On indicators of Hopf algebras

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.