Reflection Theory of Nichols Algebras over Coquasi-Hopf Algebras with Bijective Antipode

TL;DR

This paper extends the reflection theory of Nichols algebras to coquasi-Hopf algebras with bijective antipodes, establishing new categorical equivalences and classifying a specific rank three Nichols algebra as affine.

Contribution

It generalizes reflection theory to a broader class of coquasi-Hopf algebras and constructs a semi-Cartan graph from finite-dimensional Yetter-Drinfeld modules.

Findings

Established a braided monoidal equivalence between categories of rational Yetter-Drinfeld modules.

Constructed a semi-Cartan graph from modules admitting all reflections.

Proved a rank three Nichols algebra is an affine Nichols algebra.

Abstract

We investigate the reflection theory of Nichols algebras over arbitrary coquasi-Hopf algebras with bijective antipode, generalizing previous results restricted to the pointed cosemisimple setting [47]. By establishing a braided monoidal equivalence between categories of rational Yetter-Drinfeld modules via a dual pair, we demonstrate that a tuple of finite-dimensional irreducible Yetter-Drinfeld modules admitting all reflections gives rise to a semi-Cartan graph. As an application, we consider an explicit example of a rank three Nichols algebra from [41]. We show that it yields a standard Cartan graph and prove that it is, in fact, an affine Nichols algebra.