Haag duality conjecture for Kitaev models from quasi-triangular Hopf-* algebras
Let be a quasi-triangular Hopf- algebra, and consider the corresponding Kitaev model on the plane. Haag duality conjecture. The model satisfies Haag duality for cones. This would extend the known Haag-duality theorem for quantum double models of finite abelian groups to the broader class of quasi-triangular Hopf- algebras, for which the braiding of excitation operators is available. The conjecture remains open in the source.
References
Primary source
Pieter Naaijkens, “Kitaev's quantum double model from a local quantum physics point of view”, arXiv:1508.07170 (2015).
Progress summary
A known result covers only finite abelian models, while no public proof or counterexample has appeared for the broader class.
The conjecture asks whether Kitaev models built from quasi-triangular Hopf-* algebras satisfy Haag duality for cones. The established abelian quantum-double theorem does not cover this broader setting.
Known results
- Haag duality for cones is proved for finite abelian quantum-double models in the ground-state representation.
- The toric code is included as the case of the two-element cyclic group.
- Approximate Haag duality is established for abelian quantum-double models, but not for non-abelian models.
2023 literature status
A recent classification survey explicitly records the non-abelian cone-duality problem as unresolved and makes no claim addressing the quasi-triangular Hopf-* algebra conjecture. The scanned sources contain no claimed proof, counterexample, verification, or retraction.
Current status (as of August 2026): Haag duality for cones is settled in the finite abelian quantum-double case, whereas the quasi-triangular Hopf-* algebra conjecture remains open with no recorded public progress.
Sources
Solutions 0
No solutions have been posted yet.