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## "Weakly group-theoretical and solvable fusion categories", Pavel Etingof, Dmitri Nikshych, Victor Ostrik, 2009

*Reviewed March 23, 2024*

*Citation:* Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Weakly group-theoretical and solvable fusion categories." Advances in Mathematics 226.1 (2011): 176-205.

*Web:* https://arxiv.org/abs/0809.3031

*Tags:* Mathematical, Monoidal-categories, Modular-tensor-categories

This paper is one of the shining examples of the early work on fusion categories
done by Etingof, Nikshych, and Ostrik. In particular, this paper proves
a large amount of deep and useful results about faithful G-graded extensions
of fusion categories. Interestingly, both modular and super-modular (at the time called "slightly degenerate")
fusion categories ended up playing an important role.

What's great about this paper is that it isn't bogged down
with introducing definitions or proving obvious claims.
The results are by and large very much non trivial and
at times unexpected. One of the most important results
in this paper is the statement that C and D are Morita
equivalent if and only if they have equivalent centers.
This result, like all important results, is originally due
to unpublished work by Kitaev. Apparently Mueger also
came out with an unpublished proof at the same time. This
paper gives a proof of the result, as well as giving
a generalization to G-graded categories.

Another nice result from this paper is an analogue of Burnside's
theorem for fusion categories.