Reflection of Nichols Algebras over Coquasi-Hopf Algebras
Bowen Li, Gongxiang Liu
TL;DR
The paper advances the theory of Nichols algebras by extending reflection and Weyl groupoid concepts from Hopf algebras to the coquasi-Hopf setting, specifically for pointed cosemisimple structures in ${^{G}_{G} ext{YD}^{oldsymbol{ abla}}}$. It develops a robust framework using monoidal equivalences, Hopf pairings, and partial dualization to relate reflections to semi-Cartan graphs, and derives concrete finite-dimensionality criteria in terms of the associated graph. A key outcome is a new, more conceptual proof of the infinite-dimensionality for a class of Nichols algebras previously treated via computation, along with a standard mechanism to recover and classify finite-dimensional cases through the finiteness of the Weyl groupoid. The results hold potential for advancing the classification program of coquasi-Hopf algebras beyond abelian groups and provide tools for natural, categorial treatment of non-diagonal Nichols 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.