Ergodic actions of S_\mu U(2) on C*-algebras from II_1 subfactors

TL;DR

This paper constructs an ergodic action of the quantum group S_mu U(2) on C*-algebras derived from II_1 subfactors, linking subfactor theory with quantum group symmetries and spectral analysis.

Contribution

It introduces a novel ergodic C*-action of S_mu U(2) associated with II_1 subfactors, connecting subfactor indices with quantum group deformation parameters.

Findings

Identifies higher relative commutants with spectral spaces of tensor powers.

Establishes a virtual subgroup structure of S_mu U(2) from subfactor bimodules.

Relates the deformation parameter mu to the subfactor index.

Abstract

To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_\mu U(2). The deformation parameter is determined by -1<\mu<0 and [M:N]=|\mu+\mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_\mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. \mu is negative as M is a real bimodule.