  The Drinfel'd double D(A) of a finite-dimensional Hopf algebra A is a Hopf
algebraic counterpart of the monoidal center construction. Majid introduced an
important representation of the Drinfel'd double, which he called the
Schr\"odinger representation. We study this representation from the viewpoint
of the theory of monoidal categories. One of our main results is as follows: If
two finite-dimensional Hopf algebras A and B over a field k are monoidally
Morita equivalent, i.e., there exists an equivalence F from the module category
over A to the module category over B of k-linear monoidal categories, then the
equivalence between the module categories over D(A) and D(B) induced by F
preserves the Schr\"odinger representation.
  As an application, we construct a family of invariants of finite-dimensional
Hopf algebras under the monoidal Morita equivalence. This family is
parameterized by braids. The invariant associated to a braid b is, roughly
speaking, defined by "coloring'' the closure of b by the Schr\"odinger
representation. We investigate what algebraic properties this family have and,
in particular, show that the invariant associated to a certain braid closely
relates to the number of irreducible representations.
