Symplectic fermions and a quasi-Hopf algebra structure on ${\bar{U}}_{i}sl(2)$

Abstract

We consider the (finite-dimensional) small quantum group ${\bar{U}}_{q}sl(2)$ at $q=i$. We show that ${\bar{U}}_{i}sl(2)$ does not allow for an R-matrix, even though $U\otimes V\cong V\otimes U$ holds for all finite-dimensional representations $U,V$ of ${\bar{U}}_{i}sl(2)$. We then give an explicit coassociator $\Phi$ and an R-matrix $R$ such that ${\bar{U}}_{i}sl(2)$ becomes a quasi-triangular quasi-Hopf algebra. Our construction is motivated by the two-dimensional chiral conformal field theory of symplectic fermions with central charge $c=-2$. There, a braided monoidal category, $\mathcal{S}\mathcal{F}$, has been computed from the factorisation and monodromy properties of conformal blocks, and we prove that $\mathrm{Rep}({\bar{U}}_{i}sl(2),\Phi ,R)$ is braided monoidally equivalent to $\mathcal{S}\mathcal{F}$.

40pp, 11 figures; v2: few very minor corrections for the final version in Journal of Algebra

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• The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity
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