Hopf ideals of the generalized quantum double associated to skew-paired Nichols algebras

TL;DR

This paper studies Hopf ideals in a generalized quantum double constructed from Nichols algebras and Yetter-Drinfeld modules, providing classifications and explicit descriptions of certain minimal quasitriangular pointed Hopf algebras.

Contribution

It introduces the concept of thin Hopf ideals in the context of generalized quantum doubles and characterizes them, extending understanding of the structure of these Hopf algebras.

Findings

Characterization of thin Hopf ideals in generalized quantum doubles.

Explicit description of minimal quasitriangular pointed Hopf algebras in characteristic zero.

Framework for analyzing Hopf ideals in Nichols algebra-based quantum doubles.

Abstract

The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_{\tau} U^{\geq 0}_q $ associated to a natural skew pairing $ \tau : U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $. This double is generalized by $\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_{\tau} (\cmdfrakB (W) \cmddotrtimes G)$, where $F$, $G$ are abelian groups, $ V \in {}^F_F \cmdcalYD $, $ W \in {}^G_G \cmdcalYD $ are Yetter-Drinfeld modules and $ \cmdfrakB (V) $, $ \cmdfrakB (W) $ are their Nichols algebras. We prove some results on Hopf ideals of $ \cmdcalD $, including a characterization of what we call thin Hopf ideals. As an application we give an explicit description of those minimal quasitriangular pointed Hopf algebras in characteristic zero which are generated by skew primitives.