Orbifold subfactors from Hecke algebras II --- Quantum doubles and braiding ---

TL;DR

This paper demonstrates that orbifold phenomena in subfactor bimodules are general, identifying their structure with fixed point algebras in Hecke algebra subfactors, and establishes the existence of non-degenerate braiding in certain cases.

Contribution

It generalizes Ocneanu's orbifold observations by linking them to fixed point algebras in Hecke algebra subfactors and computes examples of asymptotic inclusion graphs.

Findings

Orbifold phenomena are universal in subfactor bimodules.

Identified orbifolds with fixed point algebras in Hecke algebra subfactors.

Proved existence of non-degenerate braiding on D_2n even vertices.

Abstract

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with ghosts) for SU(3)_3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu's orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D_2n, n>2.