The representation category of any compact group is the bimodule category of a II_1 factor

TL;DR

This paper demonstrates that for any compact group, one can construct a II_1 factor with a minimal group action such that the bimodule category of the fixed-point algebra matches the group's representation category, enabling explicit description of subfactors.

Contribution

It establishes a universal construction linking compact groups to II_1 factors, showing their representation categories can be realized as bimodule categories of fixed-point subfactors.

Findings

Existence of minimal actions of any compact group on a II_1 factor.

Equivalence between the bimodule category of the fixed-point algebra and the group's representation category.

Explicit description of all finite index subfactors of the fixed-point algebra.

Abstract

We prove that given any compact group G, there exists a minimal action of G on a II_1 factor M such that the bimodule category of the fixed-point II_1 factor M^G is naturally equivalent with the representation category of G. In particular, all subfactors of M^G with finite Jones index can be described explicitly.