Interpolation of the oscillator representation and Azumaya algebras in tensor categories
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910550340927488 |
|---|---|
| author | Snowden, Andrew |
| author_facet | Snowden, Andrew |
| contents | Let $\mathfrak{C}$ be a symmetric tensor category and let $A$ be an Azumaya algebra in $\mathfrak{C}$. Assuming a certain invariant $η(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor $Φ\colon \mathfrak{C} \to \mathfrak{D}$ for which $Φ(A)$ 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 $\mathfrak{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interpolation of the oscillator representation and Azumaya algebras in tensor categories Snowden, Andrew Representation Theory Category Theory Let $\mathfrak{C}$ be a symmetric tensor category and let $A$ be an Azumaya algebra in $\mathfrak{C}$. Assuming a certain invariant $η(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor $Φ\colon \mathfrak{C} \to \mathfrak{D}$ for which $Φ(A)$ 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 $\mathfrak{C}$. |
| title | Interpolation of the oscillator representation and Azumaya algebras in tensor categories |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2408.00233 |