Quantum Galois groups of subfactors

TL;DR

This paper introduces the concept of quantum Galois groups for finite-index II_1 subfactors, establishing their existence, computing examples, and extending the framework to more general algebraic structures.

Contribution

It proves the existence of a universal Hopf *-algebra acting on subfactors, computes it in key cases, and generalizes the concept to tensor categories.

Findings

Existence of quantum Galois groups for subfactors.

Explicit computation for finite-index depth-two subfactors.

Extension to universal Hopf algebras in enriched categories.

Abstract

For a finite-index $\mathrm{II}_1$ subfactor $N \subset M$, we prove the existence of a universal Hopf $\ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf $\ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.