A note on outer quantum automorphisms of finite dimensional von Neumann algebras

This is part of an ongoing project of formulating notion(s) of quantum group of outer automorphisms of a \(C^*\) or von Neumann algebra. Motivated by the fact that the group of outer automorphism of a \(II_1\) factor can be viewed as a subgroup of the group of group-like or invertible objects in the category of Hilbert bimodules of finite ranks, we explore a natural class of objects in the bimodule category of a finite dimensional (i.e. direct sum of matrix algebras) von Neumann algebra \(\mathcal{A}\) which may come from the (co-action) of a discrete quantum group. In particular, we prove that any discrete quantum group giving an outer quantum symmetry on \(\mathcal{A}\) in a sense defined by us must be a finite dimensional quantum group. We relate the analysis of such quantum groups or the corresponding fusion rings with certain combinatorial objects involving matrices with nonnegative integer entries and do some explicit computations in a few simple examples.