Higher Verlinde Categories: The Mixed Case

TL;DR

This paper explores the properties of mixed higher Verlinde categories derived from quantum groups at roots of unity, establishing tensor product formulas, embeddings, and analyzing their symmetric centers.

Contribution

It introduces the study of mixed higher Verlinde categories, extending previous work to include quantum group constructions and detailed structural analysis.

Findings

Established Steinberg tensor product formula for simple objects.

Constructed braided embeddings between categories.

Computed the symmetric center and identified Grothendieck ring.

Abstract

Over a field of characteristic $p>0$, the higher Verlinde categories $\mathrm{Ver}_{p^n}$ are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for $\mathrm{SL}_2$ has been generalized to Lusztig's quantum group for $\mathfrak{sl}_2$ and root of unity $\zeta$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^{\zeta}_{p^{(n)}}$. Inspired by the results of arXiv:2003.10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^{\zeta}_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, and identify its Grothendieck ring.