On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups

Abstract

We prove that every pre-Nichols algebra of a nondiagonal object in the twisted Yetter-Drinfeld category ${_{\k G}^{\k G} {\mathcal{YD}^\Phi}}$ has infinite Gelfand-Kirillov dimension, where $G$ is a finite abelian group and $\Phi$ is a $3$-cocycle on $G$. This leads to a complete characterization of finite GK-dimensional Nichols algebras in this category. Specifically, for any finite-dimensional $V\in {_{\k G}^{\k G} {\mathcal{YD}^\Phi}}$, we show that the Nichols algebra $B(V)$ has finite Gelfand-Kirillov dimension if and only if it is of diagonal type and its associated root system is finite, that is, an arithmetic root system. Via bosonization, this result yields a classification of finite GK-dimensional coradically graded pointed coquasi-Hopf algebras over finite abelian groups that are generated by group-like and skew-primitive elements.