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

TL;DR

This work extends Nichols algebra reflection theory from Hopf algebras to arbitrary coquasi-Hopf algebras with bijective antipode by introducing a dual-pair–driven braided monoidal equivalence between rational Yetter-Drinfeld categories. It defines rational modules in the coquasi setting, constructs a bridge between comodules and rational modules, and shows that a finite family of irreducible Yetter-Drinfeld modules admitting all reflections yields a semi-Cartan graph. The main result provides an explicit isomorphism between reflected Nichols algebras and a smash product involving the dual, via a braided equivalence, thereby generalizing prior pointed cosemisimple results. An explicit rank-3 non-diagonal Nichols algebra is analyzed to produce a standard Cartan graph and to prove its associated Tits cone is a half-plane, i.e., the algebra is affine. Overall, the paper broadens the landscape of Nichols algebra theory to non-associative coquasi-Hopf contexts and furnishes concrete affine examples with a Cartan-graph/ root-system framework.

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.

Theorems & Definitions (96)

• Theorem 1.1: Theorem \ref{['thm5.15']}
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Lemma 2.7