On indicators of Hopf algebras

TL;DR

This paper studies the properties of the $n$-th indicators of finite-dimensional Hopf algebras, revealing cyclotomic integrality, formulas for Drinfeld doubles, and applications to specific algebra classes, including gauge equivalence conditions.

Contribution

It introduces new properties of Hopf algebra indicators, such as cyclotomic integrality and explicit formulas, and applies these results to classify gauge equivalence of certain quantum groups.

Findings

Indicators are cyclotomic integers.

Explicit formula for indicators of Drinfeld doubles.

Gauge equivalence of $u_p(sl_2)$ and $u_q(sl_2)$ depends on roots of unity.

Abstract

Kashina, Montgomery and Ng introduced the $n$-th indicator $\nu_n(H)$ of a finite-dimensional Hopf algebra $H$ and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of $\nu_n$'s. In particular, we obtain the cyclotomic integrality of $\nu_n$ and a formula for $\nu_n$ of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra $u(\mathcal{D}, \lambda, \mu)$ introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of $u_q(\mathfrak{sl}_2)$ and show that if $p$ and $q$ are roots of unity of the same order, then $u_p(\mathfrak{sl}_2)$ and $u_q(\mathfrak{sl}_2)$ are gauge equivalent if and only if $q = p$, where $p$ and $q$ are roots of unity of the same odd order.