On indicators of Hopf algebras

Kashina, Montgomery and Ng introduced the n-th indicator ν_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 ν_n’s. In particular, we obtain the cyclotomic integrality of ν_n and a formula for ν_n of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra u(D, λ, µ) introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of u_q(sl₂) and show that if p and q are roots of unity of the same order, then u_p(sl₂) and u_q(sl₂) are gauge equivalent if and only if q = p, where p and q are roots of unity of the same odd order.