A quasi-Hopf algebra for the triplet vertex operator algebra

Abstract

We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet vertex operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet vertex operator algebra M(p), and our construction is parallel to extending M(p) to W(p). We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra CZ to a quasi-Hopf algebra for CZ_{2p}, which corresponds to passing from the Heisenberg vertex operator algebra to a lattice extension.

71pp

References in corpus (7)

• Logarithmic extensions of minimal models: characters and modular transformations
• Topological defects for the free boson CFT
• Braided tensor categories and extensions of vertex operator algebras
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• From boundary to bulk in logarithmic CFT
• The Verlinde formula in logarithmic CFT

Cited by in corpus (17)

• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Gluing vertex algebras
• On ribbon categories for singlet vertex algebras
• Tensor categories of affine Lie algebras beyond admissible levels
• Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories
• Constructing non-semisimple modular categories with local modules
• Categorical aspects of cointegrals on quasi-Hopf algebras
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• Davydov-Yetter cohomology and relative homological algebra
• Monadic cointegrals and applications to quasi-Hopf algebras
• Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels