A quasi-Hopf algebra for the triplet vertex operator algebra

Abstract

We give a new factorizable ribbon quasi-Hopf algebra [Formula: see text], whose underlying algebra is that of the restricted quantum group for [Formula: see text] at a [Formula: see text]th root of unity. The representation category of [Formula: see text] is conjecturally ribbon equivalent to that of the triplet vertex operator algebra (VOA) [Formula: see text]. We obtain [Formula: see text] 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 VOA [Formula: see text], and our construction is parallel to extending [Formula: see text] to [Formula: see text]. We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra [Formula: see text] to a quasi-Hopf algebra for [Formula: see text], which corresponds to passing from the Heisenberg VOA to a lattice extension.