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.