Citation: Geer, Nathan, et al. "A Hermitian TQFT from a non-semisimple category of quantum sl (2)-modules." Letters in Mathematical Physics 112.4 (2022): 74.
Web: https://arxiv.org/abs/2108.09242
Tags: Non-finite/semisimple, Modular-tensor-categories, Quantum-groups
In this paper, the authors show that the category of modules over the unrolled sl(2) quantum group can be endowed with a Hermitian ribbon structure. That is, it can be endowed with a dagger-structure that is compatible with every part of the ribbon structure. What is NOT required is that it induces a positive-definite bilinear form on hom-spaces. The compatibility implies that the induced form is non-degenerate, but it can have mixed signature.
This result is particularly interesting in light of the following two facts:
This second statement is a folklore result, which is hard to find explicitly in the literature. A good exposition is found in Dave Penneys' lecture notes. It is analogous to the fact that all unitary representations are semisimple, because the inner product allows you define an orthogonal compliment to break things down into simple pieces. In this context, it is a bit surprising that there could be a Hermitian non-semisimple category.