Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U_q(\mathfrak g)$, and by considering a certain quotient category² of the category of tilting modules³ over that Hopf algebra, one obtains a fusion category. Moreover, by using the universal $R$-matrix and the so-called charmed element, one can endow that fusion category with the structure of a modular tensor category.

Can this approach be used to further equip this category with the structure of a unitary modular tensor category?

In other words, is it possible to equip the objects of this category with some kind of extra structure (something like an inner product) such that the adjoint $f^*:W\to V$ of a morphism $f:V\to W$ makes sense, and such that the braiding and twist are unitary?

¹: That integral version is usually denoted $U_q^{res}(\mathfrak g)$. It is generated by the usual elements $K_i$, $E_i$, $F_i$, along with the divided powers $E_i^{[r]}:=\frac{E_i^r}{[r]!}$ and $F_i^{[r]}:=\frac{F_i^r}{[r]!}$.
²: This quotient category has the same objects as the original category. The hom-spaces are modded out by the subspace of negligible morphisms, where a morphisms $f:V\to W$ is called negligible if $tr_q(fg)=0$ for any $g:W\to V$.
³: A module is tilting if it admits a filtration whose associated graded pieces are Weyl modules, and also admits a filtration whose associated graded pieces are duals of Weyl modules. Here, the Weyl modules, are the ones that "look like" irreps of $\mathfrak g$.


1 Answer 1


The answer is "Sometimes."

Following the notation of Rowell's "From Quantum Groups to Unitary Modular Tensor Categories", let $\mathcal C(\mathfrak g, l, q)$ be the category corresponding to $U_q(\mathfrak g)$ such that $q=e^{\pi \imath/l}$.Denote by $m$ is the ratio of the square lengths of a long root to a short root. As mentioned there, this only gets one as far as being a ribbon fusion category (also called a pre-modular category).

The questions of unitarity and modularity for $\mathcal C(\mathfrak g, l,q)$ are addressed in section 4. Specifically $\mathcal C(\mathfrak g, l,m)$ is unitary if $m\vert l$. This is attributed to both Wenzl and Xu. Modularity in the case that $m\vert l$ is attributed to Kirillov (among others). Numerous cases where $m\not\vert l$ are treated in Rowell.

As a related but different matter, assume one has a modular tensor category $\mathcal C$ and wants to know if it can be made unitary. Then by theorems 3.2 (Every braiding of a unitary fusion category is unitary) and 3.5 (Every braiding of a unitary fusion category admits a unique unitary ribbon structure) of Galindo's "On Braided and Ribbon Unitary Fusion Categories", it suffices to determine whether or not $\mathcal C$ admits the structure of a unitary fusion category.

  • 4
    $\begingroup$ There were two open cases related to G2 in Rowell's work which we showed were unitary in arxiv.org/abs/1003.0022, see the end of Section 3.5. $\endgroup$ Feb 15, 2014 at 20:26

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