Clebsch-Gordan and 6j-coefficients for rank two quantum groups

Abstract

We calculate (q-deformed) Clebsch-Gordan and 6j-coefficients for rank two quantum groups. We explain in detail how such calculations are done, which should allow the reader to perform similar calculations in other cases. Moreover, we tabulate the q-Clebsch-Gordan and 6j-coefficients explicitly, as well as some other topological data associated with theories corresponding to rank-two quantum groups. Finally, we collect some useful properties of the fusion rules of particular conformal field theories.

43 pages. v2: minor changes and added references. For mathematica notebooks containing the various q-CG and 6j symbols, see http://arxiv.org/src/1004.5456/anc

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• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• Topological aspects of the critical three-state Potts model
• Galois conjugated tensor fusion categories and non-unitary CFT
• Composing topological domain walls and anyon mobility
• One-dimensional itinerant interacting non-Abelian anyons
• Computing associators of endomorphism fusion categories
• Quantum Gravity at the Fifth Root of Unity
• Braiding properties of paired spin-singlet and non-abelian hierarchy states
• Revisiting the symmetry-resolved entanglement for non-invertible symmetries in $1+1$d conformal field theories
• Diagrammatics, Pentagon Equations, and Hexagon Equations of Topological Orders with Loop- and Membrane-like Excitations
• Extending the planar theory of anyons to quantum wire networks
• Categorical Quantum Volume Operator