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 $u_q({\mathfrak{s}\mathfrak{l}}_2)$ and show that if $p$ and $q$ are roots of unity of the same order, then $u_p({\mathfrak{s}\mathfrak{l}}_2)$ and $u_q({\mathfrak{s}\mathfrak{l}}_2)$ are gauge equivalent if and only if $q=p$, where $p$ and $q$ are roots of unity of the same odd order.