Reflections of Nichols algebras in braided monoidal categories

Abstract

Let $\C$ be an essentially small abelian rigid braided monoidal category. We prove a reflection theorem for Nichols algebras in $\Ind(\C)$ and show that tuples admitting all reflections define connected semi-Cartan graphs. As applications, we prove that a finite Nichols algebra has a generating object of length at most one in $PSU(2)_k$ for odd $k\ge3$, in $SU(2)_{4m}$ for $m\ge1$, and in the representation categories of the Virasoro minimal models $\mathcal M(2,p)$ for odd $p\ge5$. In all three families, we prove that every finite pre-Nichols algebra is a Nichols algebra.