arXiv:2101.10016v6 [math.QA] 27 Jul 2021 WEAK QUASI-HOPF ALGEBRAS, C*-TENSOR CATEGORIES AND CONFORMAL FIELD THEORY, AND THE KAZHDAN-LUSZTIG-FINKELBERG THEOREM SEBASTIANO CARPI, SERGIO CIAMPRONE, MARCO V. GIANNONE, AND CLAUDIA PINZARI (preliminary version) Contents 1. Introduction 4 2. Preliminaries on tensor categories and their functors, generating object 22 3. Rigidity, weak tensor functors, braided symmetry, ribbon and modular category 30 4. Weak quasi-Hopf algebras 33 5. Tannaka-Krein duality, associator of CFT-type, construction of tensor equivalences, strictification with a generating object, integral weak dimension function (wdf) 40 6. w-Hopf algebras 58 7. Quasitriangular and ribbon structures 63 8. Ω-involution and C*-structure 70 9. The categories Rep h (A) and Rep + (A) 75 10. Unitary braided symmetry and involutive Tannaka-Krein duality 80 11. Unitarizability of representations and rigidity 84 12. Turning C ∗ -categories into tensor C ∗ -categories, I 89 13. Positive weak dimension and amenability 94 14. Constructing integral wdf and uniqueness of unitary tensor structure 97 15. Examples of fusion categories with different natural integral wdf 99 16. Quantum group U q (g) at roots of unity, ∗ -involution, classical limit 101 17. Fusion categories C(g,q,ℓ) and unitary ribbon wqh via integral wdf 108 18. VOAs, the Zhu algebra A(V ) and conformal nets 110 19. Kazhdan-Wenzl theory and equivalence of ribbon sl N,q,ℓ -categories 120 20. Turning C ∗ -categories into tensor C ∗ -categories, II 134 21. Coboundary categories and Deligne’s theorem 137 22. Hermitian coboundary wqh and relation with Hermitian ribbon wqh 144 23. A categorical characterization of discrete Hermitian coboundary wqh extending the Doplicher-Roberts theorem 151 24. Compatible unitary coboundary wqh, an abstract Drinfeld-Kohno, modularity154 25. Wenzl’s unitary structure of C(g,q,ℓ), square root of the quantum Casimir 157 26. Compatible unitary coboundary w-Hopf algebras U q (g) → A W (g,q,ℓ) 163 1