Interpolation of the oscillator representation and Azumaya algebras in tensor categories

TL;DR

This paper constructs a universal tensor functor linking symmetric tensor categories with Azumaya algebras to interpolation categories of symplectic groups, revealing new insights into their structure and non-semisimple cases.

Contribution

It introduces a universal tensor functor that splits Azumaya algebras in tensor categories, applicable to non-semisimple cases and clarifying the role of the Brauer group.

Findings

Existence of a universal tensor functor under certain conditions.

Application to interpolation categories of symplectic groups.

Connection between Kriz's interpolation category and the Brauer group.

Abstract

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 $\mathfrak{C}$.