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 D(A), which he called the Schrödinger representation. We study this representation from the viewpoint of the theory of tensor 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:(Formula presented.) of k-linear monoidal categories, then the equivalence (Formula presented.) induced by F preserves the Schrödinger representation. Here, (Formula presented.) for an algebra A means the category of left A-modules. 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 Schrödinger 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.
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