Reflection of Nichols Algebras over Coquasi-Hopf Algebras

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 $Φ$ 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.