Reflection of Nichols Algebras over Coquasi-Hopf Algebras

TL;DR

This paper generalizes reflection theory for Nichols algebras to coquasi-Hopf algebras, providing criteria for finite-dimensionality and a new proof for infinite-dimensional cases, advancing classification efforts.

Contribution

It develops a reflection theory for Nichols algebras over coquasi-Hopf algebras and introduces criteria linking semi-Cartan graphs to finite-dimensionality.

Findings

Established reflection theory for Nichols algebras over coquasi-Hopf algebras.

Defined semi-Cartan graphs and Weyl groupoids in this setting.

Provided criteria for finite-dimensionality and a new proof for infinite-dimensionality.

Abstract

This paper extends the foundational reflection theory of Nichols algebras to the setting of some certain coquasi-Hopf algebras. Our primary motivation arises from the classification of pointed finite-dimensional coquasi-Hopf algebras. We develop a reflection theory for tuples of simple Yetter-Drinfeld modules in the category $\GG$, where $G$ is a finite group and $\Phi$ is a 3-cocycle on $G$. We prove that such a tuple gives rise to a semi-Cartan graph if admitting all reflections. Consequently, its Weyl groupoid is well-defined. We further establish several criteria for the finite-dimensionality of Nichols algebras in terms of the associated semi-Cartan graph. As an application, we provide a new proof for the infinite-dimensionality of a specific class of Nichols algebras previously studied in \cite{huang2024classification}, bypassing extensive computational arguments.