviality of the Brauer group of $\mathfrak{C}$.