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In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").

Physically this modular fusion category describes the TQFT/anyon statistics of the Chern-Simons gauge theory.

Now I think of a modular fusion category as an $F$ and an $R$ tensor, where the $F$ tensor has $6$ indices labelled by simple objects and $4$ fusion space indices, and $R$ has $3$ indices labelled by simple objects and $2$ fusion space indices. Similarly, one can think of a Lie algebra in terms of the "structure coefficients" of the Lie bracket in some chosen basis, yielding a tensor with three indices. The level I'm not sure how to formulate in such a constructive language (this is somehow part of the question).

My question: Is there a constructive way to calculate the $F$ and $R$ tensor from the Lie algebra structure coefficients? Constructive in the sense that e.g. there is a computer program that takes tensors as inputs and tensors as outputs. And what would that construction look like?

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    $\begingroup$ The F-matrix goes by the name “quantum 6-j symbols.” The R-matrix you can compute from, well, the “R-matrix.” Neither is particularly easy to do without spending a lot of time with combinatorial representation theory. For su(2) this is all nicely worked out with pictures in Kauffman-Lins. $\endgroup$ – Noah Snyder Sep 13 '18 at 22:13
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There are two constructions of the modular fusion category. The conformal field theory approach is to take representations of the affine Kac-Moody algebra of given level and define a tensor product. This is not the naive one where you would add the levels. Then make it modular.

The other is to take the category of type I representations of the quantised enveloping algeba and specialise q to a root of unity (depending on the level). Then take the semisimplification of this category; i.e. quotient by the tensor ideal of negligible morphisms.

I believe Lusztig proved that these give the same result.

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  • $\begingroup$ "Tensor structures arising from affine lie algebras" has the equivalence $\endgroup$ – Phil Tosteson Nov 2 '18 at 23:07

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