Schr\"odinger representations from the viewpoint of monoidal categories

TL;DR

This paper explores the Schr"odinger representation of the Drinfel'd double of finite-dimensional Hopf algebras within monoidal categories, establishing invariants under monoidal Morita equivalence and linking them to algebraic properties.

Contribution

It demonstrates that the Schr"odinger representation is preserved under monoidal Morita equivalence and constructs braid-parameterized invariants of Hopf algebras based on this representation.

Findings

Invariants of Hopf algebras are constructed using braids and Schr"odinger representations.

The invariants are preserved under monoidal Morita equivalence.

One invariant relates to the number of irreducible representations.

Abstract

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.