  Two Hopf algebras are called monoidally Morita equivalent if module
categories over them are equivalent as linear monoidal categories. We introduce
monoidal Morita invariants for finite-dimensional Hopf algebras based on
certain braid group representations arising from the Drinfeld double
construction. As an application, we show, for any integer $n$, the number of
elements of order $n$ is a monoidal Morita invariant for finite group algebras.
We also describe relations between our construction and invariants of closed
3-manifolds due to Reshetikhin and Turaev.
