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## "An analogue of Radford's S4 formula for finite tensor categories", Pavel Etingof, Dmitri Nikshych, Viktor Ostrik, 2004

*Reviewed February 16, 2024*

*Citation:* Etingof, Pavel, Dmitri Nikshych, and Viktor Ostrik. "An analogue of Radford's S4 formula for finite tensor categories." International Mathematics Research Notices 2004.54 (2004): 2915-2933.

*Web:* https://arxiv.org/abs/math/0404504

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

This paper is the original article to prove that every object in a
fusion category is isomorphic to its quadruple dual. The proof
uses a generalization of Radford's S4 formula. Radford's S4 formula
is a formula for the 4th power of the antipode of a Hopf algebra
in terms of group-like elements. The generalized formula
is based on distinguished invertible objects.

This paper is also the one that introduced the categorical notion
of factorizability, and showed that Drinfeld centers of fusion categories
are factorizable. The notion of factorizable used in this paper applies to fusion categories,
and it asserts that Z(C)=C \times C^op where "op" denotes opposite category. There is a
separate notation of factorizable which applies to braided fusion categories, which asserts that Z(C)=C\times C^rev
where "rev" denotes the same category with reversed braiding. It is a well-known theorem that
being factorizable in the braided sense is equivalent to being modular. The notion
of being factorizable in the two senses above are a-priori unrelated.

It is important to note that the main result of this paper, the fact that
the identity functor is naturally isomorphic to the quadruple dual, has
since been established in other means. In particular, there is a lovely
proof using canonical string diagrams by Hagge:

> Hagge, Tobias J., and Seung-Moon Hong. "Some non-braided fusion categories of rank 3." arXiv preprint arXiv:0704.0208 (2007).