On indicators of Hopf algebras
Kenichi Shimizu

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 -th indicator of a finite-dimensional Hopf algebra 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 's. In particular, we obtain the cyclotomic integrality of and a formula for of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of and show that if and are roots of unity of the same order, then and are gauge equivalent if and only if , where and are roots of unity of the same odd order.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
