Schr\"odinger representations from the viewpoint of monoidal categories

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.