Quantum Galois groups of subfactors

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.

arXiv comment23 pages + references; to appear in IJM

References in corpus (8)

• Quantum Symmetry Groups of Finite Spaces
• Operator algebras in rigid C*-tensor categories
• A Galois correspondence for II_1 factors and quantum groupoids
• Tannaka-Krein duality for Hopf algebroids
• Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers
• Exchange relation planar algebras of small rank
• Duality for actions of weak Kac algebras and crossed product inclusions of II_1 factors
• Hopf algebra actions in tensor categories

Cited by in corpus (2)

• Braided quantum symmetries of graph ${\mathrm{C}}^{\ast }$-algebras
• Equivariant ${\mathrm{C}}^{\ast }$-correspondences and compact quantum group actions on Pimsner algebras