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## "A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules", Nathan Geer et al., 2022

*Reviewed March 22, 2024*

*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:

- The unrolled sl(2) quantum group representation category is non-semisimple;
- All unitary tensor categories are semisimple.

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.