Interpolation of the oscillator representation and Azumaya algebras in tensor categories

2024/08/01 by Andrew Snowden, Snowden, Andrew
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2408.00233

openalex publication_date 2024/08/01 · openalex created_date 2024/08/04 · openalex updated_date 2026/07/28

Abstract

Let \mathfrakC be a symmetric tensor category and let A be an Azumaya algebra in \mathfrakC. Assuming a certain invariant η(A) ∈ Pic(\mathfrakC)[2] vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor Φ\colon \mathfrakC → \mathfrakD for which Φ(A) splits. We apply this when \mathfrakC=\underlineRep(Sp_t(F_q)) is the interpolation category of finite symplectic groups and A is a certain twisted group algbera in \mathfrakC, and we show that the splitting category \mathfrakD 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 \mathfrakC.

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