Data

1. Title: The center of $\mathrm{SL}_{2}$ tilting modules
2. Authors: Daniel Tubbenhauer and Paul Wedrich
3. Status: Preprint. Last update: Tue, 21 Apr 2020 16:47:57 UTC
4. ArXiv link: https://arxiv.org/abs/2004.10146

Abstract

In this note we compute the centers of the categories of tilting modules for $G=\mathrm{SL}_{2}$ in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective $G_{g}T$-modules when $g=1,2$.

A few extra words

Let $\mathbb{K}$ denote an algebraically closed field and $\mathbf{Tilt}=\mathbf{Tilt}(\mathrm{SL}_{2})$ the additive $\mathbb{K}$-linear category of (left-)tilting modules for the algebraic group $\mathrm{SL}_{2}$. In this note we compute the (categorical) center $\mathbf{Z}(\mathbf{Tilt})$ of $\mathbf{Tilt}$, using the explicit description of the Ringel dual of $\mathrm{SL}_{2}$.
These Ringel duals are quivers living on certain infinite and fractral graphs, and are basically fractal-zigzag-type algebras on these graphs:

The full subquivers containing the first $53$ vertices of the quiver underlying $\mathrm{Z}_{\mathsf{p}}$ for $\mathsf{p}\in\{3,5,7\}$, showing from top to bottom the numbers $v$ of the vertices, the generation of $v$ (the number of non-zero digits of $v$ in its $\mathsf{p}$-adic expansion minus one) and the $\mathsf{p}$-adic expansion of $v$.

The main theorem then is that we have an (explicit) isomorphism of $\mathbb{K}$-algebras between $\mathbb{K}[X_{v}\mid v\in\mathbb{N}]/\langle X_{v}X_{w}\mid v,w\in\mathbb{N}\rangle$ and $\mathbf{Z}(\mathbf{Tilt})$.

Finally, we also compute the centers for the category of tilting modules in the quantum group case (which is generation $1$) as well as for projective $G_{g}T$-modules for $g=1,2$ (generations $1$ and $2$), both for $\mathrm{SL}_{2}$. In these cases the results are similar as for the algebraic group, but much easier and can be thought off as an approximation of order $1$ or $2$ to the group case.