Finite GK-dimensional pre-Nichols algebras and quasi-quantum groups

TL;DR

This work classifies finite GK-dimension pre-Nichols algebras in twisted Yetter-Drinfeld categories over finite abelian groups with $3$-cocycles, showing that non-diagonal objects necessarily have infinite GK-dimension. It introduces a graded, based-group framework and proves that finite GK-dim pre-Nichols algebras are twist-equivalent to diagonal-type algebras in ordinary Yetter-Drinfeld categories, enabling a complete removal of the twisted obstruction via a group $\widehat{G_V}$. Leveraging arithmetic root system theory and recent diagonal-type classifications, the authors deliver a full characterization: finite GK-dim Nichols algebras occur precisely for diagonal modules with finite arithmetic root systems; all finite GK-dim graded pre-Nichols algebras arise as twists of diagonal ones. The results extend the known diagonal-type classification to twisted settings and imply a broad class of infinite quasi-quantum groups obtained by bosonization of these twisted pre-Nichols algebras.

Abstract

In this paper, we study the classification of finite GK-dimensional pre-Nichols algebras in the twisted Yetter-Drinfeld module category $_{\k G}^{\k G} \mathcal{YD}^Φ$, where $G$ is a finite abelian group and $Φ$ is a $3$-cocycle on $G$. These algebras naturally arise from quasi-quantum groups over finite abelian groups. We prove that all pre-Nichols algebras of nondiagonal type in $_{\k G}^{\k G} \mathcal{YD}^Φ$ are infinite GK-dimensional, and every graded pre-Nichols algebra in $_{\k G}^{\k G} \mathcal{YD}^Φ$ with finite GK-dimension is twist equivalent to a graded pre-Nichols algebra in an ordinary Yetter-Drinfeld module category $_{\k \mathbb{G}}^{\k \mathbb{G}} \mathcal{YD}$, where $\mathbb{G}$ is a finite abelian group determined by $G$. In particular, we obtain a complete classification of finite GK-dimensonal Nichols algebras of finite-dimensional objects in $_{\k G}^{\k G} \mathcal{YD}^Φ$. Let $V\in {_{\k G}^{\k G} \mathcal{YD}^Φ}$ be a finite-dimensional object, we prove that the Nichols algebra $B(V)$ is finite GK-dimensional if and only if it is of diagonal type and the corresponding root system is finite, i.e., an arithmetic root system. Via bosonization, this yields a large class of infinite quasi-quantum groups over finite abelian groups.

Theorems & Definitions (96)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5
• Lemma 2.6